text: Contents 1 Motivation 1 2 The xSM 3 2.1 The Model 3 2.2 Current Phenomenological Constraints 5 → 2.3 The Electroweak Phase Transition: Benchmarks for h h h Production 7 2 1 1 − 3 Resonant Di-Higgs Production: the b¯bW +W Channel 9 3.1 Monte Carlo Generation and Object Reconstruction 10 3.2 Invariant Mass Reconstruction for h : Heavy Mass Estimator 10 2 3.3 Analysis Selection 12 3.4 Systematic Uncertainties 14 4 Prospects for LHC Run 2 and HL-LHC 16 5 Outlook 19 1 Motivation With the discovery of the Higgs boson at the Large Hadron Collider (LHC) [1, 2], exploring the thermal history associated with electroweak symmetry-breaking (EWSB) has taken on heightened interest. In the Standard Model (SM), EWSB in the early Universe occurs through a crossover transition. In contrast, beyond the Standard Model (BSM) scenarios may lead to a bona ﬁde electroweak phase transition (EWPT). If such a transition occurred and was both ﬁrst order and suﬃciently strong, it could have provided the conditions needed for generating the observed cosmic matter-antimatter asymmetry. Electroweak baryogenesis (EWBG) (for recent reviews, see [3, 4]) is one of the most widely-studied and experimentally testable scenarios for explaining the origin of the cosmic matter-antimatter asymmetry, characterized by the baryon-to-entropy density ratio Y = B n /s as (most precisely) measured by Planck [5] B ± × − Y = (8.59 0.11) 10 11 . (1.1) B Successful baryogenesis requires three ingredients in the particle physics of the early Universe, the so-called “Sakharov criteria” [6]: (i) baryon number (B)-violation; (ii) C- and CP-violation; (iii) departure from thermal equilibrium or a breakdown of CPT invariance. The SM contains the requisite B-violation in the guise of electroweak sphalerons, but it fails – 1 – ##LTLine## ##LTLine##

text: Contents

text: Motivation

text: The

text: xSM

text: The

text: Model

text: Current

text: Phenomenological

text: Constraints

text: The

text: Electroweak

text: Phase

text: Transition:

text: Benchmarks

text: for

text: h

text: h

text: h

text: Production

text: Resonant

text: Di-Higgs

text: Production:

text: the

text: b¯bW

text: +W

text: Channel

text: Monte

text: Carlo

text: Generation

text: and

text: Object

text: Reconstruction

text: Invariant

text: Mass

text: Reconstruction

text: for

text: h

text: Heavy

text: Mass

text: Estimator

text: Analysis

text: Selection

text: Systematic

text: Uncertainties

text: Prospects

text: for

text: LHC

text: Run

text: and

text: HL-LHC

text: Outlook

text: Motivation

text: With

text: the

text: discovery

text: of

text: the

text: Higgs

text: boson

text: at

text: the

text: Large

text: Hadron

text: Collider

text: (LHC)

text: exploring

text: the

text: thermal

text: history

text: associated

text: with

text: electroweak

text: symmetry-breaking

text: (EWSB)

text: has

text: taken

text: on

text: heightened

text: interest.

text: In

text: the

text: Standard

text: Model

text: (SM),

text: EWSB

text: in

text: the

text: early

text: Universe

text: occurs

text: through

text: a

text: crossover

text: transition.

text: In

text: contrast,

text: beyond

text: the

text: Standard

text: Model

text: (BSM)

text: scenarios

text: may

text: lead

text: to

text: a

text: bona

text: ﬁde

text: electroweak

text: phase

text: transition

text: (EWPT).

text: If

text: such

text: a

text: transition

text: occurred

text: and

text: was

text: both

text: ﬁrst

text: order

text: and

text: suﬃciently

text: strong,

text: it

text: could

text: have

text: provided

text: the

text: conditions

text: needed

text: for

text: generating

text: the

text: observed

text: cosmic

text: matter-antimatter

text: asymmetry.

text: Electroweak

text: baryogenesis

text: (EWBG)

text: (for

text: recent

text: reviews,

text: see

text: is

text: one

text: of

text: the

text: most

text: widely-studied

text: and

text: experimentally

text: testable

text: scenarios

text: for

text: explaining

text: the

text: origin

text: of

text: the

text: cosmic

text: matter-antimatter

text: asymmetry,

text: characterized

text: by

text: the

text: baryon-to-entropy

text: density

text: ratio

text: Y

text: B

text: n

text: /s

text: as

text: (most

text: precisely)

text: measured

text: by

text: Planck

text: B

text: Y

text: B

text: Successful

text: baryogenesis

text: requires

text: three

text: ingredients

text: in

text: the

text: particle

text: physics

text: of

text: the

text: early

text: Universe,

text: the

text: so-called

text: “Sakharov

text: criteria”

text: (i)

text: baryon

text: number

text: (B)-violation;

text: (ii)

text: C-

text: and

text: CP-violation;

text: (iii)

text: departure

text: from

text: thermal

text: equilibrium

text: or

text: a

text: breakdown

text: of

text: CPT

text: invariance.

text: The

text: SM

text: contains

text: the

text: requisite

text: B-violation

text: in

text: the

text: guise

text: of

text: electroweak

text: sphalerons,

text: but

text: it

text: fails

text: ##LTLine##

text: ##LTLine##


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text: photons

text: and

text: two

text: bottom

text: quarks,

text: Phys.

text: Rev.

text: CMS

text: collaboration,

text: CMS,

text: Search

text: for

text: a

text: low-mass

text: pseudoscalar

text: Higgs

text: boson

text: produced

text: in

text: association

text: with

text: a

text: bb

text: pair

text: in

text: pp

text: collisions

text: asqrt(s)

text: TeV,

text: Phys.

text: Lett.

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text: ##LTLine##

text: ##LTLine##


text: 1.00 0.98 0.96 0.94 0.92 0.90 0.88 0.86 0.84 0.82 0.80 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 m (GeV) 2 → → Figure 1. Present 95% C.L. excluded regions from Public ATLAS and CMS pp h ZZ → → 2 (blue) [45–48] and pp h h h (red) [37–40] searches in the (m , c ) plane, for diﬀerent values 2 1 1 → → 2 θ of λ /v . The colour gradient for pp h ZZ corresponds to values of λ /v = 0, 1, 2, 3, 4, 5 211 0 2 → → 211 0 (from lighter to darker) while the colour gradient for pp h h h corresponds to values of 2 1 1 λ /v = 3, 4, 5 (from lighter to darker). Also shown are the 95% C.L. lower limit on c from 211 0 θ Higgs signal strength measurements (horizontal solid-black line) [43] and the 95% C.L. lower limit on c (m ) from EWPO (dashed-black line), both being independent of λ /v . θ 2 211 0 O the shift in an oblique parameter can be written entirely in terms of the SM Higgs O contribution to that parameter, SM(m) (which can be found e.g. in [49, 50]), where m is either m or m . These shifts then take the form 1 2 (cid:2) (cid:3) O − O O O − O ∆ = (c2 1) SM(m ) + s2 SM(m ) = s2 SM(m ) SM(m ) . (2.15) θ 1 θ 2 θ 2 1 O The best-ﬁt values for the shifts ∆ 0 and standard deviations σ from the most recent i i post-Higgs-discovery electroweak ﬁt to the SM by the Gﬁtter group [51] (for U = 0, which (cid:32) (cid:33) is a very accurate approximation in the xSM) are given by ≡ − ± ∆S S S = 0.06 0.09 1 0.91 ≡ − SM ± ρ = (2.16) ij ∆T T T = 0.10 0.07 0.91 1 SM − being ρ the covariance matrix in the S T plane. We then perform a ∆χ2 ﬁt to ij obtain the 95% C.L. allowed region in the (m , c ) plane 2 θ – 7 – c θ ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: This completes step 2. Finally, we must show that a B(1, 1) graphon with b = 0 and d = 1, and satisfying (17), is not an entropy maximizer. Among bipodal graphons with b = 0 and d = 1 and a ﬁxed value of ε, minimizing τ and minimizing the entropy s give diﬀerent analytic equations for a and c. For all but ﬁnitely many values of ε, these equations have distinct roots, implying that the graphon that minimizes τ does not maximize the entropy. If we start at the bipodal graphon that minimizes τ for ﬁxed ε and change c to c + δ , while adjusting a to keep ε c ﬁxed, we can increase the entropy to ﬁrst order in δ while only increasing τ to second order c in δ . c To compensate for this increase in τ, we can split the ﬁrst pode in half, yielding a B(2, 1) tripodal graphon with d = 1 and b = 0, with a = a + δ and a− = a − δ . This decreases + a a τ by c3δ3, while decreasing the entropy by O(δ2). By taking δ to be of order δ2/3, we can a a a c 4/3 restore the initial value of τ at an entropy cost of O(δ ). c For δ suﬃciently small and of the correct sign, the O(δ1) gain in entropy from changing c c c and a is greater than the O(δ2) = O(δ4/3) cost in entropy from having a (cid:54)= a−, so the a c + resulting B(2, 1) graphon has higher entropy, but the same values of ε and τ, than the (cid:3) B(1, 1) graphon that minimizes τ (among B(1, 1) graphons). This completes step 3. Proof of Theorem 4.2. The proof follows the same strategy as that of the m = 1 case. In step 1 we show that the Lagrange multipliers must diverge at a continuous transition. In step 2 we show that divergent Lagrange multipliers force a B(m, 1) graphon to be a stationary point of τ for ﬁxed ε. In step 3 we show that we can perturb such a stationary B(m, 1) graphon into a C(m, 2) graphon with the same values of (ε, τ) and more entropy, implying that we are not actually at the phase boundary, which is a contradiction. We parametrize C(m, 2) graphons as follows: There are two interchangable podes, each of size c/2, and m podes of size (1 − c)/m. Let p denote the value of g(x, y) when x is in ij  the i-th pode and y is in the j-th. We deﬁne parameters {a , a−, b, d, p} such that  +    equationcolor    a if (i, j) = (1, 2) or (2, 1), + a− if (i, j) = (1, 1) or (2, 2),    p = d if i ≤ 2 < j or j ≤ 2 < i, (18) ij b if i = j > 2, p if i (cid:54)= j and i, j > 2. The transition that we are trying to rule out is one where a± approach a common value a, resulting in a B(m, 1) graphon with one pode of size c and m podes of size (1 − c)/m, with 16

text: This

text: completes

text: step

text: Finally,

text: we

text: must

text: show

text: that

text: a

text: graphon

text: with

text: b

text: and

text: d

text: and

text: satisfying

text: is

text: not

text: an

text: entropy

text: maximizer.

text: Among

text: bipodal

text: graphons

text: with

text: b

text: and

text: d

text: and

text: a

text: ﬁxed

text: value

text: of

text: ε,

text: minimizing

text: τ

text: and

text: minimizing

text: the

text: entropy

text: s

text: give

text: diﬀerent

text: analytic

text: equations

text: for

text: a

text: and

text: c.

text: For

text: all

text: but

text: ﬁnitely

text: many

text: values

text: of

text: ε,

text: these

text: equations

text: have

text: distinct

text: roots,

text: implying

text: that

text: the

text: graphon

text: that

text: minimizes

text: τ

text: does

text: not

text: maximize

text: the

text: entropy.

text: If

text: we

text: start

text: at

text: the

text: bipodal

text: graphon

text: that

text: minimizes

text: τ

text: for

text: ﬁxed

text: ε

text: and

text: change

text: c

text: to

text: c

text: δ

text: while

text: adjusting

text: a

text: to

text: keep

text: ε

text: c

text: ﬁxed,

text: we

text: can

text: increase

text: the

text: entropy

text: to

text: ﬁrst

text: order

text: in

text: δ

text: while

text: only

text: increasing

text: τ

text: to

text: second

text: order

text: c

text: in

text: δ

text: c

text: To

text: compensate

text: for

text: this

text: increase

text: in

text: τ,

text: we

text: can

text: split

text: the

text: ﬁrst

text: pode

text: in

text: half,

text: yielding

text: a

text: tripodal

text: graphon

text: with

text: d

text: and

text: b

text: with

text: a

text: a

text: δ

text: and

text: a−

text: a

text: δ

text: This

text: decreases

text: a

text: a

text: τ

text: by

text: while

text: decreasing

text: the

text: entropy

text: by

text: By

text: taking

text: δ

text: to

text: be

text: of

text: order

text: we

text: can

text: a

text: a

text: a

text: c

text: restore

text: the

text: initial

text: value

text: of

text: τ

text: at

text: an

text: entropy

text: cost

text: of

text: O(δ

text: c

text: For

text: δ

text: suﬃciently

text: small

text: and

text: of

text: the

text: correct

text: sign,

text: the

text: gain

text: in

text: entropy

text: from

text: changing

text: c

text: c

text: c

text: and

text: a

text: is

text: greater

text: than

text: the

text: cost

text: in

text: entropy

text: from

text: having

text: a

text: a−,

text: so

text: the

text: a

text: c

text: resulting

text: graphon

text: has

text: higher

text: entropy,

text: but

text: the

text: same

text: values

text: of

text: ε

text: and

text: τ,

text: than

text: the

text: graphon

text: that

text: minimizes

text: τ

text: (among

text: graphons).

text: This

text: completes

text: step

text: Proof

text: of

text: Theorem

text: The

text: proof

text: follows

text: the

text: same

text: strategy

text: as

text: that

text: of

text: the

text: m

text: case.

text: In

text: step

text: we

text: show

text: that

text: the

text: Lagrange

text: multipliers

text: must

text: diverge

text: at

text: a

text: continuous

text: transition.

text: In

text: step

text: we

text: show

text: that

text: divergent

text: Lagrange

text: multipliers

text: force

text: a

text: B(m,

text: graphon

text: to

text: be

text: a

text: stationary

text: point

text: of

text: τ

text: for

text: ﬁxed

text: ε.

text: In

text: step

text: we

text: show

text: that

text: we

text: can

text: perturb

text: such

text: a

text: stationary

text: B(m,

text: graphon

text: into

text: a

text: C(m,

text: graphon

text: with

text: the

text: same

text: values

text: of

text: (ε,

text: τ)

text: and

text: more

text: entropy,

text: implying

text: that

text: we

text: are

text: not

text: actually

text: at

text: the

text: phase

text: boundary,

text: which

text: is

text: a

text: contradiction.

text: We

text: parametrize

text: C(m,

text: graphons

text: as

text: follows:

text: There

text: are

text: two

text: interchangable

text: podes,

text: each

text: of

text: size

text: and

text: m

text: podes

text: of

text: size

text: c)/m.

text: Let

text: p

text: denote

text: the

text: value

text: of

text: g(x,

text: y)

text: when

text: x

text: is

text: in

text: ij

text: the

text: i-th

text: pode

text: and

text: y

text: is

text: in

text: the

text: j-th.

text: We

text: deﬁne

text: parameters

text: {a

text: a−,

text: b,

text: d,

text: p}

text: such

text: that

text: equationcolor

text: a

text: if

text: (i,

text: j)

text: or

text: a−

text: if

text: (i,

text: j)

text: or

text: p

text: d

text: if

text: i

text: j

text: or

text: j

text: i,

text: ij

text: b

text: if

text: i

text: j

text: p

text: if

text: i

text: j

text: and

text: i,

text: j

text: The

text: transition

text: that

text: we

text: are

text: trying

text: to

text: rule

text: out

text: is

text: one

text: where

text: a±

text: approach

text: a

text: common

text: value

text: a,

text: resulting

text: in

text: a

text: B(m,

text: graphon

text: with

text: one

text: pode

text: of

text: size

text: c

text: and

text: m

text: podes

text: of

text: size

text: c)/m,

text: with


text: − − where X = tanh 1(a) − tanh 1(b). Within the A(n, 0) phase, this second variation is negative-deﬁnite, while within the B(n − 1, 1) phase the matrix has a positive eigenvalue. The boundary is thus deﬁned (locally) by the analytic equation det H(S) = 0. One can also consider continuous changes from an A(n, 0) graphon to a graphon with a symmetry other than B(n − 1, 1). The local stability condition for such changes works out to be exactly the same as for A(n, 0) ↔ B(n − 1, 1). The upshot is that the analytic curve det H(S) = 0 deﬁnes the boundary of the region where the A(n, 0) graphon is stable against small perturbations. In Figure 11 we plot the regions where the A(3, 0) and A(4, 0) graphons are stable against small perturbations. Comparing to Figures 6 and 7, we see that the stability regions are larger than the actual A(3, 0) and A(4, 0) phases, and even overlap! Without assuming anything about the phase portrait (beyond the existence of A(3, 0) and A(4, 0) phases) this proves that some of the transitions from A(3, 0) or A(4, 0) to other phases must not be governed by the local stability condition, and so must be discontinuous. Figure 11: captioncolorThe regions of local stability for A(3, 0) and A(4, 0) graphons, show- (cid:48) ing overlap. The left plot uses coordinates (ε, τ) and the right plot uses (ε, τ = τ −ε(2ε−1)). The transition A(2, 0) ↔ B(1, 1) is a supercritical pitchfork bifurcation [16]. The ana- lytic family of entropy maximizers in the A(2, 0) phase has a natural analytic continuation into the B(1, 1) region, but no longer maximizes entropy there. The analytic family of en- tropy maximizers in the B(1, 1) phase doubles back on itself and cannot be continued into the A(2, 0) region. Power series analysis suggests that something similar happens in the transition from A(n, 0) to B(n − 1, 1). The family of graphons representing A(n, 0) entropy maximizers can be continued into the B(n − 1, 1) region but no longer maximizes entropy. The family of graphons representing B(n − 1, 1) cannot be analytically continued into the A(n, 0) region, but instead doubles back into the B(n − 1, 1) region. Unlike in the case of B(1, 1), the “doubling back” branch is not related by symmetry to the original branch, and so represents a diﬀerent set of reduced graphons, all stationary points of the entropy, but with presumably lower entropy than graphons of the original branch with the same values 19 ##LTFigure## ##LTFigure##

text: where

text: X

text: tanh

text: tanh

text: Within

text: the

text: A(n,

text: phase,

text: this

text: second

text: variation

text: is

text: negative-deﬁnite,

text: while

text: within

text: the

text: B(n

text: phase

text: the

text: matrix

text: has

text: a

text: positive

text: eigenvalue.

text: The

text: boundary

text: is

text: thus

text: deﬁned

text: (locally)

text: by

text: the

text: analytic

text: equation

text: det

text: H(S)

text: One

text: can

text: also

text: consider

text: continuous

text: changes

text: from

text: an

text: A(n,

text: graphon

text: to

text: a

text: graphon

text: with

text: a

text: symmetry

text: other

text: than

text: B(n

text: The

text: local

text: stability

text: condition

text: for

text: such

text: changes

text: works

text: out

text: to

text: be

text: exactly

text: the

text: same

text: as

text: for

text: A(n,

text: B(n

text: The

text: upshot

text: is

text: that

text: the

text: analytic

text: curve

text: det

text: H(S)

text: deﬁnes

text: the

text: boundary

text: of

text: the

text: region

text: where

text: the

text: A(n,

text: graphon

text: is

text: stable

text: against

text: small

text: perturbations.

text: In

text: Figure

text: we

text: plot

text: the

text: regions

text: where

text: the

text: and

text: graphons

text: are

text: stable

text: against

text: small

text: perturbations.

text: Comparing

text: to

text: Figures

text: and

text: we

text: see

text: that

text: the

text: stability

text: regions

text: are

text: larger

text: than

text: the

text: actual

text: and

text: phases,

text: and

text: even

text: overlap!

text: Without

text: assuming

text: anything

text: about

text: the

text: phase

text: portrait

text: (beyond

text: the

text: existence

text: of

text: and

text: phases)

text: this

text: proves

text: that

text: some

text: of

text: the

text: transitions

text: from

text: or

text: to

text: other

text: phases

text: must

text: not

text: be

text: governed

text: by

text: the

text: local

text: stability

text: condition,

text: and

text: so

text: must

text: be

text: discontinuous.

text: Figure

text: captioncolorThe

text: regions

text: of

text: local

text: stability

text: for

text: and

text: graphons,

text: show-

text: ing

text: overlap.

text: The

text: left

text: plot

text: uses

text: coordinates

text: (ε,

text: τ)

text: and

text: the

text: right

text: plot

text: uses

text: (ε,

text: τ

text: τ

text: The

text: transition

text: is

text: a

text: supercritical

text: pitchfork

text: bifurcation

text: The

text: ana-

text: lytic

text: family

text: of

text: entropy

text: maximizers

text: in

text: the

text: phase

text: has

text: a

text: natural

text: analytic

text: continuation

text: into

text: the

text: region,

text: but

text: no

text: longer

text: maximizes

text: entropy

text: there.

text: The

text: analytic

text: family

text: of

text: en-

text: tropy

text: maximizers

text: in

text: the

text: phase

text: doubles

text: back

text: on

text: itself

text: and

text: cannot

text: be

text: continued

text: into

text: the

text: region.

text: Power

text: series

text: analysis

text: suggests

text: that

text: something

text: similar

text: happens

text: in

text: the

text: transition

text: from

text: A(n,

text: to

text: B(n

text: The

text: family

text: of

text: graphons

text: representing

text: A(n,

text: entropy

text: maximizers

text: can

text: be

text: continued

text: into

text: the

text: B(n

text: region

text: but

text: no

text: longer

text: maximizes

text: entropy.

text: The

text: family

text: of

text: graphons

text: representing

text: B(n

text: cannot

text: be

text: analytically

text: continued

text: into

text: the

text: A(n,

text: region,

text: but

text: instead

text: doubles

text: back

text: into

text: the

text: B(n

text: region.

text: Unlike

text: in

text: the

text: case

text: of

text: the

text: “doubling

text: back”

text: branch

text: is

text: not

text: related

text: by

text: symmetry

text: to

text: the

text: original

text: branch,

text: and

text: so

text: represents

text: a

text: diﬀerent

text: set

text: of

text: reduced

text: graphons,

text: all

text: stationary

text: points

text: of

text: the

text: entropy,

text: but

text: with

text: presumably

text: lower

text: entropy

text: than

text: graphons

text: of

text: the

text: original

text: branch

text: with

text: the

text: same

text: values

text: ##LTFigure##

text: ##LTFigure##


text: classes: the probability of an edge, between a node v in class j with a node v in class k, is j k independent of v and v , only depending on j and k. j k The second level of symmetry concerns the equivalence classes of nodes: certain equiv- alence classes have the same sizes and edge probabilities, and others don’t. These size and probability parameters are used to distinguish distinct phases, that is, maximal open regions in the parameter space where the entropy-maximizing graphon g is unique and ε,τ varies analytically. Because of multipodality the function g restricted to a given phase can be considered ε,τ a smooth vector valued function of (ε, τ), the coordinates being the probabilities of edges between node equivalence classes, and the relative sizes of those equivalence classes. By the symmetry of a phase we refer to the symmetries among these coordinates, with the following caveat, illustrated through an example. Within the phase F (1, 1) there is a curve such that the two node equivalence classes have the same size. In a narrow sense this might have signalled a higher symmetry. However there is no singular behavior as (ε, τ) crosses this curve so the curve is simply part of F (1, 1) and does not aﬀect the ‘symmetry’ of the phase. Each of the phases has a diﬀerent symmetry and we conjecture that they all fall into the three families: A(n, 0), B(n, 1) and C(n, 2), or F (1, 1), in which the notation completely speciﬁes the symmetry (except for the pairs F (1, 1), B(1, 1), and C(1, 2), B(2, 1)). An important result of this paper is the conjectured phase diagram, Figure 1. The other goal is to show how knowledge of the structure of optimizing graphons in phases can be helpful in understanding the role of symmetry in speciﬁc features of phase transitions. Consider the following, paraphrased from P.W. Anderson [1], a picture he attributes to Landau [8]: The First Theorem of solid-state physics states that it is impossible to change symmetry gradually. A symmetry element is either there or it is not; there is no way for it to grow imperceptibly. This intuitive picture has been applied, for instance by Landau, to understand why there is no critical point for the ﬂuid/solid transition [1], though it has been diﬃcult to make the argument rigorous: This is the theoretical argument, which has appeared to some to be a little too straightforward to be absolutely convincing [11]. We suggest that network models such as the edge/triangle model of this paper provide a useful framework for enabling a rigorous study of such symmetry principles. This was done in [16] speciﬁcally for the issue of existence of a critical point. Landau’s symmetry principles are commonly applied to the issue of whether phases are continuous at a transition [8], which is also related to uniqueness of entropy optimizers. We have proven that in the edge/triangle model certain transitions cannot be continu- ous, and evidence suggests that certain other transitions are continuous. It is worthwhile discussing how these transitions are approached at the micro-level, that is, in terms of the multipodal parameters, the probabilities of edges between various type of nodes. In this 21

text: classes:

text: the

text: probability

text: of

text: an

text: edge,

text: between

text: a

text: node

text: v

text: in

text: class

text: j

text: with

text: a

text: node

text: v

text: in

text: class

text: k,

text: is

text: j

text: k

text: independent

text: of

text: v

text: and

text: v

text: only

text: depending

text: on

text: j

text: and

text: k.

text: j

text: k

text: The

text: second

text: level

text: of

text: symmetry

text: concerns

text: the

text: equivalence

text: classes

text: of

text: nodes:

text: certain

text: equiv-

text: alence

text: classes

text: have

text: the

text: same

text: sizes

text: and

text: edge

text: probabilities,

text: and

text: others

text: don’t.

text: These

text: size

text: and

text: probability

text: parameters

text: are

text: used

text: to

text: distinguish

text: distinct

text: phases,

text: that

text: is,

text: maximal

text: open

text: regions

text: in

text: the

text: parameter

text: space

text: where

text: the

text: entropy-maximizing

text: graphon

text: g

text: is

text: unique

text: and

text: ε,τ

text: varies

text: analytically.

text: Because

text: of

text: multipodality

text: the

text: function

text: g

text: restricted

text: to

text: a

text: given

text: phase

text: can

text: be

text: considered

text: ε,τ

text: a

text: smooth

text: vector

text: valued

text: function

text: of

text: (ε,

text: τ),

text: the

text: coordinates

text: being

text: the

text: probabilities

text: of

text: edges

text: between

text: node

text: equivalence

text: classes,

text: and

text: the

text: relative

text: sizes

text: of

text: those

text: equivalence

text: classes.

text: By

text: the

text: symmetry

text: of

text: a

text: phase

text: we

text: refer

text: to

text: the

text: symmetries

text: among

text: these

text: coordinates,

text: with

text: the

text: following

text: caveat,

text: illustrated

text: through

text: an

text: example.

text: Within

text: the

text: phase

text: F

text: there

text: is

text: a

text: curve

text: such

text: that

text: the

text: two

text: node

text: equivalence

text: classes

text: have

text: the

text: same

text: size.

text: In

text: a

text: narrow

text: sense

text: this

text: might

text: have

text: signalled

text: a

text: higher

text: symmetry.

text: However

text: there

text: is

text: no

text: singular

text: behavior

text: as

text: (ε,

text: τ)

text: crosses

text: this

text: curve

text: so

text: the

text: curve

text: is

text: simply

text: part

text: of

text: F

text: and

text: does

text: not

text: aﬀect

text: the

text: ‘symmetry’

text: of

text: the

text: phase.

text: Each

text: of

text: the

text: phases

text: has

text: a

text: diﬀerent

text: symmetry

text: and

text: we

text: conjecture

text: that

text: they

text: all

text: fall

text: into

text: the

text: three

text: families:

text: A(n,

text: B(n,

text: and

text: C(n,

text: or

text: F

text: in

text: which

text: the

text: notation

text: completely

text: speciﬁes

text: the

text: symmetry

text: (except

text: for

text: the

text: pairs

text: F

text: and

text: An

text: important

text: result

text: of

text: this

text: paper

text: is

text: the

text: conjectured

text: phase

text: diagram,

text: Figure

text: The

text: other

text: goal

text: is

text: to

text: show

text: how

text: knowledge

text: of

text: the

text: structure

text: of

text: optimizing

text: graphons

text: in

text: phases

text: can

text: be

text: helpful

text: in

text: understanding

text: the

text: role

text: of

text: symmetry

text: in

text: speciﬁc

text: features

text: of

text: phase

text: transitions.

text: Consider

text: the

text: following,

text: paraphrased

text: from

text: P.W.

text: Anderson

text: a

text: picture

text: he

text: attributes

text: to

text: Landau

text: The

text: First

text: Theorem

text: of

text: solid-state

text: physics

text: states

text: that

text: it

text: is

text: impossible

text: to

text: change

text: symmetry

text: gradually.

text: A

text: symmetry

text: element

text: is

text: either

text: there

text: or

text: it

text: is

text: not;

text: there

text: is

text: no

text: way

text: for

text: it

text: to

text: grow

text: imperceptibly.

text: This

text: intuitive

text: picture

text: has

text: been

text: applied,

text: for

text: instance

text: by

text: Landau,

text: to

text: understand

text: why

text: there

text: is

text: no

text: critical

text: point

text: for

text: the

text: ﬂuid/solid

text: transition

text: though

text: it

text: has

text: been

text: diﬃcult

text: to

text: make

text: the

text: argument

text: rigorous:

text: This

text: is

text: the

text: theoretical

text: argument,

text: which

text: has

text: appeared

text: to

text: some

text: to

text: be

text: a

text: little

text: too

text: straightforward

text: to

text: be

text: absolutely

text: convincing

text: We

text: suggest

text: that

text: network

text: models

text: such

text: as

text: the

text: edge/triangle

text: model

text: of

text: this

text: paper

text: provide

text: a

text: useful

text: framework

text: for

text: enabling

text: a

text: rigorous

text: study

text: of

text: such

text: symmetry

text: principles.

text: This

text: was

text: done

text: in

text: speciﬁcally

text: for

text: the

text: issue

text: of

text: existence

text: of

text: a

text: critical

text: point.

text: Landau’s

text: symmetry

text: principles

text: are

text: commonly

text: applied

text: to

text: the

text: issue

text: of

text: whether

text: phases

text: are

text: continuous

text: at

text: a

text: transition

text: which

text: is

text: also

text: related

text: to

text: uniqueness

text: of

text: entropy

text: optimizers.

text: We

text: have

text: proven

text: that

text: in

text: the

text: edge/triangle

text: model

text: certain

text: transitions

text: cannot

text: be

text: continu-

text: ous,

text: and

text: evidence

text: suggests

text: that

text: certain

text: other

text: transitions

text: are

text: continuous.

text: It

text: is

text: worthwhile

text: discussing

text: how

text: these

text: transitions

text: are

text: approached

text: at

text: the

text: micro-level,

text: that

text: is,

text: in

text: terms

text: of

text: the

text: multipodal

text: parameters,

text: the

text: probabilities

text: of

text: edges

text: between

text: various

text: type

text: of

text: nodes.

text: In

text: this


text: W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, noop Numerical Recipes, 3rd ed. (Cambridge University Press, Cambridge, 2007). L. F. Shampine, Math. Comp. 46, 135 (1986). W. G. Harter and C. W. Patterson, J. Chem. Phys. 80, 4241 (1984). F. Haake, Quantum Signatures of Chaos, 3rd ed. (Springer Verlag, Berlin, 2010). O. Bohigas, M. J. Giannoni, and C. Schmit, Phys. Rev. Lett. 52, 1 (1984). M. D. Poulsen, E. P´eronne, H. Stapelfeldt, C. Z. Bisgaard, S. S. Viftrup, E. Hamilton, and T. Seideman, noop J. Chem. Phys. 121, 783 (2004). For alternative chaos time scales see, e.g., J. W. Duﬀ and P. Brumer, J. Chem. Phys. 71, 2693 (1979). S. Tomsovic and J. H. Lefebvre, Phys. Rev. Lett. 79, 3629 (1997). J. Floß and I. Sh. Averbukh, Phys. Rev. Lett. 113, 043002 (2014). P. Brumer, noop J. Comput. Phys. 14, 391 (1974). J. W. Duﬀ and P. Brumer, noop J. Chem. Phys. 65, 3566 (1976). D. M. Villeneuve, S. A. Aseyev, P. Dietrich, M. Spanner, M. Yu. Ivanov, and P. B. Corkum, noop Phys. Rev. Lett. 85, 542 (2000). L. Yuan, C. Toro, M. Bell, and A. S. Mullin, Faraday Discuss. 150, 101 (2011). A. Korobenko, A. A. Milner, and V. Milner, Phys. Rev. Lett. 112, 113004 (2014). A. Korobenko and V. Milner, Phys. Rev. Lett. 116, 183001 (2016) . 30

text: W.

text: H.

text: Press,

text: S.

text: A.

text: Teukolsky,

text: W.

text: T.

text: Vetterling,

text: and

text: B.

text: P.

text: Flannery,

text: noop

text: Numerical

text: Recipes,

text: ed.

text: (Cambridge

text: University

text: Press,

text: Cambridge,

text: L.

text: F.

text: Shampine,

text: Math.

text: Comp.

text: W.

text: G.

text: Harter

text: and

text: C.

text: W.

text: Patterson,

text: J.

text: Chem.

text: Phys.

text: F.

text: Haake,

text: Quantum

text: Signatures

text: of

text: Chaos,

text: ed.

text: (Springer

text: Verlag,

text: Berlin,

text: O.

text: Bohigas,

text: M.

text: J.

text: Giannoni,

text: and

text: C.

text: Schmit,

text: Phys.

text: Rev.

text: Lett.

text: M.

text: D.

text: Poulsen,

text: E.

text: P´eronne,

text: H.

text: Stapelfeldt,

text: C.

text: Z.

text: Bisgaard,

text: S.

text: S.

text: Viftrup,

text: E.

text: Hamilton,

text: and

text: T.

text: Seideman,

text: noop

text: J.

text: Chem.

text: Phys.

text: For

text: alternative

text: chaos

text: time

text: scales

text: see,

text: e.g.,

text: J.

text: W.

text: Duﬀ

text: and

text: P.

text: Brumer,

text: J.

text: Chem.

text: Phys.

text: S.

text: Tomsovic

text: and

text: J.

text: H.

text: Lefebvre,

text: Phys.

text: Rev.

text: Lett.

text: J.

text: Floß

text: and

text: I.

text: Sh.

text: Averbukh,

text: Phys.

text: Rev.

text: Lett.

text: P.

text: Brumer,

text: noop

text: J.

text: Comput.

text: Phys.

text: J.

text: W.

text: Duﬀ

text: and

text: P.

text: Brumer,

text: noop

text: J.

text: Chem.

text: Phys.

text: D.

text: M.

text: Villeneuve,

text: S.

text: A.

text: Aseyev,

text: P.

text: Dietrich,

text: M.

text: Spanner,

text: M.

text: Yu.

text: Ivanov,

text: and

text: P.

text: B.

text: Corkum,

text: noop

text: Phys.

text: Rev.

text: Lett.

text: L.

text: Yuan,

text: C.

text: Toro,

text: M.

text: Bell,

text: and

text: A.

text: S.

text: Mullin,

text: Faraday

text: Discuss.

text: A.

text: Korobenko,

text: A.

text: A.

text: Milner,

text: and

text: V.

text: Milner,

text: Phys.

text: Rev.

text: Lett.

text: A.

text: Korobenko

text: and

text: V.

text: Milner,

text: Phys.

text: Rev.

text: Lett.


text: 11 |Y − | ≤ − (cid:0) (cid:1) where (a) follows from b 1 b 1 and from the convexity of x log x and Jensen’s inequality, (b) follows from E|Y − | − − − b 1 = A 1 (1 1/A)b 1 since the Yb−1 are chosen uniformly with rep(cid:40)lacement from the set X of cardinality A, and (c) follows (cid:88) from − ≤ B − − B(2A B + 1)/(2A), if B A, (1 (b 1)/A)+ = (A + 1)/2, if B > A. b=1 ≥ { } 0.5 min A, B and from − − − ≥ − − ≥ 1 (1 1/A)b 1 1 exp( 1) 0.5 ≥ for b A + 1. (cid:88) From (1) and (8), we obtain − ≤ B P ∈ Y − R R(cid:63) 2 log(BL) + 3 + 2B + (Y / b 1) FL b ≤ b=1 2 log(BL) + 3 + 3B. Combining this with (9) yields R − ≤ 2 log(BL) + 3 + 3B FL 1 − { } − . R(cid:63) 0.5(L 1) min A, B + 0.5(B A)+ log(A/2) For B large enough, this can be simpliﬁed as R − ≤ log(L) + B FL 1 7 − { } − , R(cid:63) (L 1) min A, B + (B A)+ log(A/2) concluding the proof. APPENDIX B ANALYSIS OF FIXED-LENGTH DEDUPLICATION WITH VARIABLE SOURCE-SYMBOL LENGTH (EXAMPLE 5) This appendix contains the formal analysis for Example 5. We start with an upper bound on R(cid:63). Since R(cid:63) is the rate of the optimal preﬁx-free code, it is upper bounded as ≤ R(cid:63) H(S) + 1 (cid:0) (cid:1) (see, e.g., [28, Theorem 5.4.1]). Now, (cid:1) ≤ H(S) H S, YB | = H(YB) + H(S YB = H(YB) ≤ X H(YB, ) X | X = H( ) + H(YB ) ≤ A(L + 2) + B log(A). Thus, ≤ R(cid:63) A(L + 2) + B log(A) + 1. (10) ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: |Y

text: where

text: (a)

text: follows

text: from

text: b

text: b

text: and

text: from

text: the

text: convexity

text: of

text: x

text: log

text: x

text: and

text: Jensen’s

text: inequality,

text: (b)

text: follows

text: from

text: E|Y

text: b

text: A

text: since

text: the

text: are

text: chosen

text: uniformly

text: with

text: from

text: the

text: set

text: X

text: of

text: cardinality

text: A,

text: and

text: (c)

text: follows

text: from

text: B

text: B

text: if

text: B

text: A,

text: (b

text: (A

text: if

text: B

text: A.

text: min

text: A,

text: B

text: and

text: from

text: exp(

text: for

text: b

text: A

text: From

text: and

text: we

text: obtain

text: B

text: P

text: Y

text: R

text: log(BL)

text: (Y

text: b

text: FL

text: b

text: log(BL)

text: Combining

text: this

text: with

text: yields

text: R

text: log(BL)

text: FL

text: min

text: A,

text: B

text: A)+

text: For

text: B

text: large

text: enough,

text: this

text: can

text: be

text: simpliﬁed

text: as

text: R

text: log(L)

text: B

text: FL

text: (L

text: min

text: A,

text: B

text: (B

text: A)+

text: concluding

text: the

text: proof.

text: APPENDIX

text: B

text: ANALYSIS

text: OF

text: FIXED-LENGTH

text: DEDUPLICATION

text: WITH

text: VARIABLE

text: SOURCE-SYMBOL

text: LENGTH

text: (EXAMPLE

text: This

text: appendix

text: contains

text: the

text: formal

text: analysis

text: for

text: Example

text: We

text: start

text: with

text: an

text: upper

text: bound

text: on

text: Since

text: is

text: the

text: rate

text: of

text: the

text: optimal

text: preﬁx-free

text: code,

text: it

text: is

text: upper

text: bounded

text: as

text: H(S)

text: (see,

text: e.g.,

text: Theorem

text: Now,

text: H(S)

text: H

text: S,

text: YB

text: H(YB)

text: H(S

text: YB

text: H(YB)

text: X

text: H(YB,

text: X

text: X

text: H(

text: H(YB

text: A(L

text: B

text: log(A).

text: Thus,

text: A(L

text: B

text: log(A)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 II. PROBLEM SETTING A. Source Description X |X | We consider a source alphabet , of size = A, generated randomly as follows. Fix a distribution N E P over with ﬁnite mean (L). Generate A independently and identically distributed (i.i.d.) random L ∈ { } variables L , L , . . . , L from P . We next generate a sequence of binary strings X 0, 1 La for each ∈ { 1 2 } A L { } a\{ } a 1, 2, . . . , A : Starting with a = 1, choose X uniformly at random from 0, 1 La X , X , . . . , X − . a 1 2 a 1 Thus, each X is a binary string of length L , called a source symbol in the following. Finally, the source X ⊂a { }∗ { }∗ a alphabet 0, 1 (where 0, 1 denotes the set of all ﬁnite-length binary sequences) is deﬁned as the union of all the source symbols, X (cid:44) { } X A . (cid:0) (cid:1) a a=1 |X | Note that is equal to A by construction of the X . To simplify some of the derivations later on, we a P E ≤ ≤ E assume that L is tightly concentrated around its mean, speciﬁcally that (L)/2 L 2 (L) = 1. ≤ Furthermore, to ensure that the source alphabet generation is always well deﬁned, we assume that 2 ≤ E − A 2 (L)/2 1. X From this randomly generated source alphabet , we now generate a random source sequence S as X follows. Choose B elements Y , Y , . . . , Y i.i.d. uniformly at random from . Thus, each Y is an { }∗ 1 2 B b element of 0, 1 . The sequence Y is referred to as a source block. Construct the source sequence S as · · · b { }∗ the concatenation Y Y Y of Y , Y , . . . , Y . We emphasize that S is an element of 0, 1 , in other 1 2 B 1 2 B words, the boundaries of the source blocks Y , Y , . . . , Y are not preserved. Denote by (cid:96)(S) the (random) 1 2 B length of S. X { } Example 1. Assume the source alphabet is randomly generated as = 1, 00, 10, 01100, 001100 . From this, B = 2 elements are drawn i.i.d. uniformly at random, say Y = 10 and Y = 01100. The resulting 1 2 source sequence is then S = Y Y = 1001100. Note that, from S alone, it is not clear if S was generated 1 2 as the concatenation of 10 and 01100 or as the concatenation of 1 and 001100. There is a third possible X parsing of S into the ﬁve elements 1, 00, 1, 1, and 00 from . However, knowing B = 2, this third ♦ parsing is not valid. Our goal is to compress the source sequence S, knowing only P , A, and B. In particular, the source X L alphabet and the parsing into the source blocks Y , Y , . . . , Y are not known. More formally, we are 1 2 B looking for a preﬁx-free source code for the random variable S. We have a complete statistical description E of S and (since (L) is ﬁnite) its entropy H(S) is well deﬁned. The expected rate R(cid:63) of the optimal preﬁx-free source code for S is thus bounded as ≤ ≤ H(S) R(cid:63) H(S) + 1. Example 2. We illustrate the order and relative size of the various quantities in the problem setting using data from a recent large-scale primary (i.e., non backup) data deduplication study [5]. The length (cid:96)(S) of the source sequence ranges from several hundreds of GB to a few TB. The expected value of the length L = (cid:96)(X ) of the source symbols is a bit harder to quantify (since the notion of source symbol itself is a a a model abstraction), but the experiments in [5] suggest that reasonable numbers range from a few kB to a few MB. The number B of source blocks is consequently on the order of perhaps 105 to 109. As in our model, the data in [5] indicates that duplicates are not localized, but occur over the entire source string. In other words, the source has long-range dependence. The number of distinct source symbols A is again hard to quantify, but the results in [5], [12] suggest that, depending on the scenario, A should be ♦ somewhere in the range 0.01B to B. There are several differences between the source model introduced here and the standard source coding → ∞ setup. First, the source considered here is nonergodic in the following sense. As B , we obtain { } a stochastic process S = (S , S , . . . ) with each element S taking values in 0, 1 . This process is not 1 2 i

text: II.

label: section

text: PROBLEM

text: SETTING

text: A.

text: Source

text: Description

text: X

text: |X

text: We

text: consider

text: a

text: source

text: alphabet

text: of

text: size

text: A,

text: generated

text: randomly

text: as

text: follows.

text: Fix

text: a

text: distribution

text: N

text: E

text: P

text: over

text: with

text: ﬁnite

text: mean

text: (L).

text: Generate

text: A

text: independently

text: and

text: identically

text: distributed

text: (i.i.d.)

text: random

text: L

text: variables

text: L

text: L

text: L

text: from

text: P

text: We

text: next

text: generate

text: a

text: sequence

text: of

text: binary

text: strings

text: X

text: La

text: for

text: each

text: A

text: L

text: a\{

text: a

text: A

text: Starting

text: with

text: a

text: choose

text: X

text: uniformly

text: at

text: random

text: from

text: La

text: X

text: X

text: X

text: a

text: a

text: Thus,

text: each

text: X

text: is

text: a

text: binary

text: string

text: of

text: length

text: L

text: called

text: a

text: source

text: symbol

text: in

text: the

text: following.

text: Finally,

text: the

text: source

text: X

text: ⊂a

text: a

text: alphabet

text: (where

text: denotes

text: the

text: set

text: of

text: all

text: ﬁnite-length

text: binary

text: sequences)

text: is

text: deﬁned

text: as

text: the

text: union

text: of

text: all

text: the

text: source

text: symbols,

text: X

text: X

text: A

text: a

text: |X

text: Note

text: that

text: is

text: equal

text: to

text: A

text: by

text: construction

text: of

text: the

text: X

text: To

text: simplify

text: some

text: of

text: the

text: derivations

text: later

text: on,

text: we

text: a

text: P

text: E

text: E

text: assume

text: that

text: L

text: is

text: tightly

text: concentrated

text: around

text: its

text: mean,

text: speciﬁcally

text: that

text: L

text: (L)

text: Furthermore,

text: to

text: ensure

text: that

text: the

text: source

text: alphabet

text: generation

text: is

text: always

text: well

text: deﬁned,

text: we

text: assume

text: that

text: E

text: A

text: X

text: From

text: this

text: randomly

text: generated

text: source

text: alphabet

text: we

text: now

text: generate

text: a

text: random

text: source

text: sequence

text: S

text: as

text: X

text: follows.

text: Choose

text: B

text: elements

text: Y

text: Y

text: Y

text: i.i.d.

text: uniformly

text: at

text: random

text: from

text: Thus,

text: each

text: Y

text: is

text: an

text: B

text: b

text: element

text: of

text: The

text: sequence

text: Y

text: is

text: referred

text: to

text: as

text: a

text: source

text: block.

text: Construct

text: the

text: source

text: sequence

text: S

text: as

text: b

text: the

text: concatenation

text: Y

text: Y

text: Y

text: of

text: Y

text: Y

text: Y

text: We

text: emphasize

text: that

text: S

text: is

text: an

text: element

text: of

text: in

text: other

text: B

text: B

text: words,

text: the

text: boundaries

text: of

text: the

text: source

text: blocks

text: Y

text: Y

text: Y

text: are

text: not

text: preserved.

text: Denote

text: by

text: the

text: (random)

text: B

text: length

text: of

text: S.

text: X

text: Example

text: Assume

text: the

text: source

text: alphabet

text: is

text: randomly

text: generated

text: as

text: From

text: this,

text: B

text: elements

text: are

text: drawn

text: i.i.d.

text: uniformly

text: at

text: random,

text: say

text: Y

text: and

text: Y

text: The

text: resulting

text: source

text: sequence

text: is

text: then

text: S

text: Y

text: Y

text: Note

text: that,

text: from

text: S

text: alone,

text: it

text: is

text: not

text: clear

text: if

text: S

text: was

text: generated

text: as

text: the

text: concatenation

text: of

text: and

text: or

text: as

text: the

text: concatenation

text: of

text: and

text: There

text: is

text: a

text: third

text: possible

text: X

text: parsing

text: of

text: S

text: into

text: the

text: ﬁve

text: elements

text: and

text: from

text: However,

text: knowing

text: B

text: this

text: third

text: parsing

text: is

text: not

text: valid.

text: Our

text: goal

text: is

text: to

text: compress

text: the

text: source

text: sequence

text: S,

text: knowing

text: only

text: P

text: A,

text: and

text: B.

text: In

text: particular,

text: the

text: source

text: X

text: L

text: alphabet

text: and

text: the

text: parsing

text: into

text: the

text: source

text: blocks

text: Y

text: Y

text: Y

text: are

text: not

text: known.

text: More

text: formally,

text: we

text: are

text: B

text: looking

text: for

text: a

text: preﬁx-free

text: source

text: code

text: for

text: the

text: random

text: variable

text: S.

text: We

text: have

text: a

text: complete

text: statistical

text: description

text: E

text: of

text: S

text: and

text: (since

text: (L)

text: is

text: ﬁnite)

text: its

text: entropy

text: H(S)

text: is

text: well

text: deﬁned.

text: The

text: expected

text: rate

text: of

text: the

text: optimal

text: preﬁx-free

text: source

text: code

text: for

text: S

text: is

text: thus

text: bounded

text: as

text: H(S)

text: H(S)

text: Example

text: We

text: illustrate

text: the

text: order

text: and

text: relative

text: size

text: of

text: the

text: various

text: quantities

text: in

text: the

text: problem

text: setting

text: using

text: data

text: from

text: a

text: recent

text: large-scale

text: primary

text: (i.e.,

text: non

text: backup)

text: data

text: deduplication

text: study

text: The

text: length

text: of

text: the

text: source

text: sequence

text: ranges

text: from

text: several

text: hundreds

text: of

text: GB

text: to

text: a

text: few

text: TB.

text: The

text: expected

text: value

text: of

text: the

text: length

text: L

text: of

text: the

text: source

text: symbols

text: is

text: a

text: bit

text: harder

text: to

text: quantify

text: (since

text: the

text: notion

text: of

text: source

text: symbol

text: itself

text: is

text: a

text: a

text: a

text: model

text: abstraction),

text: but

text: the

text: experiments

text: in

text: suggest

text: that

text: reasonable

text: numbers

text: range

text: from

text: a

text: few

text: kB

text: to

text: a

text: few

text: MB.

text: The

text: number

text: B

text: of

text: source

text: blocks

text: is

text: consequently

text: on

text: the

text: order

text: of

text: perhaps

text: to

text: As

text: in

text: our

text: model,

text: the

text: data

text: in

text: indicates

text: that

text: duplicates

text: are

text: not

text: localized,

text: but

text: occur

text: over

text: the

text: entire

text: source

text: string.

text: In

text: other

text: words,

text: the

text: source

text: has

text: long-range

text: dependence.

text: The

text: number

text: of

text: distinct

text: source

text: symbols

text: A

text: is

text: again

text: hard

text: to

text: quantify,

text: but

text: the

text: results

text: in

text: suggest

text: that,

text: depending

text: on

text: the

text: scenario,

text: A

text: should

text: be

text: somewhere

text: in

text: the

text: range

text: to

text: B.

text: There

text: are

text: several

text: differences

text: between

text: the

text: source

text: model

text: introduced

text: here

text: and

text: the

text: standard

text: source

text: coding

text: setup.

text: First,

text: the

text: source

text: considered

text: here

text: is

text: nonergodic

text: in

text: the

text: following

text: sense.

text: As

text: B

text: we

text: obtain

text: a

text: stochastic

text: process

text: S

text: (S

text: S

text: with

text: each

text: element

text: S

text: taking

text: values

text: in

text: This

text: process

text: is

text: not

text: i


text: 21 tail(X1) head(X2) tail(X3) head(X4) Fig. 4. Duplicate boundary deduplication chunks arising from the concatenation of different source symbols X X and X X . Compare to 1 2 3 4 Fig. 2 in Appendix . tail(X ) head(X ) = tail(X ) head(X ), then one of head(X ), head(X ) is a substring of the other, and 1 2 3 4 2 4 one of tail(X ), tail(X ) is a substring of the other. 1 3 ≥ Consider a source symbol X , and assume M 10 for now. With probability at least a − · − · − − · − ≥ − − 1 2 2M 5 2 M 2 2 M/2 1 2 3 { − } it has head(X ) of length at least min 2M 5, L/2 containing a least one symbol 1 in the ﬁrst M/2 bits a { − } and it has tail(X ) of length at least min 2M 5, L/2 and containing a least one symbol 1 in the last M/2 a X bits. A short calculation shows that this implies that with probability at least 3/4, the source alphabet has at least A/8 symbols with this property. − − − X Moreover, with probability at least 1 (AL)22 2M 5 the source alphabet has no repeating, nonover- − lapping substrings of size 2M 5. This argument is reported with more detail in Appendix . In particular, (cid:0) (cid:1) (cid:0) (cid:1) if ≥ M 5 + log log 8(AL)2 = 5 + log log 8B2 , (37) X then with probability at least 3/4 the source alphabet has no repeating, nonoverlapping substrings { − } of size min 2M 5, L/2 . Combining the two arguments shows that with probability at least 1/2 there are at least A/8 source symbols with both long, duplicate-free heads and tails. Further, each of these heads contains a symbol one within the ﬁrst M/2 bits, and each of these tails contains a symbol one within its ﬁrst M/2 bits. If this event holds, then A2/64 = B/64 of all A2 possible concatenations X X produce unique boundary { − } a a˜ chunks of length at least min 2M 4, L . (cid:0) (cid:1) (cid:0) (cid:1) Finally, since B = A2, we will see at least 1/2 of these possible unique boundary chunks in ex- pectation. Therefore, the expected number of bits needed to store just the boundary chunks is at least { } ≥ { } Ω min 2M, L B Ω min 2M, B1/2 B , assuming (37) i(cid:0)s satisﬁed. (cid:1) Combining these arguments, the rate R of variable-length deduplication is lower bounded as VL ≥ E { } R (C) + I{ ≥ }Ω min 2M, B1/2 B , VL M 5+log log(8B2) (cid:0) (cid:1) E − − The expected number of chunks (C) is lower bounded by 2 MBL/M = 2 MB3/2/M, so that ≥ − { } R 2 MB3/2/M + I{ ≥ }Ω min 2M, B1/2 B/M . VL M 5+log log(8B2) This lower bound is minimized by (cid:44) M 1 log(B) + O(1), 4 which results in the bound ≥ − R Ω(B5/4 log 1(B)) (38) VL ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Fig.

text: Duplicate

text: boundary

text: deduplication

text: chunks

text: arising

text: from

text: the

text: concatenation

text: of

text: different

text: source

text: symbols

text: X

text: X

text: and

text: X

text: X

text: Compare

text: to

text: Fig.

text: in

text: Appendix

text: tail(X

text: head(X

text: tail(X

text: head(X

text: then

text: one

text: of

text: head(X

text: head(X

text: is

text: a

text: substring

text: of

text: the

text: other,

text: and

text: one

text: of

text: tail(X

text: tail(X

text: is

text: a

text: substring

text: of

text: the

text: other.

text: Consider

text: a

text: source

text: symbol

text: X

text: and

text: assume

text: M

text: for

text: now.

text: With

text: probability

text: at

text: least

text: a

text: M

text: it

text: has

text: head(X

text: of

text: length

text: at

text: least

text: min

text: containing

text: a

text: least

text: one

text: symbol

text: in

text: the

text: ﬁrst

text: bits

text: a

text: and

text: it

text: has

text: tail(X

text: of

text: length

text: at

text: least

text: min

text: and

text: containing

text: a

text: least

text: one

text: symbol

text: in

text: the

text: last

text: a

text: X

text: bits.

text: A

text: short

text: calculation

text: shows

text: that

text: this

text: implies

text: that

text: with

text: probability

text: at

text: least

text: the

text: source

text: alphabet

text: has

text: at

text: least

text: symbols

text: with

text: this

text: property.

text: X

text: Moreover,

text: with

text: probability

text: at

text: least

text: the

text: source

text: alphabet

text: has

text: no

text: repeating,

text: nonover-

text: lapping

text: substrings

text: of

text: size

text: This

text: argument

text: is

text: reported

text: with

text: more

text: detail

text: in

text: Appendix

text: In

text: particular,

text: if

text: M

text: log

text: log

text: log

text: log

text: X

text: then

text: with

text: probability

text: at

text: least

text: the

text: source

text: alphabet

text: has

text: no

text: repeating,

text: nonoverlapping

text: substrings

text: of

text: size

text: min

text: Combining

text: the

text: two

text: arguments

text: shows

text: that

text: with

text: probability

text: at

text: least

text: there

text: are

text: at

text: least

text: source

text: symbols

text: with

text: both

text: long,

text: duplicate-free

text: heads

text: and

text: tails.

text: Further,

text: each

text: of

text: these

text: heads

text: contains

text: a

text: symbol

text: one

text: within

text: the

text: ﬁrst

text: bits,

text: and

text: each

text: of

text: these

text: tails

text: contains

text: a

text: symbol

text: one

text: within

text: its

text: ﬁrst

text: bits.

text: If

text: this

text: event

text: holds,

text: then

text: of

text: all

text: possible

text: concatenations

text: X

text: X

text: produce

text: unique

text: boundary

text: a

text: a˜

text: chunks

text: of

text: length

text: at

text: least

text: min

text: L

text: Finally,

text: since

text: B

text: we

text: will

text: see

text: at

text: least

text: of

text: these

text: possible

text: unique

text: boundary

text: chunks

text: in

text: ex-

text: pectation.

text: Therefore,

text: the

text: expected

text: number

text: of

text: bits

text: needed

text: to

text: store

text: just

text: the

text: boundary

text: chunks

text: is

text: at

text: least

text: Ω

text: min

text: L

text: B

text: Ω

text: min

text: B

text: assuming

text: satisﬁed.

text: Combining

text: these

text: arguments,

text: the

text: rate

text: R

text: of

text: variable-length

text: deduplication

text: is

text: lower

text: bounded

text: as

text: VL

text: E

text: R

text: (C)

text: I{

text: }Ω

text: min

text: B

text: VL

text: M

text: E

text: The

text: expected

text: number

text: of

text: chunks

text: (C)

text: is

text: lower

text: bounded

text: by

text: MBL/M

text: so

text: that

text: R

text: I{

text: }Ω

text: min

text: B/M

text: VL

text: M

text: This

text: lower

text: bound

text: is

text: minimized

text: by

text: M

text: log(B)

text: which

text: results

text: in

text: the

text: bound

text: R

text: log

text: VL

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 24 (cid:18) (cid:19) For the seco(cid:88)nd-to-la(cid:88)st te(cid:0)rm in (40)(cid:1), we h(cid:88)ave(cid:16)the upper bou(cid:0)nd (cid:1) (cid:0) (cid:1)(cid:17) B E ≤ B E| C | E E 5 + (cid:96)(Z ) 5 ∂ + (cid:96) head(Y ) + (cid:96) tail(Y ) . (41) c b b b ∈ C b=1 c ∂ b=1 b Here, head(Y ) are the bits from the start of Y forward until the end of the ﬁrst (counting forwards) b ∈ { b} − − − occurrence of the string u10M such that u 0, 1 2M 1 M 1 does not contain 0M. And tail(Y ) are the b bits from the end of Y backwards until the end of the ﬁrst (counting backwards) occurrence of the string b −(cid:0) − (cid:1) 0M plus an additional 2M 1 M bits. If no such substring occurs, head and tail denote all the bits in Y b in either case. (cid:0) (cid:1) The expected value of (cid:96) tail(Y ) is upper bounded as b (cid:0) (cid:1) E ≤ − − − ≤ (cid:96) tail(Y ) 2M+1 2 + 2M 1 M 2M+2 b E as before by [30, Theorem 8.2] (see again Appendix ). The quantity (cid:96) head(Y ) can be upper bounded b by replacing Y with an inﬁnite-length Bernoulli(1/2) process. The head of that Bernoulli process is then b · · · − − a concatenation of sub-chunks of the form U 0MU 0M U 0M with (cid:96)(U ) < 2M 1 M for n < N and ≥ − − 1 2 N n with (cid:96)(U ) 2M 1 M. Consider the sequence of lengths (cid:96)(U ), (cid:96)(U ), . . . . This sequence forms an N 1 2 i.i.d. stochastic process, and N is a stoppi(cid:0)ng time w(cid:1)ith respect to this process. Thus, we can apply Wald’s equation together with [30, Theorem 8.2] to obtain E ≤ E (cid:96) head(Y ) (N)2M+1. b − The random variable N is geometrically di(cid:0)stributed w(cid:1)ith probability of success lower bounded by 1 − · − E ≤ 2M 1 2 M = 1/2. Thus (N) 2 and E ≤ (cid:96) head(Y ) 2M+2. b The same argument also shows that E| C | ≤ E ≤ ∂ (N) + 1 3. b (cid:18) (cid:19) Substituting the last three equa(cid:88)tions in(cid:88)to (4(cid:0)1), we ob(cid:1)tain (cid:0) (cid:1) B E ≤ 5 + (cid:96)(Z ) B 15 + 2M+3 . (42) c ∈ C b=1 c ∂ b E P E For the last term 12B (L) ( ) in (40), we need to upper bound the probability that there is at least one interior chunk in the source alphabet that is either equal to another interior chunk of the source − alphabet or to a boundary chunk of the source sequence. Since all chunks have length at least 2M 1 by construction, whenever this last event holds, then X , X , . . . , X contains a nonoverlapping duplicate − 1 2 A substring of length 2M 2 (where the additional factor 1/2 accounts for the boundary chunks). Consider a source symbol, say X . The probability that it contains a nonoverlapping duplicate substring − 1 E of length 2M 2 is upper bounded by the probability that a Bernoulli(1/2) process of length 2 (L) contains such a substring. Consider next two source symbols, say X and X . Condition on their lengths (cid:96)(X ) 1 2 1 and (cid:96)(X ). If these lengths are distinct, then X and X are independent Bernoulli(1/2) processes of 2 1 2 given length. If the lengths are the same, then X and X are not independent, since they are chosen 1 2 − without replacement. However, the probability of X containing a duplicate substring of length 2M 2 1 from X is upper bounded by drawing them with replacement. Further, in both cases, the probability of 2 the event under consideration is increased if we increase the length of the source symbols. In summary, X (cid:0) (cid:1) − the probability that the source alphabet contains a nonoverlapping duplicate substring of length 2M 2 E is upper bounded by the probability that A independent Bernoulli(1/2) processes of length 2 (L) contain such a duplicate. This probability can in turn be upper bounded by 2AE(L) 22−2M−2. Therefore,

label: equation

text: For

text: the

text: in

text: we

text: upper

text: B

text: E

text: B

text: E|

text: C

text: E

text: E

text: head(Y

text: tail(Y

text: c

text: b

text: b

text: b

text: C

text: c

text: b

text: Here,

text: head(Y

text: are

text: the

text: bits

text: from

text: the

text: start

text: of

text: Y

text: forward

text: until

text: the

text: end

text: of

text: the

text: ﬁrst

text: (counting

text: forwards)

text: b

text: b}

text: occurrence

text: of

text: the

text: string

text: such

text: that

text: u

text: M

text: does

text: not

text: contain

text: And

text: tail(Y

text: are

text: the

text: b

text: bits

text: from

text: the

text: end

text: of

text: Y

text: backwards

text: until

text: the

text: end

text: of

text: the

text: ﬁrst

text: (counting

text: backwards)

text: occurrence

text: of

text: the

text: string

text: b

text: plus

text: an

text: additional

text: M

text: bits.

text: If

text: no

text: such

text: substring

text: occurs,

text: head

text: and

text: tail

text: denote

text: all

text: the

text: bits

text: in

text: Y

text: b

text: in

text: either

text: case.

text: The

text: expected

text: value

text: of

text: tail(Y

text: is

text: upper

text: bounded

text: as

text: b

text: E

text: tail(Y

text: M

text: b

text: E

text: as

text: before

text: by

text: Theorem

text: (see

text: again

text: Appendix

text: The

text: quantity

text: head(Y

text: can

text: be

text: upper

text: bounded

text: b

text: by

text: replacing

text: Y

text: with

text: an

text: inﬁnite-length

text: process.

text: The

text: head

text: of

text: that

text: Bernoulli

text: process

text: is

text: then

text: b

text: a

text: concatenation

text: of

text: sub-chunks

text: of

text: the

text: form

text: U

text: U

text: with

text: M

text: for

text: n

text: N

text: and

text: N

text: n

text: with

text: M.

text: Consider

text: the

text: sequence

text: of

text: lengths

text: This

text: sequence

text: forms

text: an

text: N

text: i.i.d.

text: stochastic

text: process,

text: and

text: N

text: is

text: a

text: time

text: respect

text: to

text: this

text: process.

text: Thus,

text: we

text: can

text: apply

text: Wald’s

text: equation

text: together

text: with

text: Theorem

text: to

text: obtain

text: E

text: E

text: head(Y

text: b

text: The

text: random

text: variable

text: N

text: is

text: geometrically

text: probability

text: of

text: success

text: lower

text: bounded

text: by

text: E

text: M

text: Thus

text: (N)

text: and

text: E

text: head(Y

text: b

text: The

text: same

text: argument

text: also

text: shows

text: that

text: E|

text: C

text: E

text: (N)

text: b

text: Substituting

text: the

text: last

text: three

text: we

text: B

text: E

text: B

text: c

text: C

text: c

text: b

text: E

text: P

text: E

text: For

text: the

text: last

text: term

text: (L)

text: in

text: we

text: need

text: to

text: upper

text: bound

text: the

text: probability

text: that

text: there

text: is

text: at

text: least

text: one

text: interior

text: chunk

text: in

text: the

text: source

text: alphabet

text: that

text: is

text: either

text: equal

text: to

text: another

text: interior

text: chunk

text: of

text: the

text: source

text: alphabet

text: or

text: to

text: a

text: boundary

text: chunk

text: of

text: the

text: source

text: sequence.

text: Since

text: all

text: chunks

text: have

text: length

text: at

text: least

text: by

text: construction,

text: whenever

text: this

text: last

text: event

text: holds,

text: then

text: X

text: X

text: X

text: contains

text: a

text: nonoverlapping

text: duplicate

text: A

text: substring

text: of

text: length

text: (where

text: the

text: additional

text: factor

text: accounts

text: for

text: the

text: boundary

text: chunks).

text: Consider

text: a

text: source

text: symbol,

text: say

text: X

text: The

text: probability

text: that

text: it

text: contains

text: a

text: nonoverlapping

text: duplicate

text: substring

text: E

text: of

text: length

text: is

text: upper

text: bounded

text: by

text: the

text: probability

text: that

text: a

text: process

text: of

text: length

text: (L)

text: contains

text: such

text: a

text: substring.

text: Consider

text: next

text: two

text: source

text: symbols,

text: say

text: X

text: and

text: X

text: Condition

text: on

text: their

text: lengths

text: and

text: If

text: these

text: lengths

text: are

text: distinct,

text: then

text: X

text: and

text: X

text: are

text: independent

text: processes

text: of

text: given

text: length.

text: If

text: the

text: lengths

text: are

text: the

text: same,

text: then

text: X

text: and

text: X

text: are

text: not

text: independent,

text: since

text: they

text: are

text: chosen

text: without

text: replacement.

text: However,

text: the

text: probability

text: of

text: X

text: containing

text: a

text: duplicate

text: substring

text: of

text: length

text: from

text: X

text: is

text: upper

text: bounded

text: by

text: drawing

text: them

text: with

text: replacement.

text: Further,

text: in

text: both

text: cases,

text: the

text: probability

text: of

text: the

text: event

text: under

text: consideration

text: is

text: increased

text: if

text: we

text: increase

text: the

text: length

text: of

text: the

text: source

text: symbols.

text: In

text: summary,

text: X

text: the

text: probability

text: that

text: the

text: source

text: alphabet

text: contains

text: a

text: nonoverlapping

text: duplicate

text: substring

text: of

text: length

text: E

text: is

text: upper

text: bounded

text: by

text: the

text: probability

text: that

text: A

text: independent

text: processes

text: of

text: length

text: (L)

text: contain

text: such

text: a

text: duplicate.

text: This

text: probability

text: can

text: in

text: turn

text: be

text: upper

text: bounded

text: by

text: Therefore,


text: 7 chunks takes on all L possible offsets with respect to the source block boundaries. Due to these L possible starting points, the ﬁxed-length deduplication scheme encounters Θ(L) distinct chunks instead of the only A = 2 distinct source symbols, resulting in the factor Θ(L) = Θ(B) overhead compared to the optimal ♦ scheme. This argument is made precise in Appendix . From Example 5, we see that the (general) deduplication problem is fundamentally one of synchronizing deduplication chunks with source blocks. The variable-length deduplication scheme, described in Section , utilizes marker sequences to achieve this synchronization. The next theorem bounds its performance. Theorem 2. Consider the source model with B source blocks drawn with replacement from the A source E symbols of expected length (L). The performa(cid:0)nce (cid:112)of the (cid:1)varia(cid:0)ble-leng(cid:1)th deduplication scheme with optimized marker length M satisﬁes then (cid:0) (cid:1) E E ≤ R ≤ 5B 1 + (L) log B (L) VL 1 R(cid:63) 1 + min{A, B}(E(L) − 1) + (B − A)+ log(A/2) − 2B log(2E(L)) + , for B large enough. The proof of Theorem 2 is reported in Appendix . We illustrate this result with two examples. Example 4 (Continued). Consider again the scenario with A = 105 source symbols and with B = 106 source blocks. This time, the source symbols are not of constant length, but have the same expected length E (L) = 106 bits as before. Theorem 2 shows then that variable-length deduplication has performance satisfying ≤ ≤ R(cid:63) R 3R(cid:63), VL i.e., is within a factor 3 of optimal. By numerically optimizing the value of the marker length M, this ♦ factor can be further reduced to 1.6. Example 5 (Continued). Consider again the scenario with A = 2 source symbols with symbol-length distribution P assigning equal mass to the values L and L + 1, and with B = 3L source blocks. Recall L that the ﬁxed-length deduplication scheme had a rate at least order B times larger than the optimal scheme: R ≥ FL Ω(B). R(cid:63) On the other hand, by tightening the arguments in the proof of Theorem 2 for the case B > A2 (see Appendix ), the rate of the variable-length deduplication scheme satisﬁes R ≤ VL O(log3 B). R(cid:63) Thus, variable-length deduplication is only suboptimal by at most a polylogarithmic as opposed to a linear ♦ factor. The analysis in the proof of Theorem 2 in Appendix indicates that the optimal choice of the marker length M, governing the expected chunk length of the variable-length deduplication scheme, balances two competing requirements. On the one hand, for each already encountered chunk, we need to encode a pointer into the dictionary. A smaller chunk length increases both the number of chunks that need to be encoded and the size of the pointers. On the other hand, the chunks covering the boundaries of two source blocks will usually not be contained in the dictionary (since there are A2 such possible boundaries (cid:112) for A source symbols), and will have to be encoded directly. Hence, a larger chunk length increases (cid:112) the amount of bits contained in inefﬁciently encodable boundary chunks. The choice of marker length ≈ E E M 0.5 log (L) in the proof of Theorem 2 splits each source block into an average of about (L) E chunks of expected length about (L), which balances these two detrimental effects. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: chunks

text: takes

text: on

text: all

text: L

text: possible

text: offsets

text: with

text: respect

text: to

text: the

text: source

text: block

text: boundaries.

text: Due

text: to

text: these

text: L

text: possible

text: starting

text: points,

text: the

text: ﬁxed-length

text: deduplication

text: scheme

text: encounters

text: Θ(L)

text: distinct

text: chunks

text: instead

text: of

text: the

text: only

text: A

text: distinct

text: source

text: symbols,

text: resulting

text: in

text: the

text: factor

text: Θ(L)

text: Θ(B)

text: overhead

text: compared

text: to

text: the

text: optimal

text: scheme.

text: This

text: argument

text: is

text: made

text: precise

text: in

text: Appendix

text: From

text: Example

text: we

text: see

text: that

text: the

text: (general)

text: deduplication

text: problem

text: is

text: fundamentally

text: one

text: of

text: synchronizing

text: deduplication

text: chunks

text: with

text: source

text: blocks.

text: The

text: variable-length

text: deduplication

text: scheme,

text: described

text: in

text: Section

text: utilizes

text: marker

text: sequences

text: to

text: achieve

text: this

text: synchronization.

text: The

text: next

text: theorem

text: bounds

text: its

text: performance.

text: Theorem

text: Consider

text: the

text: source

text: model

text: with

text: B

text: source

text: blocks

text: drawn

text: with

text: replacement

text: from

text: the

text: A

text: source

text: E

text: symbols

text: of

text: expected

text: length

text: (L).

text: The

text: the

text: deduplication

text: scheme

text: with

text: optimized

text: marker

text: length

text: M

text: satisﬁes

text: then

text: E

text: E

text: R

text: (L)

text: log

text: B

text: (L)

text: VL

text: min{A,

text: B}(E(L)

text: (B

text: A)+

text: for

text: B

text: large

text: enough.

text: The

text: proof

text: of

text: Theorem

text: is

text: reported

text: in

text: Appendix

text: We

text: illustrate

text: this

text: result

text: with

text: two

text: examples.

text: Example

text: (Continued).

text: Consider

text: again

text: the

text: scenario

text: with

text: A

text: source

text: symbols

text: and

text: with

text: B

text: source

text: blocks.

text: This

text: time,

text: the

text: source

text: symbols

text: are

text: not

text: of

text: constant

text: length,

text: but

text: have

text: the

text: same

text: expected

text: length

text: E

text: (L)

text: bits

text: as

text: before.

text: Theorem

text: shows

text: then

text: that

text: variable-length

text: deduplication

text: has

text: performance

text: satisfying

text: R

text: VL

text: i.e.,

text: is

text: within

text: a

text: factor

text: of

text: optimal.

text: By

text: numerically

text: optimizing

text: the

text: value

text: of

text: the

text: marker

text: length

text: M,

text: this

text: factor

text: can

text: be

text: further

text: reduced

text: to

text: Example

text: (Continued).

text: Consider

text: again

text: the

text: scenario

text: with

text: A

text: source

text: symbols

text: with

text: symbol-length

text: distribution

text: P

text: assigning

text: equal

text: mass

text: to

text: the

text: values

text: L

text: and

text: L

text: and

text: with

text: B

text: source

text: blocks.

text: Recall

text: L

text: that

text: the

text: ﬁxed-length

text: deduplication

text: scheme

text: had

text: a

text: rate

text: at

text: least

text: order

text: B

text: times

text: larger

text: than

text: the

text: optimal

text: scheme:

text: R

text: FL

text: Ω(B).

text: On

text: the

text: other

text: hand,

text: by

text: tightening

text: the

text: arguments

text: in

text: the

text: proof

text: of

text: Theorem

text: for

text: the

text: case

text: B

text: (see

text: Appendix

text: the

text: rate

text: of

text: the

text: variable-length

text: deduplication

text: scheme

text: satisﬁes

text: R

text: VL

text: B).

text: Thus,

text: variable-length

text: deduplication

text: is

text: only

text: suboptimal

text: by

text: at

text: most

text: a

text: polylogarithmic

text: as

text: opposed

text: to

text: a

text: linear

text: factor.

text: The

text: analysis

text: in

text: the

text: proof

text: of

text: Theorem

text: in

text: Appendix

text: indicates

text: that

text: the

text: optimal

text: choice

text: of

text: the

text: marker

text: length

text: M,

text: governing

text: the

text: expected

text: chunk

text: length

text: of

text: the

text: variable-length

text: deduplication

text: scheme,

text: balances

text: two

text: competing

text: requirements.

text: On

text: the

text: one

text: hand,

text: for

text: each

text: already

text: encountered

text: chunk,

text: we

text: need

text: to

text: encode

text: a

text: pointer

text: into

text: the

text: dictionary.

text: A

text: smaller

text: chunk

text: length

text: increases

text: both

text: the

text: number

text: of

text: chunks

text: that

text: need

text: to

text: be

text: encoded

text: and

text: the

text: size

text: of

text: the

text: pointers.

text: On

text: the

text: other

text: hand,

text: the

text: chunks

text: covering

text: the

text: boundaries

text: of

text: two

text: source

text: blocks

text: will

text: usually

text: not

text: be

text: contained

text: in

text: the

text: dictionary

text: (since

text: there

text: are

text: such

text: possible

text: boundaries

text: for

text: A

text: source

text: symbols),

text: and

text: will

text: have

text: to

text: be

text: encoded

text: directly.

text: Hence,

text: a

text: larger

text: chunk

text: length

text: increases

text: the

text: amount

text: of

text: bits

text: contained

text: in

text: inefﬁciently

text: encodable

text: boundary

text: chunks.

text: The

text: choice

text: of

text: marker

text: length

text: E

text: E

text: M

text: log

text: (L)

text: in

text: the

text: proof

text: of

text: Theorem

text: splits

text: each

text: source

text: block

text: into

text: an

text: average

text: of

text: about

text: (L)

text: E

text: chunks

text: of

text: expected

text: length

text: about

text: (L),

text: which

text: balances

text: these

text: two

text: detrimental

text: effects.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##