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Submitted on 23 Jul 2012

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Towards the parallelization of Reversible Jump MarkovChain Monte Carlo algorithms for vision problems

Yannick Verdie, Florent Lafarge

To cite this version:Yannick Verdie, Florent Lafarge. Towards the parallelization of Reversible Jump Markov Chain MonteCarlo algorithms for vision problems. [Research Report] RR-8016, INRIA. 2012. <hal-00720005>

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RESEARCH

REPORT

N° 8016July 2012

Project-Teams Geometrica, Ayin

Towards the

parallelization of

Reversible Jump Markov

Chain Monte Carlo

algorithms for vision

problems.

Yannick Verdié , Florent Lafarge

RESEARCH CENTRE

SOPHIA ANTIPOLIS – MÉDITERRANÉE

2004 route des Lucioles - BP 93

06902 Sophia Antipolis Cedex

♦rs t ♣r③t♦♥ ♦ rs ♠♣

r♦ ♥ ♦♥t r♦ ♦rt♠s ♦r

s♦♥ ♣r♦♠s

❨♥♥ ❱ré ∗ ♦r♥t r ∗

Pr♦t♠s ♦♠tr ②♥

sr ♣♦rt ♥ ② ♣s

strt P♦♥t ♣r♦sss ♠♦♥strt ♥② ♥ ♦♠♣tt♥ss ♥ rss♥♦t r♦♥t♦♥ ♣r♦♠s ♥ s♦♥ ♦r s♠t♥ ts ♠t♠t ♠♦s s tts s♣② ♦♥ r s♥s ①st♥ s♠♣rs sr r♦♠ r ♣r♦r♠♥s ♥ tr♠s ♦♦♠♣tt♦♥ t♠ ♥ stt② ❲ ♣r♦♣♦s ♥ s♠♣♥ ♣r♦r s ♦♥ ♦♥t r♦♦r♠s♠ r ♦rt♠ ①♣♦ts r♦♥ ♣r♦♣rts ♦ ♣♦♥t ♣r♦sss t♦ ♣r♦r♠ t s♠♣♥♥ ♣r s ♣r♦r s ♠ ♥t♦ tr♥ ♠♥s♠ s tt t ♣♦♥ts r♥♦♥♥♦r♠② strt ♥ t s♥ ♣r♦r♠♥s ♦ t s♠♣r r ♥②③ tr♦ st♦ ①♣r♠♥ts ♦♥ r♦s ♦t r♦♥t♦♥ ♣r♦♠s r♦♠ r s♥s ♥ tr♦ ♦♠♣rs♦♥st♦ t ①st♥ ♦rt♠s

②♦rs ♠ ♥ s♦♥ t♦st ♦ ♦♥t r♦ ♥r② ♠♥♠③t♦♥ r♦♥♦♠

∗ ♦♣ ♥t♣♦s r♥

❱rs ♣rést♦♥ s ♦rt♠s ♦♥t r♦ ♣r

♥ r♦ à sts rérss ♣♦r s ♣r♦è♠s ♥

s♦♥

és♠é s ♣r♦sss ♣♦♥ts ♠rqés s s♦♥t ♠♦♥trés ①trê♠♠♥t ♣r♦r♠♥ts ♣♦rtrtr s ♣r♦è♠s tt♦♥ ❵♦ts t r♦♥♥ss♥ ♦r♠s ♥ s♦♥ ♣r ♦r♥tr♣♥♥t s♠t♦♥ s ♠♦ès ♠té♠tqs st ♥ t ♥ ♣rtr ♣♦rs sè♥s rs s é♥t♦♥♥rs ①st♥t s♦r♥t ♣r♦r♠♥s ♠♦②♥♥ ♥ tr♠ t♠♣s t stté ♦s ♣r♦♣♦s♦♥s ♥ ♥♦ ♦rt♠ ①♣♦t♥t s ♣r♦♣rétés♠r♦♥♥s s ♣r♦sss ♣♦♥ts ♥ tr ♥ é♥t♦♥♥ ♥ ♣rè tt♠ét♦ st ♥téré à ♥ ♠♥s♠ é ♣r s ♦♥♥és ♥ ❵♦t♥r ♥ stt♦♥ ♥♦♥♥♦r♠ s ♣♦♥ts ♥s sè♥ s ♣r♦r♠♥s ♥♦ é♥t♦♥♥r s♦♥t ♥②sésà trrs ♣srs ①♣ér♥s sr rs sè♥s t ♦♠♣rés ① ♦rt♠s ①st♥ts

♦tsés ♠ t s♦♥ ♠♦è st♦stq ♦♥t r♦ ♠♥♠st♦♥ ❵♥r ♠♣ r♦

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

♥tr♦t♦♥

r♦ ♣♦♥t ♣r♦sss ♦♥sttt ♥ ♦t♦r♥t ①t♥s♦♥ ♦ trt♦♥ r♦ ♥♦♠s ❲rs s rss ♥ ♣r♦♠s ♦♥ stt r♣s r♦ ♣♦♥t ♣r♦sss ♥ t ♦t r♦♥t♦♥ ♣r♦♠s ② rt② ♠♥♣t♥ ♣r♠tr ♥tts ♦♥②♥♠ r♣s s ♣r♦st ♠♦s ♥tr♦ ② ② t ❬❪ ①♣♦t r♥♦♠rs ♦s r③t♦♥s r ♦♥rt♦♥s ♦ ♣r♠tr ♦ts ♦t ♥ ss♥t♦ ♣♦♥t ♣♦st♦♥ ♥ t s♥ ♥♠r ♦ ♦ts s ts r♥♦♠ r ♥ ts♠st ♥♦t st♠t ♦r s♣ ② ♥ sr ♥♦tr str♥t ♦ r♦ ♣♦♥t ♣r♦sss str t② t♦ t ♥t♦ ♦♥t ♦♠♣① s♣t ♥trt♦♥s t♥ t ♦ts ♥ t♦ ♠♣♦s♦ rr③t♦♥ ♦♥str♥ts

P♦♥t ♣r♦sss ♦r s♦♥ ♣r♦♠s

♥② r♥t ♦rs ①♣♦t♥ ♣♦♥t ♣r♦sss ♥ ♣r♦♣♦s t♦ rss r rt② ♦♦♠♣tr s♦♥ ♣r♦♠s ❬ ❪ r♦♥ ♥trst ♥ ts ♣r♦st♠♦s s ♠♦tt ② t ♥ t♦ ♠♥♣t ♣r♠tr ♦ts ♥trt♥ ♥ s♥s Pr♠tr ♦ts ♥ ♥ ♥ srt ♥♦r ♦♥t♥♦s ♦♠♥s ② s② ♦rrs♣♦♥t♦ ♦♠tr ♥tts s♠♥ts rt♥s rs ♦r ♣♥s t ♥ ♠♦r ♥r② ♥②t②♣ ♦ ♠t♠♥s♦♥ ♥t♦♥ ♣♦♥t ♣r♦ss rqrs t ♦r♠t♦♥ ♦ ♣r♦t②♥st② ♥ ♦rr t♦ ♠sr t qt② ♦ ♦♥rt♦♥ ♦ ♦ts ♥st② s t②♣②♥ s ♦♠♥t♦♥ ♦ tr♠ ssss♥ t ♦♥sst♥② ♦ ♦ts t♦ t t ♥ tr♠t♥ ♥t♦ ♦♥t s♣t ♥trt♦♥s t♥ ♦ts ♥ r♦♥ ♦♥t①t ♦♣t♠♦♥rt♦♥ ♠①♠③♥ ts ♥st② s s② sr ♦r ② ♦♥t r♦ s s♠♣r♣ ♦ ①♣♦r♥ t ♦ ♦♥rt♦♥ s♣ ♥ ♠♦st ss r♦ ♥ ♦♥t r♦ ♦rt♠ ❬ ❪

s♦♠s t ❬❪ ♣r♦♣♦s ♣♦♥t ♣r♦ss ♦r ♦♥t♥ ♣♦♣t♦♥s r♦♠ r ♠s ♥tt② ♥ ♣tr ② ♥ ♣s t ❬❪ ♣rs♥t ♣♦♥t ♣r♦ss ♦r s♠r ♣♣t♦♥t t t♦ r♦ tt♦♥ r♦♠ r♦♥s ♣♦t♦s ♦r ♦ts r ♥ s st ♦ ♦② s♣ t♠♣ts r♥ r♦♠ tr♥♥ t t ♠s r s ② ❯tst ❬❪ t♦ tt ♣♦♣ ② ♣♦♥t ♣r♦ss ♥ r t ♦ts r s♣ ② ②♥rs♥ t ❬❪ ♥ ♦st t ❬❪ ♣r♦♣♦s ♣♦♥t ♣r♦sss ♦r ①trt♥ ♥ ♥t♦rs r♦♠♠s ② t♥ ♥t♦ ♦♥t s♣t ♥trt♦♥s t♥ ♥s t♦ ♦r t ♦t ♦♥♥①♦♥r t ❬❪ ♣rs♥t ♥r ♠♦ ♦r ①trt♥ r♥t t②♣s ♦ ♦♠tr trsr♦♠ ♠s ♥♥ ♥ rt♥s ♦r ss ♠①tr ♦ ♦t ♥trt♦♥s r s♦ ♦♥sr s tt t ♣r♦ss ♥ rss r♥t ♣r♦♠s r♥♥ r♦♠ ♣♦♣t♦♥ ♦♥t♥t♦ ♥ ♥t♦r ①trt♦♥ tr♦ t♦ t①tr r♦♥t♦♥ s♦t ❬❪ ♦♣s ♣♦♥t ♣r♦ss♦r tr♥ rt♥r ♦ts r♦♠ ♦ ♠♦♥♦♠♥s♦♥ ♣♦♥t ♣r♦ss s ♣r♦♣♦s ②t t ❬❪ ♦r ♠♦♥ s♥s ② ♠①trs ♦ ♣r♠tr ♥t♦♥s ♠♣♦s♥♣②s ♦♥str♥ts t♥ t ♠♦s

♦tt♦♥s

rsts ♦t♥ ② ts ♣♦♥t ♣r♦sss r ♣rtr② ♦♥♥♥ ♥ ♦♠♣tt tt ♣r♦r♠♥s r♠♥ ♠t ♥ tr♠s ♦ ♦♠♣tt♦♥ t♠ ♥ ♦♥r♥ stt② s♣② ♦♥ r s♥s s rs ①♣♥ ② ♥str② s ♥ rt♥t ♥t ♥♦ t♦♥trt ts ♠t♠t ♠♦s ♥ tr ♣r♦ts ♥ t ♣♦♥t ♣r♦sss ♣rs♥t ♥t♦♥ ♠♣s③ ♦♠♣① ♠♦ ♦r♠t♦♥s ② ♣r♦♣♦s♥ ♦ts ♣r♠tr② s♦♣stt ❬ ❪ ♥ t♥qs t♦ t ♦ts t♦ t t ❬❪ ♥ ♥♦♥tr s♣t ♥trt♦♥s

♥

❨♥♥ ❱ré ♦r♥t r

t♥ ♦ts ❬ ❪ ♦r ts ♦rs ♦♥② st② rss t ♦♣t♠③t♦♥ sssr♦♠ s ♦♠♣① ♠♦s

♠♣s♦♥ ♦rt♠s ❬ ❪ ♥ s♥ t♦ s♣♣ t s♠♣♥② ♥srt♥ s♦♥ ②♥♠s ♦r t ①♣♦rt♦♥ ♦ t ♦♥t♥♦s ss♣s s s♠♣rsr ♥♦rt♥t② rstrt t♦ s♣ ♥st② ♦r♠s t ♦♥srt♦♥s s♦ ♥ st♦ r t s♠♣♥ t ♠♦r ♥② ❬❪ ♦r ♠ s♠♥tt♦♥ ♣r♦♠s ♦♠ ♦rs s♦ ♣r♦♣♦s ♣r③t♦♥ ♣r♦rs ② s♥ ♠t♣ ♥s s♠t♥♦s② ❬❪ ♦r♦♠♣♦st♦♥ s♠s ♥ ♦♥rt♦♥s s♣s ♦ ① ♠♥s♦♥ ❬ ❪ ♦r t② r♠t ② ♦rr ts ♥ r ♥♦t s♥ t♦ ♣r♦r♠ ♦♥ r s♥s ♥ t♦♥ t①st♥ ♦♠♣♦st♦♥ s♠s ♥♥♦t s ♦r ♦♥rt♦♥ s♣s ♦ r ♠♥s♦♥ ♠♥s♠ s ♦♥ ♠t♣ rt♦♥ ♥ strt♦♥ ♦ ♦ts s ♥ s♦ ♦♣♦r rss♥ ♣♦♣t♦♥ ♦♥t♥ ♣r♦♠s ❬ ❪ rtss ♦t rt♦♥s rqr tsrt③t♦♥ ♦ t ♣♦♥t ♦♦r♥ts ♥s s♥♥t ♦ss ♦ r②

s tr♥t rs♦♥s ♦ t ♦♥♥t♦♥ s♠♣r ♦② ♦ t ♠♣r♦♠♥t♦ ♦♣t♠③t♦♥ ♣r♦r♠♥s ♥ s♣ ♦♥t①tst s t ♥s ♥ tr♠s ♦ ♦♠♣tt♦♥t♠ r♠♥ ♥ r s② r③ t t ①♣♥s ♦ s♦t♦♥ stt② ♥♥ st♥t s♠♣r ♦r ♥r r♦ ♣♦♥t ♣r♦sss r② r♣rs♥ts ♥♥ ♣r♦♠

♦♥trt♦♥s

❲ ♣rs♥t s♦t♦♥s t♦ rss ts ♣r♦♠ ♥ t♦ rst② r ♦♠♣tt♦♥ t♠s r♥t②♥ qt② ♥ stt② ♦ t s♦t♦♥ r ♦rt♠ ♣rs♥ts sr ♠♣♦rt♥t♦♥trt♦♥s t♦ t

❼ ♠♣♥ ♥ ♣r ♦♥trr② t♦ t ♦♥♥t♦♥ s♠♣r ♠s ts♦t♦♥ ♦ ② sss ♣rtrt♦♥s ♦r ♦rt♠ ♥ ♣r♦r♠ r ♥♠r♦ ♣rtrt♦♥s s♠t♥♦s② s♥ ♥q ♥ r♦♥ ♣r♦♣rt② ♦ ♣♦♥t♣r♦sss s ①♣♦t t♦ ♠ t ♦ s♠♣♥ ♣r♦♠ s♣t② ♥♣♥♥t ♥ ♥♦r♦♦

❼ ♦♥ ♥♦r♠ ♣♦♥t strt♦♥s P♦♥t ♣r♦sss ♠♥② s ♥♦r♠ ♣♦♥t strt♦♥s r ♦♠♣tt♦♥② s② t♦ s♠t t ♠ t s♠♣♥ ①tr♠② s♦❲ ♣r♦♣♦s ♥ ♥t ♠♥s♠ ♦♥ t ♠♦t♦♥s rt♦♥s ♦r r♠♦s ♦♦ts ② t♥ ♥t♦ ♦♥t ♥♦r♠t♦♥ ♦♥ t ♦sr s♥s ♦♥trr② t♦ t tr♥ s♦t♦♥s ♣r♦♣♦s ② ❬❪ ♥ ❬❪ ♦r ♥♦♥♥♦r♠ strt♦♥ s ♥♦t t rt②r♦♠ ♠ ♦♦ t s rt s♣♣rtt♦♥♥ trs t♦ ♥sr t s♠♣♥♣r③t♦♥

❼ ♥t P❯ ♠♣♠♥tt♦♥ ❲ ♣r♦♣♦s ♥ ♠♣♠♥tt♦♥ ♦♥ P❯ s♥♥t② rs ♦♠♣t♥ t♠s t rs♣t t♦ ①st♥ ♦rt♠s ♥rs♥stt② ♥ ♠♣r♦♥ t qt② ♦ t ♦t♥ s♦t♦♥

❼ ♦t tt♦♥ ♠♦ r♦♠ ♣♦♥t ♦ ♦ t t ♦rt♠ ♥ ♣r♦♣♦s ♥ ♦r♥ ♣♦♥t ♣r♦ss t♦ tt ♣r♠tr ♦ts r♦♠ ♣♦♥t ♦ss ♠♦ s ♣♣ t♦ tr r♦♥t♦♥ r♦♠ sr s♥s ♦ r r♥ ♥ ♥tr♥r♦♥♠♥ts ♦ ♦r ♥♦ t s t rst ♣♦♥t ♣r♦ss s♠♣r t♦ t t♦ ♣r♦r♠♥ s ② ♦♠♣① stt s♣s

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

P♦♥t Pr♦ss r♦♥

♣♦♥t ♣r♦ss srs r♥♦♠ ♦♥rt♦♥s ♦ ♣♦♥ts ♥ ♦♥t♥♦s ♦♥ st K t♠t② s♣♥ ♣♦♥t ♣r♦ss Z s ♠sr ♠♣♣♥ r♦♠ ♣r♦t② s♣ (Ω,A,P)t♦ t st ♦ ♦♥rt♦♥s ♦ ♣♦♥ts ♥ K s tt

∀ω ∈ Ω, pi ∈ K,Z(ω) = p1, ..., pn(ω)

r n(ω) s t ♥♠r ♦ ♣♦♥ts ss♦t t t ♥t ω ❲ ♥♦t ② P t s♣ ♦♦♥rt♦♥s ♦ ♣♦♥ts ♥K ♠♦st ♥tr ♣♦♥t ♣r♦ss s t ♦♠♦♥♦s P♦ss♦♥ ♣r♦ss♦r t ♥♠r ♦ ♣♦♥ts ♦♦s srt P♦ss♦♥ strt♦♥ rs t ♣♦st♦♥ ♦ t♣♦♥ts s ♥♦r♠② ♥ ♥♣♥♥t② strt ♥ K P♦♥t ♣r♦sss ♥ s♦ ♣r♦ ♠♦r♦♠♣① r③t♦♥s ♦ ♣♦♥ts ② ♥ s♣ ② ♥st② h(.) ♥ ♥ P ♥ rr♥♠sr µ(.) ♥r t ♦♥t♦♥ tt t ♥♦r♠③t♦♥ ♦♥st♥t ♦ h(.) s ♥t

∫

p∈P

h(p)dµ(p) < ∞

♠sr µ(.) ♥ t ♥st② h(.) s s② ♥ t ♥t♥st② ♠sr ν(.) ♦ ♥♦♠♦♥♦s P♦ss♦♥ ♣r♦ss ♣②♥ ♥st② h(.) ♦s t ♥srt♦♥ ♦ t ♦♥sst♥②♥ s♦ t rt♦♥ ♦ s♣t ♥trt♦♥s t♥ t ♣♦♥ts ♥ ♣rtr t r♦♥♣r♦♣rt② ♥ s ♥ ♣♦♥t ♣r♦sss s♠r② t♦ r♥♦♠ s t♦ rt s♣t ♥♣♥♥ ♦ t ♣♦♥ts ♥ ♥♦r♦♦ ♦t s♦ tt h(.) ♥ ①♣rss ② s ♥r②U(.) s tt

h(.) ∝ exp−U(.)

r♦♠ ♣♦♥ts t♦ ♣r♠tr ♦ts ❲t ♠s ♣♦♥t ♣r♦sss ttrt ♦r s♦♥ s t♣♦sst② ♦ ♠r♥ ♣♦♥t pi ② t♦♥ ♣r♠trs mi s tt t ♣♦♥t ♦♠sss♦t t ♥ ♦t xi = (pi,mi) ❲ ♥♦t ② C t ♦rrs♣♦♥♥ s♣ ♦ ♦t♦♥rt♦♥s r ♦♥rt♦♥ s ♥ ② x = x1, ..., xn(x) ♦r ①♠♣ ♣♦♥t♣r♦ss ♦♥ K ×M t K ⊂ R

2 ♥ t t♦♥ ♣r♠tr s♣ M =]− π2 ,

π2 ]× [lmin, lmax]

♥ s♥ s r♥♦♠ ♦♥rt♦♥s ♦ ♥s♠♥ts s♥ ♥ ♦r♥tt♦♥ ♥ ♥t r t♦ ♣♦♥t s ♣♦♥t ♣r♦sss r s♦ ♠r ♣♦♥t ♣r♦sss ♥t trtr

♠♦st ♣♦♣r ♠② ♦ ♣♦♥t ♣r♦sss ♦rrs♣♦♥s t♦ t r♦ ♣♦♥t ♣r♦sss ♦ ♦tss♣ ② s ♥rs ♦♥ C ♦ t ♦r♠

∀x ∈ C, U(x) =∑

xi∈x

D(xi) +∑

xi∼xj

V (xi, xj)

r ∼ ♥♦ts t s②♠♠tr ♥♦r♦♦ rt♦♥s♣ ♦ t r♦ ♣♦♥t ♣r♦ss D(xi)s ♥tr② t tr♠ ♠sr♥ t qt② ♦ ♦t xi t rs♣t t♦ t ♥ V (xi, xj) ♣rs ♥trt♦♥ tr♠ t♥ t♦ ♥♦r♥ ♦ts xi ♥ xj ∼rt♦♥s♣ ss② ♥ ♠t st♥ ǫ t♥ ♣♦♥ts s tt

xi ∼ xj = (xi, xj) ∈ x2 : i > j, ||pi − pj ||2 < ǫ

♥ t sq ♦♥sr r♦ ♣♦♥t ♣r♦sss ♦ ts ♦r♠ ♦t tt ts ♥r② ♦r♠s s♠rts t t st♥r ♥ ♥rs ♦r s s t ♥ ♣♣♥① ♦r♣r♦♠ ♥ s♥ s ♥r③t♦♥ ♦ ts ♠♦s

♥

❨♥♥ ❱ré ♦r♥t r

r r♦♠ t t♦ rt r③t♦♥s ♦ ♣♦♥t ♣r♦ss ♥ ♦ r♦ ♣♦♥t ♣r♦ss ♥♦ r♦ ♣♦♥t ♣r♦ss ♦ ♥s♠♥ts r② s ♥s r♣rs♥t t ♣rs ♦ ♣♦♥ts♥trt♥ t rs♣t t♦ t ♥♦r♥ rt♦♥s♣ s s♣ r ② ♠t st♥ǫ t♥ t♦ ♣♦♥ts

♠t♦♥ P♦♥t ♣r♦sss r s② s♠t ② rs ♠♣ s♠♣r ❬❪t♦ sr ♦r t ♦♥rt♦♥ ♠♥♠③s t ♥r② U s s♠♣r ♦♥ssts ♦ s♠t♥ srt r♦ ♥ (Xt)t∈N ♦♥ t ♦♥rt♦♥ s♣ C ♦♥r♥ t♦rs ♥ ♥r♥t♠sr s♣ ② U t trt♦♥ t rr♥t ♦♥rt♦♥ x ♦ t ♥ s ♦②♣rtr t♦ ♦♥rt♦♥ y ♦r♥ t♦ ♥st② ♥t♦♥ Q(x → .) s♦ r♥ ♣rtrt♦♥s r ♦ ♠♥s tt x ♥ y r ♦s ♥ r ② ♥♦ ♠♦r t♥ ♦♥♦t ♦♥rt♦♥ y s t♥ ♣t s ♥ stt ♦ t ♥ t rt♥ ♣r♦t②♣♥♥ ♦♥ t ♥r② rt♦♥ t♥ x ♥ y ♥ r①t♦♥ ♣r♠tr Tt r♥Q ♥ ♦r♠t s ♠①tr ♦ sr♥s Qm ♦s♥ t ♣r♦t② qm s tt

Q(x → .) =∑

m

qmQm(x → .)

sr♥ s s② t t♦ s♣ t②♣s ♦ ♠♦s s t rt♦♥r♠♦ ♦ ♥♦t rt ♥ t r♥ ♦r t ♠♦t♦♥ ♦ ♣r♠trs ♦ ♥ ♦t tr♥st♦♥tt♦♥ ♦r r♦tt♦♥ r♥s r♥ ♠①tr ♠st ♦ ♥② ♦♥rt♦♥ ♥ C t♦ r r♦♠ ♥② ♦tr ♦♥rt♦♥ ♥ ♥t ♥♠r ♦ ♣rtrt♦♥s rrtt② ♦♥t♦♥♦ t r♦ ♥ ♥ sr♥ s t♦ rrs t♦ ♣r♦♣♦s t ♥rs♣rtrt♦♥

s♠♣r s ♦♥tr♦ ② t r①t♦♥ ♣r♠tr Tt t t♠♣rtr♣♥♥ ♦♥ t♠ t ♥ ♣♣r♦♥ ③r♦ s t t♥s t♦ ♥♥t② t♦ ♦rt♠ rs♦ Tt s ♥ssr② t♦ ♥sr t ♦♥r♥ t♦ t ♦ ♠♥♠♠ r♦♠ ♥② ♥t ♦♥rt♦♥♦♥ ss str ♦♠tr rs s ♥ ♣♣r♦①♠t s♦t♦♥ ♦s t♦ t ♦♣t♠♠❬❪

s♠♣♥ ♣r♦r

♠t♥♦s ♠t♣ ♣rtrt♦♥s

♦♥♥t♦♥ s♠♣r ♣r♦r♠s sss ♣rtrt♦♥s ♦♥ ♦ts ♣r♦r s ♦♦s② ♦♥ ♥ st♦s s♣② ♦r r s ♣r♦♠s ♥tr t st

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

♦rt♠ s♠♣r ❬❪

♥t③ X0 = x0 ♥ T0 t t = 0 t trt♦♥ t t Xt = x

❼ ♦♦s sr♥ Qm ♦r♥ t♦ ♣r♦t② qm

❼ Prtr x t♦ y ♦r♥ t♦ Qm(x → .)

❼ ♦♠♣t t r♥ rt♦

R =Qm(y → x)

Qm(x → y)exp

(

U(x)− U(y)

Tt

)

❼ ♦♦s Xt+1 = y t ♣r♦t② min(1, R) ♥ Xt+1 = x ♦trs

♥①♣♦r ♦r r♦ ♣♦♥t ♣r♦sss ♦♥ssts ♥ s♠♣♥ ♦ts ♥ ♣r ② ①♣♦t♥ tr♦♥t♦♥ ♥♣♥♥ ♦ts t ♦ ♥♦r♦♦ strt② ♠♣s ♣rtt♦♥♥t s♣ K s♦ tt s♠t♥♦s ♣rtrt♦♥s r ♣r♦r♠ t ♦t♦♥s r ♥♦ ♣rt t♦♥♦t ♥trr ♥ r t ♦♥r♥ ♣r♦♣rts

r♦♠ sq♥t t♦ ♣r s♠♣♥ t (Xt)t∈N r♦ ♥ s♠t♥ r♦♣♦♥t ♣r♦ss ♥ K s♥ ②♥♠s ♥ cs ♣rtt♦♥ ♦ t s♣ K r ♦♠♣♦♥♥t cs s ♦ s c1 ♥ c2 r s ♥♣♥♥t ♦♥ X t tr♥st♦♥♣r♦t② ♦r ♥② r♥♦♠ ♣rtrt♦♥ ♥ ♥ c1 ♥ t ♥② t♠ t ♦s ♥♦t ♣♥ ♦♥ tr♦ts ♦r ♣rtrt♦♥s ♥ ♥ c2 ♥ rs

❲ ♥ ♠♦♥strt tt t tr♥st♦♥ ♣r♦t② ♦ t♦ sss ♣rtrt♦♥s ♥ ♥♥♣♥♥t s ♥r t t♠♣rtr Tt s q t♦ t ♣r♦t ♦ t tr♥st♦♥ ♣r♦ts ♦ ♣rtrt♦♥ ♥r t s♠ t♠♣rtr ♥ ♦tr ♦rs r③♥ t♦ sss♣rtrt♦♥s ♦♥ ♥♣♥♥t s t t s♠ t♠♣rtr s q♥t t♦ ♣r♦r♠♥ t♠ ♥♣r

t x r③t♦♥ ♦ t ♣♦♥t ♣r♦ss s tt x = (x1, x2, u) r x1 rs♣t②x2 r♣rs♥ts t st ♦ ♣♦♥ts ♥ ♥ t c1 rs♣t② c2 ♥ u s t r♠♥♥st ♦ ♣♦♥ts ♥ ♥ K − c1, c2 t y ♥ ♦♥rt♦♥ ♦ ♣♦♥ts ♦t♥ r♦♠ x

② t♦ ♣rtrt♦♥s ♦♥ t s c1 ♥ c2 s♦ tt y = (y1, y2, u) ♣r♦t②Pr[Xt+2 = y|Xt = x] ♥ ①♣rss s

Pr[Xt+2 = y|Xt = x] =

Pr[Xt+2 = (y1, y2, u)|Xt+1 = (y1, x2, u)]× Pr[Xt+1 = (y1, x2, u)|Xt = x]+Pr[Xt+2 = (y1, y2, u)|Xt+1 = (x1, y2, u)]× Pr[Xt+1 = (x1, y2, u)|Xt = x]

♥

❨♥♥ ❱ré ♦r♥t r

r strt♦♥ ♦ t q♥ t♥ t♦ sss ♣rtrt♦♥s ♦♥ ♥♣♥♥ts c1 ♥ c2 ♥ t♦ s♠t♥♦s ♣rtrt♦♥s ♦♥

♦r s t t♠♣rtr ♣r♠tr s ♦♥st♥t t♥ t ♥ t+ 2

Pr[Xt+2 = y|Xt+1] = (y1, x2, u)

= Q((y1, x2, u) → y)×min[

1, Q(y→(y1,x2,u))Q((y1,x2,u)→y) exp

(

U((y1,x2,u))−U(y)Tt+1

)]

= Q((y1, x2, u) → y)×min[

1, Q(y→(y1,x2,u))Q((y1,x2,u)→y) exp

(

U((y1,x2,u))−U(y)Tt

)]

= Pr[Xt+1 = y|Xt = (y1, x2, u)]

s c1 ♥ c2 ♥ ♥♣♥♥t t tr♥st♦♥ ♣r♦t② ♦r t ♣rtrt♦♥ y2 ♥♥ c2 ♦s ♥♦t ♣♥ ② ♥t♦♥ ♦♥ x1 ♥ y1 s ♥ ♣rtr

Pr[Xt+1 = (y1, y2, u)|Xt = (y1, x2, u)] = Pr[Xt+1 = (x1, y2, u)|Xt = (x1, x2, u)]

♦t♥

Pr[Xt+2 = y|Xt+1 = (y1, x2, u)] = Pr[Xt+1 = (x1, y2, u)|Xt = x]

♠r② ♥ ♠♦♥strt tt

Pr[Xt+2 = y|Xt+1 = (x1, y2, u)] = Pr[Xt+1 = (y1, x2, u)|Xt = x]

♥② ② ♥srt♥ q ♥ ♥ q ♦t♥ t ①♣t rst

Pr[Xt+2 = y|Xt = x] =

2! Pr[Xt+1 = (y1, x2, u)|Xt = x]× Pr[Xt+1 = (x1, y2, u)|Xt = x]

r 2! s t ♦♠♥t♦r ♦♥t ♦rrs♣♦♥♥ t♦ t ♥♠r ♦ ♣r♠tt♦♥s ♦ ♣rtrt♦♥s ♥ t sq♥t ♥

♥sr♥ ♥♣♥♥ ♦ ♥♣♥♥t s ♠st ♦t t ♠♥♠♠ st♥ r♦♠ ♦tr s strt ♥ ts st♥ ♠st t ♥t♦ ♦♥t ♦t t

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

r ♥♣♥♥ ♦ s ♥ t t s t t ♦ t c2 s ♥♦t r ♥♦t♦ ♥sr t ♥♣♥♥ ♦ t s c1 ♥ c3 t t♦ r② ♣♦♥ts ♥ c1 ♥ c3 ♥♥♦t ♣rtr t t s♠ t♠ ♥ t rt s t s c1 ♥ c3 r ♥♣♥♥t s q ssts

t ♦ t ♥♦r♥ rt♦♥s♣ ǫ ♥ t ♥t ♦ t st ♠♦ ♦ s ♦t♣rtrt♦♥ ♥♦t ② δmax ♥♣♥♥ t♥ t♦ s cs ♥ cs′ s t♥ r♥t

minp∈cs, p′∈cs′

||p− p′||2 ≥ ǫ+ 2δmax

♥♣♥♥t tr♥ ♥tr ♦♥ssts ♦ ♣rtt♦♥♥ t s♣ K ♥t♦ rr ♠♦s ♦ s t s③ rtr t♥ ♦r q t♦ ǫ + 2δmax s ♥ t♥ rr♦♣ ♥t♦ 2dimK sts s tt s ♥t t♦ s ♦♥♥ t♦ r♥t sts strts t rr♦♣♥ s♠ ♦r dimK = 2 ♥ dimK = 3 s rr♦♣♥ s♠♥srs t ♠t ♥♣♥♥ t♥ t s ♦ s♠ st ♥ t sq s st s ♠st st ♦ t② ♥♣♥♥t s

♦♥♥♦r♠ ♣♦♥t strt♦♥s

♠♣♥ ♦ts ♥ ♣r rr ♣rtt♦♥♥ s strt ♦♥ & s ♦r ♥♦t ♦♣t♠ s t s♣t ♣♦♥t strt♦♥ s ♥ssr② ♥♦r♠ ♥ ♦s ♥♦t t♥t♦ ♦♥t t rtrsts ♦ ♦sr s♥s ♦ ♦r♦♠ ts ♣r♦♠ ♥♦♥rr♣rtt♦♥♥ ♦ t s♥ s rt ② ①♣♦t♥ ts ♥♦

♦♥♥♦r♠ r♥ r♦♠ ♥♦r♠ sr♥s ①trs ♦ sr♥s r rq♥t②s t♦ s♠t ♣♦♥t ♣r♦sss ② ②♥♠s sr♥ ♦rrs♣♦♥♥ t♦ ♣rtrt♦♥ t②♣ rt ♥ t tr♥st♦♥ r♦tt♦♥ t ♦r t ♦♥sst♥ ♦♠t♥ sr♥s t s♣t rstrt♦♥s t♦ rt ♥♦♥♥♦r♠ ♣♦♥t strt♦♥s s♥♦t ♥ ①♣♦t ♥ t trtr t cs

(1), .., cs(L) L ♣rtt♦♥s ♦ t s♣ K s

tt cs(i) s s ♣rtt♦♥ ♦ cs

(i−1) L ♣rtt♦♥s ♥ s♣♣rtt♦♥♥tr ♥♦t K ♥ ♦s s ♦rrs♣♦♥ t♦ ♣rtt♦♥ ♦ K ② ss♦t♥ t ♦♥t♥ ♥ K ♥♦r♠ sr♥ s♣t② rstrt t♦ t ss♣ s♣♣♦rt♥ ts ♥♦♥♥♦r♠ r♥ ♥ rt ② ♠t♦♥ s ♥ ♥ q ♥ strt ♥

tr♥ s♣♣rtt♦♥♥ tr rr 1t♦2dimK rr ss♦♥ s♠s ♦♥sr t♦ ♣rtt♦♥♥ tr t②♣② qtr ♥ ♠♥s♦♥ t♦ ♥ ♥ ♦tr ♥

♥

❨♥♥ ❱ré ♦r♥t r

r r ♣rtt♦♥♥ s♠ ♦ K ♥ ♠♥s♦♥ t♦ t t s r sqrs rr♦♣ ♥t♦ 4 ♠sts ②♦ r ♥ r♥ s ♥t t♦ s ♦♥♥ t♦r♥t ♠sts ♥ ♠♥s♦♥ tr rt t s r s rr♦♣ ♥t♦ 8 ♠sts

♠♥s♦♥ tr ss♦♥ ♦ t s s r♥ ② t t ❲ ss♠ tt ss ♦♥trst ♥ K ♥ t ♦ts ♣r♦t② t♦ ♦♥ t♦ ♥ r♦② st♥s r♦♠ t t ①trt♦♥ ♦ s ss s ♥♦t rss r ♥ s s♣♣♦s t♦ ♦♥ ② s♠♥tt♦♥ ♦rt♠ ♦ t trtr ♣t t♦ t ♦♥sr ♣♣t♦♥ t ♥ ♦ t tr s ♥t♦ 2dimK s t t ♥①t t ♦r♣s t t ♥ss ♦ ♥trst rr ♦♠♣♦st♦♥ s st♦♣♣ ♥ t ♠♥♠ s③ ♦ t ♦♠s ♥r♦r t♦ ǫ+2δmax ♥ t ♥♣♥♥ ♦♥t♦♥ q s ♥♦t ♦♥r

♣rtt♦♥♥ tr ♦s t rt♦♥ ♦ ♥♦♥ ♥♦r♠ ♣♦♥t strt♦♥ ♥tr② ♥♥t② ♥ t ♥st② ♣r♦rss② rss ♥ ♠♦♥ r r♦♠ t ss ♦ ♥trsts s♦♥ ♥ ♥ ♥sr t♦ ♥♦♥♥ ♣♦♥t strt♦♥ s ts ♥♦t rt②t ♥ t ss ♦ ♥trst s ♥rt② ①trt

r♥ ♦r♠t♦♥ ♥ s♣♣rtt♦♥♥ tr K ♦♠♣♦s ♦ L s ♥ 2dimK

♠sts ♦r ♥ ♦r♠t ♥r r♥ Q s ♠①tr ♦ ♥♦r♠ sr♥sQc,t sr♥ ♥ ♥ ♦♥ t c ♦ K ② t ♣rtrt♦♥ t②♣ t ∈ T s tt

∀x ∈ C, Q(x → .) =∑

c∈K

∑

t∈T

qc,t Qc,t(x → .)

r qc,t s t ♣r♦t② ♦ ♦♦s♥ sr♥ Qc,t(x → .) ♥ ②

qc,t =Pr(t)

#s ♥ K

♦r t②♣s ♦ r♥s r s② ♦♥sr ♥ ♣rt s♦ tt T = rt ♥ ttr♥st♦♥ r♦tt♦♥ s st♥ r♥ ♥ s♦ s ♥ ♦ts sr♣♦ss t②♣s ♦t tt t r♥ Q s rrs s s♠ ♦ rrs sr♥s ♦t s♦

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

r ♣ ♣rtt♦♥♥ tr ♥ ♠♥s♦♥ t♦ ss ♦ ♥trst r s st♠tr♦♠ ♥ ♥♣t ♠ qtr s rt s♦ tt t s r rrs② ♣rtt♦♥♦r♥ t♦ t ss ♦ ♥trst s ♦♠♣♦s ♦ ♦r ♠sts ②♦ r ♥r♥ sts ♦ s t♦ r♥t② t s♠♣♥ ♣r③t♦♥ ♠t♦♥ t tr♥t s s♥s ♥♦♥ ♥♦r♠ strt♦♥ ♦t ♦ t strt♦♥ ♥tr② srst ss ♦ ♥trst ② ♣r♦rss② rs♥ t ♥st② ♥ ♠♦♥ ②

tt ts r♥ ♦s s t♦ st t ♦ ♦♥rt♦♥ s♣ C s♥ t s r♥t ② tsr♥s ♦ t ♦rsst ♦ K

♠♣r ♦r♠t♦♥

r♥ ♥ ♥ q s ♠ ♥t♦ t ②♥♠s s♦ tt t ♥ s♠♣r♦s t ss♦t♦♥ ♦ ♠t♣ ♣rtrt♦♥s ♣r♦r♠ ♥ ♣r t r♥t ♥♦♥♥♦r♠♣♦♥t strt♦♥s s ♦♥ tr♥ s♣ ♣rtt♦♥♥ trs ♦rt♠ ts t s♠♣♥ ♣r♦r

♦rt♠ r ♣r s♠♣r

♥t③ X0 = x0 ♥ T0 t t = 0♦♠♣t s♣♣rtt♦♥♥ tr Kt trt♦♥ t t Xt = x

❼ ♦♦s ♠st Smic ∈ K ♥ r♥ t②♣ t ∈ T ♦r♥ t♦ ♣r♦t②∑

c∈Smic

qc,t

❼ ♦r c ∈ Smic

Prtr x ♥ t c t♦ ♦♥rt♦♥ y ♦r♥ t♦ Qc,t(x → .)

t t r♥ rt♦

R =Qc,t(y → x)

Qc,t(x → y)exp

(

U(x)− U(y)

Tt

)

♦♦s Xt+1 = y t ♣r♦t② min(1, R) ♥ Xt+1 = x ♦trs

♦t tt t t♠♣rtr ♣r♠tr s ♣t tr srs ♦ s♠t♥♦s ♣rtrt♦♥ss tt t t♠♣rtr rs s q♥t t♦ ♦♦♥ s ② ♣t ♥ st♥r

♥

❨♥♥ ❱ré ♦r♥t r

sq♥t s♠♣♥ ♦t s♦ tt t rr ♣rtt♦♥♥ ♦ K ♣r♦tts t s♠♣ r♦♠ ♠♦s ts ♥ ♣rt t s♠♣♥ s st♦♣♣ ♥ ♥♦ ♣rtrt♦♥ s ♥♣t r♥ rt♥ ♥♠r ♦ trt♦♥s

①♣r♠♥ts

♣r♦♣♦s ♦rt♠ s ♥ ♠♣♠♥t ♦♥ P❯ ♦r dimK = 2 ♥ dimK = 3 ♥tst ♦♥ r♦s ♣r♦♠s ♥♥ ♣♦♣t♦♥ ♦♥t♥ ♥♥t♦r ①trt♦♥ r♦♠ ♠s♥ ♦t r♦♥t♦♥ r♦♠ ♣♦♥t ♦s ♦t tt t♦♥ rsts ♥ ♦♠♣rs♦♥s r♥ ♥ s s ts ♦♥ ♥r② ♦r♠t♦♥s

♠♣♠♥tt♦♥

♦rt♠ s ♥ ♠♣♠♥t ♦♥ P❯ s♥ ❯ tr s t t♦ s♠t♥♦s ♣rtrt♦♥ s♦ tt ♦♣rt♦♥s r ♣r♦r♠ ♥ ♣r ♦r ♦ ♠st s♠♣r s ts t ♠♦r ♥t s t ♠st ♦♥t♥s ♠♥② s ♥ ♥r②s♣♥ s t s♥ s♣♣♦rt ②K s r ♦r♦r t ♦ s ♥ ♣r♦r♠♠ t♦ ♦t♠♦♥s♠♥ ♦♣rt♦♥s ♥ ♣rtr t trs ♦ ♥♦t ♦♠♠♥t t♥ ♦tr♥ ♠♠♦r② ♦s♥ ♣r♠ts st ♠♠♦r② ss ♠♠♦r② tr♥sr t♥ P❯ ♥P❯ s s♦ ♥ ♠♥♠③ ② ♥①♥ t ♣r♠tr ♦ts ①♣r♠♥ts ♣rs♥t♥ ts st♦♥ ♥ ♣r♦r♠ ♦♥ 2.5 ③ ❳♦♥ ♦♠♣tr t r♣s rr♦ rttr

P♦♥t ♣r♦sss ♥

♦rt♠ s ♥ t ♦♥ ♣♦♣t♦♥ ♦♥t♥ ♣r♦♠s r♦♠ rs ♠s s♥ ♣♦♥t ♣r♦ss ♥ t dimK = 2 ♣r♦♠ ♣rs♥t ♥ ♦♥ssts ♥ tt♥♠rt♥ rs t♦ ①trt ♥♦r♠t♦♥ ♦♥ tr ♥♠r tr s③ ♥ tr s♣t ♦r♥③t♦♥ ♣♦♥t ♣r♦ss s ♠r ② ♣ss r s♠♣ ♦♠tr ♦ts ♥ ② ♣♦♥t♥tr ♦ ♠ss ♥ t♦♥ ♣r♠trs ♥ r ♣t t♦ ♣tr t r ♦♥t♦rs ♥r② s s♣ ② ♥tr② t tr♠ s ♦♥ t ttr②② st♥ t♥t r♦♠tr② ♥s ♥ ♦ts ♦ t ♦t ♥ ♣rs ♥trt♦♥ ♣♥③♥ t str♦♥♦r♣♣♥ ♦ ♦ts s ♣♣♥①

♦♠♣tt♦♥ t♠ qt② ♦ t r ♥r② ♥ stt② r t tr ♠♣♦rt♥t rtrs t♦ t ♥ ♦♠♣r t ♣r♦r♠♥ ♦ s♠♣rs s s♦♥ ♦♥ ♦r ♦rt♠ ♦t♥s t st rsts ♦r ♦ t rtr ♦♠♣r t♦ t ①st♥ s♠♣rs ♥♣rtr r ttr ♥r② s ♦r ❬❪ ♥ ♦r ❬❪ s♥♥t②r♥ ♦♠♣tt♦♥ t♠s s s s ♦r ❬❪ ♥ 2.8 × 106 s ♦r ❬❪ ♦t tt♦r t rs♦♥s ♠♥t♦♥ ♥ t♦♥ ts ♣ ♥ ♣r♦r♠♥ ♥rss ♥ t ♥♣t s♥♦♠s rr s♦ ♥r♥s ♥ ♠♣♦rt♥t ♠tt♦♥ ♦ t rr♥ ♣♦♥t ♣r♦sss♠♣r ♦r ♣♦♣t♦♥ ♦♥t♥ ❬❪ ♦♠♣r t♦ ♦r ♦rt♠ ♥ t srt③t♦♥ ♦ t♦t ♣r♠trs rqr ♥ ❬❪ ss ♣♣r♦①♠t tt♦♥ ♥ ♦③t♦♥ ♦ ♦ts ①♣♥s t r qt② ♦ t r ♥r② stt② s ♥②③ ② t ♦♥t♦ rt♦♥ ♥ s t st♥r t♦♥ ♦r ♠♥ ♥ ♥♦♥ t♦ r♥t sttst♠sr ♦r ♦♠♣r♥ ♠t♦s ♥ r♥t ♠♥s r s♠♣r ♣r♦s ttr stt②t♥ t ①st♥ ♦rt♠s

♠♣t ♦ t tr♥ s♣ ♣rtt♦♥♥ tr s s♦ ♠sr ② ♣r♦r♠♥ tsts t

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

r r ♦♥t♥ ② ♣♦♥t ♣r♦ss ♦ ♣ss rt ♦r t♥ t♥ t♦s♥ rsr ①trt ② ♦r ♦rt♠ ♥ ♠♥ts r♦♠ t r s r ♠ ♠ qtr strtr s s t♦ rt ♥♦♥♥♦r♠ ♣♦♥t strt♦♥ ♦t ♦♥ t r♦♣♣♣rts ♦ t rs r rt② ♣tr ② ♣ss ♥ s♣t ♦ t ♦ qt② ♦ t ♠♥ t ♣rt ♦r♣♣♥ ♦ rs

♦♥t ♦ rt♦♥♥r② t♠ ♦ts

❬❪ 7.3% 4.2% 1.7%♠t♣ rt ♥ 5.0% 2.1% 1.3%t ❬❪♦r s♠♣r t♦t 7.4% 6.2% 1.6%♣rtt♦♥♥ tr♦r s♠♣r t 4.4% 1.8% 1.1%♣rtt♦♥♥ tr

Pr♦r♠♥s ♦ t r♦s s♠♣rs t ♣rs♥ts t ♦♥ts ♦ rt♦♥♦ t ♥r② t♠ ♥ ♥♠r ♦ ♦ts r t t ♦♥r♥ ♦r 50 s♠t♦♥s

♥

❨♥♥ ❱ré ♦r♥t r

r Pr♦r♠♥s ♦ t r♦s s♠♣rs r♣ srs t ♥r② rs ♦rt♠ r♦♠ t r ♠ ♣rs♥t ♥ ♠ s r♣rs♥t s♥ ♦rt♠ s♦t tt t ♦rt♠ ❬❪ s s♦ s♦ tt t ♦♥r♥ s ♥♦t s♣② ♦♥ tr♣ 2.9× 107 trt♦♥s r rqr rss 1.8× 104 ♦r ♦r ♦rt♠

♦r s♠♣r ♠♣s♦♥ ❬❪ ❬❪ t ♦♥t♦r ❬❪♠ ♠♥t rtt♦♥ 0.45% 2.06% 0.06% 2.28%❯♥rtt♦♥ 32.7% 52.7% 17% 58.9%♣rs♥tt♦♥ ♥s♠♥t ♥s♠♥t ♥s♠♥t ♣①s

♦♠♣rs♦♥s t ①st♥ ♥♥t♦r ①trt♦♥ ♠t♦s r♦♠ t ♠ ♣rs♥t♥

♥♦r♠ ♣♦♥t strt♦♥s ♣r♦r♠♥s rs t r♠♥ ttr t♥ t ①st♥♦rt♠s ♥ ♣rtr t s♠♣r ♦ss stt② ♥ t ♦ts r tt ♥ ♦tss rt② t♥ t t ♣rtt♦♥♥ tr

♦rt♠ s s♦ ♥ tst ♦♥ ♥♥t♦r ①trt♦♥ r♦♠ ♠s ♣r♠tr♦ts r r ♥s♠♥ts ♥ ② ♣♦♥t ♥tr ♦ ♠ss ♥ t♦ t♦♥ ♣r♠trs♥t ♥ ♦r♥tt♦♥ ♦♥trr② t♦ t ♣♦♣t♦♥ ♦♥t♥ ♠♦ t ♣rs ♣♦t♥t♥s ♦♥♥①♦♥ ♥trt♦♥ ♦r ♥♥ t ♥s♠♥ts r s♦s r♦ ♥t♦r①trt♦♥ rst ♦t♥ r♦♠ stt ♠ rs ♣r♦s ♠♥ts ♦ ♦♠♣rs♦♥s t ①st♥ ♠t♦s rst qt② ♥ tr♠s ♦ r♦ ♥r♦rtt♦♥ s ♦②s♠r t♦ ①st♥ ♣♣r♦s r t♥ ❬❪ ♥ ❬❪ ♥ ♦r t♥ ❬❪ t ♦r ♦rt♠s♥♥t② ♠♣r♦s t ♦♠♣t♥ t♠s 16 s♦♥s r rqr ♥ ♦r s ♦♠♣r t♦7 ♠♥ts ② ♠♣s♦♥ ♦rt♠ ❬❪ 155 ♠♥ts ♦r ♠t♦ ❬❪ ♥ 60♠♥ts ♦r ♥ t ♦♥t♦r s ♣♣r♦ ❬❪

P♦♥t ♣r♦sss ♥

❲ tst ♦r ♦rt♠ t dimK = 3 ♦♥ ♥ ♦r♥ ♦t r♦♥t♦♥ ♣r♦♠ r♦♠ srs♥s ♦ s t♦ ①trt trs r♦♠ ♥strtr ♣♦♥t ♦s ♦♥t♥♥ ♦t ♦ ♦trs♥♦s ♥ ♦tr r♥t ♦ts ♥s r♦♥ rs ♥s rs t ♥ t♦ r♦♥③

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

r ♥♥t♦r ①trt♦♥ ② ♣♦♥t ♣r♦ss ♦ ♥s♠♥ts ♠ ♥ t ♣♦♥tstrt♦♥ s r♦② st♠t rt t r♦ ♥t♦r s r♦r r s♠♥ts ② ♦r♦rt♠ ♥ 16 s♦♥s r♦♠ t stt ♠ ♦t tt s s♦♥ ♦♥ t ♦s♣ s♦♠♣rts ♦ t ♥t♦r ♥ ♦♠tt ♥ r♦s r ♥ ② trs t s♦♠ ♦t♦♥s ①st♥♠t♦s s♦ ♥♦♥tr ts ♥ s ss

tr s♣s ♥ t②♣s ♦ts ss♦t t t ♣♦♥t ♣r♦ss ♦rrs♣♦♥ t♦ rr②♦ r♥t t♠♣ts ♦ trs t ♥ ♣♣♥① ♥tr② t tr♠ ♦ t ♥r②♠srs t st♥ r♦♠ ♣♦♥ts t♦ t♠♣t rs t ♣rs ♥trt♦♥ ts ♥t♦♦♥t ♦♥str♥ts ♦♥ ♦t ♦r♣♣♥ s s ♦♥ tr t②♣ ♦♠♣tt♦♥ ♦♠♣r t♦t ♦r♠r ♣♣t♦♥s t ♦♥rt♦♥ s♣ C s ♦ r ♠♥s♦♥ s♥ t ♦ts r♣r♠tr② ♠♦r ♦♠♣① s ♦s ♦r ♦rt♠ t♦ ①♣♦t ♠♦r ♣② ts ♣♦t♥t r♦tt♦♥ r♥ s ♥♦t s r s♥ t ♦ts r ♥r♥t ② r♦tt♦♥ ♦r s st♥ r♥ ♥ ♦rr t♦ ①♥ t t②♣ ♦ ♥ ♦t

s♦s rsts ♦t♥ r♦♠ sr s♥s ♦ r r♥ ♥ ♥tr ♥r♦♥♠♥ts 30rs♣t② 5.4 t♦s♥ trs r ①trt ♥ 96 rs♣ 53 ♠♥ts ♦♥ t 3.7♠2 ♠♦♥t♥r rs♣ 1♠2 r♥ r r♦♠ 13.8 rs♣ 2.3 ♠♦♥ ♥♣t ♣♦♥ts ♦♠♣tt♦♥ t♠s♥ ♣♣r t ♥♥ ♥♦♥tr ♦ts ♥ s rs s♥s ② ♣♦♥t ♣r♦sss s ♥ t♦ ♦r ♥♦ s ♥♦t ♥ ♥t ♥♦ t♦ t ①tr♠ ♦♠♣①t② ♦ t stt s♣ ♦t s♦ tt t ♣r♦r♠♥s ♦ ♠♣r♦ ② r♥ t s♣C t ♣♦♥t ♣r♦ss ♦♥ ♠♥♦s r t ③♦♦r♥t ♦ ♣♦♥ts s tr♠♥ ②♥ st♠t r♦♥ sr

t♥ t tt♦♥ qt② t r② ♦r ts ♣♣t♦♥ s t ts s♥ ♥♦r♦♥ trt ①sts s strt ♦♥ t r♦♣♣ ♣rt ♥ ♠♥② ♥① ttrs ♦♥ r♥t ③♦♥s r♦♠ r ♠s qr t t sr s♥s ♦ts r ♦② ♦t ♥ tt t♦ t ♥♣t ♣♦♥ts t ♦♠ss♦♥s ♥ ♥ trs r srr♦♥② ♦tr t②♣s ♦ r♥ ♥tts s s ♥s ♥♦♥♦r♣♣♥ ♦♥str♥t ♦ t ♥r②♦s s t♦ ♦t♥ stst♦r② rsts ♦r rs t tr ♥st② rr♦rs rq♥t② ♦r

♥

❨♥♥ ❱ré ♦r♥t r

♥ st♥s♥ t tr t②♣ ♥ s♣t ♦ t tr ♦♠♣tt♦♥ tr♠ ♦ t ♥r②

r r r♦♥t♦♥ r♦♠ ♣♦♥t ♦s ② ♣♦♥t ♣r♦ss s♣ ② ♣r♠tr♠♦s ♦ trs r ♦rt♠ tts trs ♥ r♦♥③s tr s♣s ♥ rs t ♥♣ts♥ ♣♦♥ts ♥tr ♥ t♦♣ rt ♥♣t s♥ ♣♦♥ts r♥ ♥r♦♥♠♥ts ♥s♣t ♦ ♦tr t②♣s ♦ r♥ ♥tts ♥s r ♥ ♥s ♦♥t♥ ♥ ♥♣t ♣♦♥t♦s r ♦t s♥s ♥ r ♠ s ♦♥ t♦ ♦tt♦♠ rt t r♦♣♣ ♣rt t♦ ♣r♦ ♠♦r ♥tt r♣rs♥tt♦♥ ♦ t s♥ ♥ t tr ♦t♦♥ ♦t ♦♥ t r♦♣♣ ♣rt ♦t ♣r♠tr ♠♦s t t♦ t ♥♣t ♣♦♥ts ♦rrs♣♦♥♥ t♦ trs ♥ ♦ t ♥trt♦♥♦ tr ♦♠♣tt♦♥ ♦s t rr③t♦♥ ♦ t tr t②♣ ♥ ♥♦r♦♦

♦♥s♦♥

❲ ♣r♦♣♦s ♥ ♦rt♠ t♦ s♠♣ ♣♦♥t ♣r♦sss ♦s str♥ts ♥ ♦♥ t ①♣♦tt♦♥♦ r♦♥ ♣r♦♣rts t♦ ♥ t s♠♣♥ t♦ ♣r♦r♠ ♥ ♣r ♥ t ♥trt♦♥♦ tr♥ ♠♥s♠ ♦♥ ♥t strt♦♥s ♦ t ♣♦♥ts ♥ t s♥ r♦rt♠ ♠♣r♦s t ♣r♦r♠♥s ♦ t ①st♥ s♠♣rs ♥ tr♠s ♦ ♦♠♣t♥ t♠s ♥stt② s♣② ♦♥ r s♥s r t ♥ s r② ♠♣♦rt♥t t ♥ s t♦t♣rtr rstrt♦♥s ♦♥trr② t♦ ♠♦st s♠♣rs ♥ ♥ ♣♣rs s ♥ ♥trst♥ tr♥tt♦ t st♥r ♦♣t♠③t♦♥ t♥qs ♦r ♥ ♣r♦♠s s ①♣♥ ♥ ♣♣♥① ♥ ♣rtr ♦♥ ♥ ♥s s♥ t ♠♦ ♣r♦♣♦s ♥ t♦♥ t♦ ①trt ♥② t②♣♦ ♣r♠tr ♦ts r♦♠ r ♣♦♥t ♦s ♥ tr ♦rs t ♦ ♥trst♥ t♦♠♣♠♥t t ♦rt♠ ♦♥ ♦tr P❯ rttrs ♠♦r ♣t t♦ t ♠♥♣t♦♥ ♦ t strtrs rttr s♦ tt t ♣r♦r♠♥s ♦r ♣♦♥t ♣r♦sss ♥ ♦ ♠♣r♦

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

♥♦♠♥ts

s ♦r s ♣rt② ♥ ② t r♦♣♥ sr ♦♥ trt♥ r♥t ♦st♦♠tr② Pr♦ss♥ r♥t r♠♥t t♦rs t♥ ♠ss♦ s♦t r♥ ♣♣♥ ♥② t ♦r ❱t ♥ t ♦r ♣r♦♥♥ ttsts

♣♣♥s

ts ♦♥ t ♣♦♣t♦♥ ♦♥t♥ ♠♦ r♦♠ ♠s

♥ ts ♣♣♥① t t ♥r② ♠♦ s ♦r t ♣♦♥t ♣r♦ss ♦ ♣ss tt s♥ ♣rs♥t ♥ st♦♥ ♦ t ♣♣r x r♣rs♥ts ♦♥rt♦♥ ♦ ♣ss ♥tr ♦♠ss p ♦ ♥ ♣s s ♦♥t♥ ♥ t ♦♠♣t st K s♣♣♦rt♥ t t ♠ s

r ♣s ♣r♠trs ♥ ♣s s ♥ ② ♣♦♥t p ∈ K ♥tr ♦ ♠ss ♦ t♦t ♥ 3 t♦♥ ♣r♠trs r t s♠♠♦r ①s b t s♠♠♥♦r ①s a♥ t ♥ θ ♥s rs♣t② ♦rr♥ ♦♠ ♦ t ♦t s ♥♦t ② Sin

rs♣t② Sout

♥r② ♦♦s t ♦r♠ ♥ ② q ♦ t ♣♣r

U(x) =∑

xi∈x

D(xi) +∑

xi∼xj

V (xi, xj)

r t ♥tr② t tr♠ D(xi) ♥ t ♣♦t♥t V (xi, xj) r ♥ ②

D(xi) =

1− d(xi)d0

d(xi) < d0

exp(d0−d(xi)d0

)− 1 ♦trs

V (xi, xj) = βA(xi ∩ xj)

min(A(xi), A(xj))

• d(xi) r♣rs♥ts t ttr②② st♥ t♥ t r♦♠tr② ♥s ♥ ♦ts t♦t xi

d(xi) =m2

in −m2out

4(σin − σout)−

1

2ln(

2σinσout

σ2in + σ2

out

)

♥

❨♥♥ ❱ré ♦r♥t r

r t tr♠ D(xi) ♥ ♥t♦♥ ♦ t ttr②② st♥ d(xi) r t♦♥t d0 t ♠♦r st t ♦t tt♥

r min ♥ σin rs♣t② mout ♥ σout r t ♥t♥st② ♠♥ ♥ st♥rt♦♥ ♥ Sin rs♣t② ♥ Sout

• d0 s ♦♥t ①♥ t s♥stt② ♦ t ♦t tt♥

• A(xi) s t r ♦ ♦t xi

• β s ♦♥t t♥ t ♥♦♥♦r♣♣♥ ♦♥str♥t t rs♣t t♦ t t tr♠

rs ♥ sr t ♦t♦♥ ♦ t ♦ts r♥ t s♠♣♥ ♥ s ♦ r♦♥t♥ ♦t tt s ♠t♠t tt♦♥ s s t♦ r♦② ①trt t ss ♦♥trst r♦♠ t ♠ ♦ rs s

ts ♦♥ t ♥♥t♦r ①trt♦♥ ♠♦ r♦♠ ♠s

❲ t r t ♥r② ♠♦ s ♦r t ♣♦♥t ♣r♦ss ♦ ♥s♠♥ts tt s ♥♣rs♥t ♥ st♦♥ ♦ t ♣♣r ♠♦ ♦r♠t♦♥ s rt② s♠r t♦ t ♣♦♣t♦♥♦♥t♥ ♣r♦♠ t ♥ ♣♣♥① s strt ♦♥ ♥s♠♥t s ♥ ② ♣r♠trs ♥♥ t ♣♦♥t ♦rrs♣♦♥♥ t♦ t ♥tr ♦ ♠ss ♦ t ♦t♠r② t♦ t ♣♦♣t♦♥ ♦♥t♥ ♠♦ t tt♥ qt② t rs♣t t♦ t t s s♦♥ t ttr②② st♥ ♥tr② t tr♠ D(xi) ♦ t ♥r② s ♥ ② q ♣♦t♥t V (xi, xj) ♣♥③s str♦♥ ♦t ♦r♣s s q ♦r t ♣♦t♥t s♦ts ♥t♦ ♦♥t ♦♥♥t♦♥ ♥trt♦♥ ♥ ♦rr t♦ ♦r t ♥♥ ♦ t ♥s♠♥ts ♣♦t♥t tr♠ s ♥ ②

V (xi, xj) = β1A(xi ∩ xj)

min(A(xi), A(xj))+ xi∼ncxj

× β2f(xi, xj)

• β1 ♥ β2 r t♦ ♦♥t t♥ rs♣t② t ♥♦♥♦r♣♣♥ ♥ ♦♥♥t♦♥♦♥str♥ts t rs♣t t♦ t t tr♠

• ∼nc s t ♥♦♥♦♥♥t♦♥ rt♦♥s♣ t♥ t♦ ♦ts xi ∼nc xj t ♥♦r rs♦ xi ♥ xj s ♦ ♥♦t ♦r♣

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

r ♦t♦♥ ♦ t ♦t ♦♥rt♦♥ r♥ t s♠♣♥ ♥ ♠ ♣t r♣rs♥t♥ r ♣♦♣t♦♥ s r♦② s♠♥t ② ♠t♠t tt♦♥ ♥t♦ ♥r②♠ t ♦♦r ♦rrs♣♦♥s t♦ t ss ♦ ♥trst qtr s rt r♦♠ ts♠♥t ♠ ♥r② ♣r♦rss② rss r♥ t s♠♣♥ ♦tt♦♠ t♦ts ♥ t rr♥t ♦♥rt♦♥ ♦♠ ♠♦r ♥ ♠♦r r♥t

r ♦t♦♥ ♦ t ♥♠r ♦ ♦ts r♥ t s♠♣♥ r♦♠ t r ♠ ♣rs♥t♥ ♦ t ♣♣r ♦t tt t♠ s r♣rs♥t s♥ ♦rt♠ s ♥ tt ♦rt♠ ❬❪ s s♦ s♦ tt t ♦♥r♥ s ♥♦t s♣② ♦♥ t r♣ r ♥ ♥tst ♥♠r ♦ rs ♦♥ ② ♥ ①♣rt r♦♥ trt ♥♠r ♦ ♦ts ♦♥ ② ♦rs♠♣r s r② ♦s t♦ t ①♣rt ♥♠r s ♥♦t t s ♦ t ♦tr s♠♣rs ♦ttt st♠t♥ t ♦rrt ♥♠r ♦ ♦t ♦s ♥♦t ♠♥ tt t ♦ts r ♦rrt② ttt♦ t t t t s ♥ ♠♣♦rt♥t rtr♦♥ ♦r ♣♦♣t♦♥ ♦♥t♥ ♣r♦♠s s ♠♥t♦♥ ♥❬❪

♥

❨♥♥ ❱ré ♦r♥t r

r ♥s♠♥t ♣r♠trs ♥s♠♥t s ♥ ② ♣♦♥t p ∈ K ♥tr ♦♠ss ♦ t ♦t ♥ 3 t♦♥ ♣r♠trs r t s♠♥t b t s♠t a♥ t ♦r♥tt♦♥ θ ♦t tt t s♠t a ♥ ♣r♠♥r② ① ♣♥♥ ♦♥ t♣♣t♦♥ ♦r ♠♦♥♦s r♦♥t♦r ①trt♦♥ ♥s rs♣t② ♦rr♥♦♠ ♦ t ♦t s ♥♦t ② Sin rs♣t② Sout ♥t ♦ t ♦♥♥t♦♥ r cs♦ t ♥♦r r s ♣r♠tr ♦ t ♠♦ ♦r♠t♦♥ ♥♦rs r ♥♦t② A1 ♥ A2

• condition s t ♥t ♥t♦♥ rtr♥♥ ♦♥ ♥ ♦♥t♦♥ s ♥ ③r♦ ♦trs

• f(xi, xj) s s②♠♠tr ♥t♦♥ t♥ t ♣♥③t♦♥ ♦ t♦ ♥♦♥♦♥♥t ♦tsxi ♥ xj t rs♣t t♦ tr r tt♥ qt② ♥t♦♥ f s ♥tr♦ t♦st② r① t ♦♥♥t♦♥ ♦♥str♥t ♥ t t♦ ♦ts r ♦ r② ♦♦ qt②

s ♦r t r ♦♥t♥ ♣r♦♠ s s ♠t♠t tt♦♥ t♦ r♦② ①trt tss ♦ ♥trst r♦♠ t r ♠ ♦ r♦♥t♦r s♦♥ ♦♥ r ♦ t ♣♣r ♥t r♦ ♣①s ♥ ts ♠ r rt② rt ♦♠♣r t♦ t r♦♥ s♠♥trst s ♥t② ♥♦t ♦♣t♠ t s♥t t♦ rt ♥ ♥t ♣rtt♦♥♥ tr s r ♠ ♦ t ♣♣r

ts ♦♥ t tr r♦♥t♦♥ ♠♦ r♦♠ ♣♦♥t ♦s

♥ ts ♣♣♥① t t ♥r② ♠♦ s ♦r t ♣♦♥t ♣r♦ss ♦ trs tt s ♥♣rs♥t ♥ st♦♥ ♦ t ♣♣r x r♣rs♥ts ♦♥rt♦♥ ♦ ♠♦s ♦ trs r♦♠ t♠♣t rr② sr ♥ ♥tr ♦ ♠ss p ♦ tr s ♦♥t♥ ♥ t ♦♠♣t st K s♣♣♦rt♥ t ♦♥♥♦① ♦ t ♥♣t ♣♦♥t ♦ s ❲ ♥♦t ② ∂xi t sr ♦ t ♦t xi ♥② Cxi t ②♥r ♦♠ ♥ rt ①s ♣ss♥ tr♦ t ♥tr ♦ ♠ss ♦ xi ♥ t ♥♣t ♣♦♥ts r ♦♥sr t♦ ♠sr t qt② ♦ xi ♥r② ♦♦s t ♦r♠ ♥ ② q ♦ t ♣♣r t t ♥tr② t tr♠ D(xi) ♥

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

r rr② ♦ tr ♠♦s t ♦ts r s♣② ② ♣♦♥t ♥tr ♦ ♠ssstrt ② r ♦t ♥ t♦♥ ♣r♠trs rr♦s ♥♥ t ♥♦♣② t②♣♦s s♣ ♥ ♦♥♦ ♣♥ ♦r r ♣s♦ ♣♦♣r ♦r t ♦r s♠♣s♦ ♦ ♦r ♠♣

r r ♣r♠trs tr s ♥ ② ♣♦♥t p ∈ K ♥tr ♦ ♠ss ♦ t ♦t t②♣ t ∈ ♦♥♦ ♣s♦ s♠♣s♦ ♥ 3 t♦♥ ♣r♠trs r t♥♦♣② t a t tr♥ t b ♥ t ♥♦♣② ♠tr c ②♥r ♦♠ Cxi

r♣rs♥ts t ttrt♦♥ s♣ ♦ ♦t xi ♥ t ♥♣t ♣♦♥ts r s t♦ ♠sr tqt② ♦ ts ♦t

♥

❨♥♥ ❱ré ♦r♥t r

t ♣rs ♣♦t♥t V (xi, xj) ♥ ②

D(xi) =1

|Cxi|

∏

pc∈Cxi

γ(d(pc, ∂xi))

V (xi, xj) = β1Voverlapping(xi, xj) + β2Vcompetition(xi, xj)

• |Cxi| s ♦♥t ♥♦r♠③♥ t ♥tr② t tr♠ t rs♣t t♦ t ♥♠r ♦ ♥♣t♣♦♥ts ♦♥t♥ ♥ Cxi

• d(pc, ∂xi) s st♥ ♠sr♥ t ♦r♥ ♦ t ♣♦♥t pc t rs♣t t♦ t ♦tsr ∂xi d s ♥♦t t ♦♥♥t♦♥ ♦rt♦♦♥ st♥ r♦♠ ♣♦♥t t♦ sr s sr trs ♦ ♥♦t sr ♣s♦♦♥♦ s♣s ♥♣t ♣♦♥ts r ♥♦t ♦♠♦♥♦s②strt ♦♥ t ♦t sr r d s ♥ s t ♦♠♥t♦♥ ♦ t ♣♥♠trst♥ t ♣r♦t♦♥ ♥ t ♣♥ ♦ qt♦♥ z = 0 ♦ t ♥ st♥ ♥t t♠tr rt♦♥ s tt ♣♦♥ts ♦ts t ♦t r ♠♦r ♣♥③ t♥ ♥s♣♦♥ts strts t ♦r ♦ d ♥ t ❳❩♣♥ ♦t tt d s ♥r♥t ②r♦tt♦♥ r♦♥ ❩①s

• γ(.) ∈ [−1, 1] s qt② ♥t♦♥ s strt② ♥rs♥

• Voverlapping s t ♣rs ♣♦t♥t ♣♥③♥ str♦♥ ♦r♣♣♥ t♥ t♦ ♦ts♥ ♥ ②

Voverlapping(xi, xj) =A(xi ∩ xj)

min(A(xi), A(xj))

r A(xi) s t r ♦ t ♦t xi ♣r♦t ♦♥t♦ t ♣♥ ♦ qt♦♥ z = 0

• Vcompetition s t ♣rs ♣♦t♥t ♦r♥ s♠r tr t②♣ t ♥ ♥♦r♦♦

Vcompetition(xi, xj) = ti 6=tj

r . s t ♥t ♥t♦♥

• β1 ♥ β2 r t♦ ♦♥ts t♥ rs♣t② t ♥♦♥♦r♣♣♥ ♦♥str♥t ♥t ♦♠♣tt♦♥ tr♠ t rs♣t t♦ t t tr♠

♥ ♦rr t♦ r♦② ①trt t ss ♦ ♥trst r♦♠ ♣♦♥t ♦ sttr sr♣t♦r ❬❪ s st♦ ♥t② t ♣♦♥ts ♣♦t♥t② ♦rrs♣♦♥ t♦ trs

Pr♦r♠♥ tsts ♦♥ ♦♥♥t♦♥ r♦ ♥♦♠

♠♦s

s ♠♥t♦♥ ♥ t ♣♣r ♦r ♦♥t r♦ s♠♣r ♥ s ♦r ♦♣t♠③♥ ♦♥♥t♦♥♠t ♥rs ♦ t ♦r♠

U(l) =∑

i∈V

Di(li) +∑

(i,j)∈E

V (li, lj)

r V s t st ♦ rts ♦r sts ♥ s ♦ ♠s E s t st ♦ s ♣rs ♦ ♥♦r♥ rts ♥ l ∈ [1, N ]card(V) s ♦♥rt♦♥ ♦ s ♦r V t N t ♥♠r ♦

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

r ♦r ♦ d ♥ t ❳❩♣♥ s♦rs r♣rs♥t t st♥s r♦♠ ♣♦♥ts t♦t s♠♣s♦ sr ♦♦r ♥ t

s ♦ t ♣r♦♠ ♦r ts ♦♥r♥♥ ts t②♣ ♦ ♥r② ♠♥♠③t♦♥ ♣r♦♠s ♥ ♦♥ ♥ ❬❪

♥ t ♣♦♥t ♣r♦sss ♥ s♥ s ♥r③t♦♥ ♦ ts ♦♥♥t♦♥ ♠♦s r t ♠♥s♦♥ ♦ t ♦♥rt♦♥ s♣ s r r♣ strtr s ♥♦t stt t ②♥♠ s r ♥ ♥ srt ♦r♥ ♦♥t♥♦s ♦♠♥s s♦ tt ♦♠♣① ♣r♠tr ♦ts♥ ♥ tr s ♥♦ ♦♥str♥t ♦♥ t ♦r♠ ♦ t ♣rs ♥trt♦♥ tr♠ V ② t ♠♥♠③t♦♥t♥qs

♥♦r♠ strt♦♥ ♦ t s s ♦♥sr r ♥ tr s ♥♦ ♣♦♥t t♦ ♦♠♣t ♣rtt♦♥♥ tr r♦♠ ♣r♥ s♠♥tt♦♥ s t ♣r♦♠ s t♦ s♠♥t t t rr ♣rtt♦♥♥ s t♥ s s strt ♦♥ r ♥ ♦ t ♣♣r t ♦ s ♥ ② t ♥♣♥♥ ♦♥t♦♥ s q ♥ t ♣♣r s δmax = 0 ♦tsr ① t t ♦ s ♠st s♣r♦r ♦r q t♦ ǫ t t ♦ t ♥♦r♦♦rt♦♥s♣ ♦r ♥st♥ ♥ s ♦ ♥ ♠ ♥ ♣r♦♠ t ♦r ♦♥♥①t② ♥♦r♦♦ t ♦♥t♦♥ ♠♣s tt ♥ ♦♥t♥ ♦♥ ♥q ♣① ♣rtt♦♥♥ s♦♦s② ♣r♦♠s♥ s ♦♥ qrtr ♦ t ♣①s ♦ t ♠ ♣rtr s♠t♥♦s②t trt♦♥ ♦ t s♠♣♥

♣r♦r♠♥s ♦ ♦r s♠♣r ♥ tst r♦♠ s ♠♦ ♦ ♠ s♠♥tt♦♥ ♥tr② t tr♠ Di(li) ♠srs t qt② ♦ li t ♣① i ss♥ strt♦♥s♦r ♣rs② t r♦♠tr② strt♦♥ ♦ ss s ♠♦ ② ss♥ ♦s♠♥ ♥ st♥r t♦♥ r ♠♦ ♣r♠trs t♦ st♠t ♦r ① ② ♥ sr ♣♦t♥t V ♦rrs♣♦♥s t♦ t ♦♥♥t♦♥ P♦tts ♠♦ ❬❪ s♦ tt t ♥ s s♠♦♦t♥ ♦ ♥♦r♦♦ ♦♥♥①t②

♥ r s♦ t ♣r♦r♠♥s ♦t♥ r♦♠ r♦s ♠s ♥ ♣r♦ ♦♠♣rs♦♥s t t st♥r ♦♣t♠③t♦♥ t♥qs ♠①♣r♦t Pr♦♣t♦♥ P❬❪ r♣t s ♦rt♠s ❬❪ ♥ trt ♦♥t♦♥ ♦s ❬❪ r s♠♣r♦♠♣ts t t ♦tr ♦rt♠s r ♥r② s s② st② r t♥② s♥ α①♣♥s♦♥ αβ s♣ ♦r P t t ♦♠♣tt♦♥ t♠ r ♦r ♥ ♦ t♠

♥

❨♥♥ ❱ré ♦r♥t r

r ♠ s♠♥tt♦♥ t♦♣ ♥♣t ♠s r♦♠ t t♦ rt ♦rs ♣♥t r♥ ♠t s♦♥ r♦ sts ♦t♥ ② ♦r s♠♣r ② tr r♦ α①♣♥s♦♥ ♥② st r♦ ♦ r s ♥♦t t♦ t t s♠♥tt♦♥ ♠♦ s ♦♦s②♥♦t ♦♣t♠ t t♦ ♦♠♣r t rsts r♦♠ r♦s ♦♣t♠③t♦♥ t♥qs ♥ ♣rtr r♦♠α①♣♥s♦♥ ♥ ♣r♦ t ♦st ♥ st ♥rs ♥ t ♠♥ rs♣t②

♦r s♠♣r α①♣♥s♦♥ αβ s♣ P ♥r② t♠ ♥r② t♠ ♥r② t♠ ♥r② t♠ ♥r② t♠

♦rs

s ♣①s

♣♥t

s ♣①s

r

s ♣①s

♠t

s ♣①s

Pr♦r♠♥s ♦ r♦s ♦♣t♠③t♦♥ ♦rt♠s ♥ tr♠s ♦ r ♥r② ♥ ♦♠♣tt♦♥ t♠ ①♣rss ♥ s♦♥ r♦♠ t ♠s s♦♥ ♥

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

♦♠s r② ttrt ♥ t ♥♠r ♦ s ♥ t ♠ s③ ♥rs ♥ ♣rtrt rsts ♦t♥ r♦♠ ♠t s s♠t ♠ ♦ 8 sss s ♥♦rr♣t ② ss♥ ♥♦s s♦ tt ♦r s♠♣r ♥ ♥ ♥trst♥ tr♥t t♦ tst♥r ♦♣t♠③t♦♥ t♥qs ② str♦♥② r♥ ♦♠♣tt♦♥ t♠s r♥ ♥ ♥r② ♦ s♠r qt② ♦t tt t ♥r② r♦♠ t ♦rt♠ s r② ♦♥ ts♠ 4.1186 ♥ ts st ♥ ♦ ♠♥♠ s strt ♥ ♦tt♦♠ rt♠ r t ♥tr rt str s ♥ ♥♦rrt② ss ♦♦r ♥st ♦ rr②s ♣r♠♥r② ①♣r♠♥ts ♣rs♥t ♥ ts ♣♣♥① r ♣r♦♠s♥ t ♥ t♦ ♦♣ ♥ ♣rtr ♦♠♣rs♦♥s t ♣r③ rs♦♥s ♦ r♣t s ♦rt♠s ❬❪ ♠st t♦ t ♠♦r ♣② t ♣♦t♥t ♦ t s♠♣r ♦r s♦♥ ♠t ♣r♦♠s

t♦♥ rsts ♥ ♦♠♣rs♦♥s

❲ ♣rs♥t ♥ ts ♣♣♥① s♦♠ t♦♥ rsts ♥ ♦♠♣rs♦♥s ♦♥ ♣♦♣t♦♥ ♦♥t♥♥ ♥♥t♦r ①trt♦♥ ♣r♦♠s ♦t tt t ♦♦♥ ♠s ♥ ♣tr ♥ rs♦t♦♥ rr s ♥t t♦ ③♦♦♠ ♥ t ♠s ♥ ♦rr t♦ s ♠♦r ts

• ♥ ♣r♦♣♦s rsts ♦♥ ♦♥t♥ r♦♠ ♠r♦s♦♣ ♠s s♠s ♥ s♠t ② ❬❪ ♥ r ♣r♦ t r♦♥ trt t ①t♥♠r ♥ ♦t♦♥ ♦ s r ♥♦♥ ♦r ♠ r ♦rt♠ s ♥ ♦♠♣rt♦ t s♣rs ♣♣r♦ ♣r♦♣♦s ② ♠♣ts② t ❬❪ ♥ ♦t tt♦♥trr② t♦ ♦r ♦rt♠ ts s♣rs ♣♣r♦ st rs ♥ st♠t ♥♠r ♦s ♣r ♠s t♦t ♦t♥ ♥ ♥t♥ t♠

• ♥ ♣rs♥t t♦♥ ♥♥t♦r rsts s♦s s ♦♠♣rs♦♥s t r♥t ♣♣r♦s ❬ ❪ r♦♠ ♦t t r♦ ♠ ♣rs♥t ♥ ♦t ♣♣r ♥ ♥ r ♠ ♦ rr♥t♦r ♠r② t♦ ♦ t ♣♣r ♣r♦s ♦♠♣rs♦♥ t ❬ ❪ r♦♠ t rr ♠ ♥ tr♠s ♦ ♦♠♣tt♦♥ t♠♥ ♥t♦r ①trt♦♥ r②

• ♥ s♦ rsts ♦♥ r♥t ♦t ①trt♦♥ ♣r♦♠s r♦♠ ♠s

♥

❨♥♥ ❱ré ♦r♥t r

r ♦♥t♥ r♦♠ t t ♠r♦s♦♣ ♠ ♠ r♦♥ rt s♦st ♦t♦♥ ♦ tr♦ r♦ss rt r ♣♦♥t ♣r♦ss ♦ ♣ss ♣trst s t r② rr♦rs s strt ♦♥ t rt r♦♣ ♦♠ss♦♥s ♥ ♣♣r ♥ ♠♥②s r rr♦♣ ♥ t♥② r ♦t tt ♥ s s t s r② t t♦ s② ttt s ♥ ♦r ♥ ①♣rt

r♦♥ rt r ♦rt♠ ♠♣ts② t ❬❪ L1rrst♦♥♠♣ts② t ❬❪ ♦♥♦rrst♦♥

♦♠♣rs♦♥s t t ♦♥t♥ ♣♣r♦ ♣r♦♣♦s ② ♠♣ts② t ❬❪ r♦♠t ♠r♦s♦♣ ♠s ♦ ♥ s ♦rrs♣♦♥ t♦ t ♥♠r ♦ s ♥t ♦♥sr ♠ r ♦rt♠ ♣r♦s ttr st♠t♦♥ ♦ t ♥♠r ♦ s t♥♦t t L1 ♥ ♦♥♦rr③t♦♥ rs♦♥s ♦ ❬❪ ♥ ♣rtr ♦r ♦rt♠ s ♠♦rrt ♥ s ♦ ♦♥rt♦♥s ♦ s ② ♦♥♥trt ♥ s♠ rs

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

r ♦♥t♥ r♦♠ st ♦ s① ♠r♦s♦♣ ♠s ♠r② t♦ r♦♥rt s r♣rs♥t ② ♠ t r♦sss rs ♦r rst s ♥ ② ♦tt♦♠ tr ♣ss

♥

❨♥♥ ❱ré ♦r♥t r

r ♥♥t♦r ①trt♦♥ r ♦rt♠ s ♠♦r rt rsts t♥ t ♠t♦s♣r♦♣♦s ② ♦r② t ❬❪ ♥ r t ❬❪ r♦♠ t♦♣ t r♦ ♥ ♦tt♦♠ rr♥t♦r ♠s t ♦s ♥♦t ♣r♦r♠ ttr t♥ ♦st t ❬❪ ♣r♦♣♦s ♠♦r ♦♠♣①♠♦ t ♠♥② ♦♠tr ♥trt♦♥s t♦ strtr t ♥s♠♥ts t♦tr s♦ ♠♣②♥♠♥② ♠♦ ♣r♠trs t♦ t♥ s t ♥ ♦ t ♣♣r ♥ ♦rs♠♣r s♥♥t② rs t ♦♠♣tt♦♥ t♠ ♦♠♣r t♦ ts tr ♠t♦s

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

♦r s♠♣r r t ❬❪ ♦st t ❬❪ ♦r② t ❬❪♠ ♠♥t rtt♦♥ 0.78% 1.36% 0.22% 1.68%❯♥rtt♦♥ 30.5% 35.4% 32% 39.8%♣rs♥tt♦♥ ♥s♠♥t ♥s♠♥t ♥s♠♥t ♣①s

♦♠♣rs♦♥ t ①st♥ ♥♥t♦r ①trt♦♥ ♠t♦s r♦♠ t rr ♠ ♣rs♥t ♥ ♦tt♦♠ ♠r② t♦ t ♦♠♣rs♦♥ r♦♠ t r♦ ♠ t rst qt②♥ tr♠s ♦ r♦ ♥r♦rtt♦♥ s r t♥ ❬❪ ♥ ❬❪ t ♦r t♥ ❬❪ r s♠♣r s r② str t♥ t ♦tr ♣♣r♦s ♦t ♦r tt t ♥ ♦ t♠ s ♥♦t s♦♠♣♦rt♥t t♥ ♦r t r♦ ♠ s ♠ r ♠

r t♦♠t ♦♥t♥ ♦♥t♥ ♥ ①trt♥ ♦♣♥ st♦♠t ♦♥ ♣r♠ts s t♦♥②③ t rs ♥r♦♥♠♥t ♦♥t♦♥ ♥ t ♣♥t rts t♦ strss ♣♥♦♠♥s s r ♣♦t♦♥ ② ♦s♥ st♦♠t r ♣♦♥t ♣r♦ss ♣trs ♦♣♥ st♦♠t ♠rs ② ♣ss r♦♠ ♠♦s ♠s 757 st♦♠t r tt ♥ s♦♥s r♦♠ t t♠ ♦t t rt ♦ ♦♣♥ st♦♠t r ♦rrt② ①trt ♥ s♣t ♦ t ♥st② ♦t ♦s st♦♠t ♦ ♦♥trst ♠rs

♥

❨♥♥ ❱ré ♦r♥t r

r ① tt♦♥ r♥ tr ♦ t①s s ♥②③ ② ♦♥t♥ ♥ ①trt♥②♦ s r♦♠ r ♠s r ♣♦♥t ♣r♦ss ♣trs t①s ② ♣ss ♦t tt tttr② st♥ s ♥ ♣t t♦ ♦r t ②♦ ♦ts ♥ t ♠ 87 t①s rtt ♥ s♦♥s r♦♠ t t ♠ s s♦♥ ♦♥ t t ♦r ♦rt♠ ♦s t♦③t♦♥ ♦ t①s ♥ s♣t ♦ t ♣rs♥ ♦ ♦tr s

r ①trt♦♥ ♦♥t♥ ♥ tr♥ s ♥ s ♦s s t♦ ♥②③ t♦r ♦ s r♦♠ ♥r ♣♦♥t ♦ r ♣♦♥t ♣r♦ss ♣trs ♦s ②♣ss r♦♠ ♠s ♦ s s r tt ♥ ♠♥ts r♦♠ t t ♠ ss♦♥ ♦♥ t r♦♣♣ ♠ t s r ♦② tt ♥ s♣t ♦ t ♥st② ♥ ♦r♣ ♦t tt t ♦♠♣t♥ t♠ s ♦♠♣r t♦ ♦tr ♣♣t♦♥s s t♣rtt♦♥♥ s♠ ♦♥t♥s s

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

r♥s

❬❪ ② s♦t ❱ t♦st ♦♠tr② ♠♦s ♥ s♦♥ ♦r♥♦ ♣♣ ttsts

❬❪ s♦♠s ❳ ♥♦s ❩③♥ t ①trt♦♥ s♥ st♦st rt♥t ②♥♠s ♥ ♦♥t♥♠ ♦r♥ ♦ t♠t ♠♥ ♥ ❱s♦♥

❬❪ ❲ ♦♥s r ♣♦♥t ♣r♦sss ♦r r♦ ♦♥t♥ ♥ ❱P ♠ ❯

❬❪ r ♠r s♦♠s ❳ ♦♠tr tr ①trt♦♥ ② ♠t♠r♣♦♥t ♣r♦ss P

❬❪ s♦t ❱ ♣t ♠♣ t♦♥ ♦r r ♥♠r ♦ ♠♦♥ ♦ts s♥♠r♦ sq♥t ♦t ♣r♦sss P

❬❪ t r ♦① ♦r ❯ rtr ♣ ♠r ♣♦♥t♣r♦ss ♦r ♠♦♥ r ♦r♠s P

❬❪ ♦st s♦♠ ❳ ❩r P♦♥t ♣r♦sss ♦r ♥s♣rs ♥ ♥t♦r①trt♦♥ ♥ r♠♦t s♥s♥ P

❬❪ ♥ ♥ ❩♥ r ♣♦♥t ♣r♦ss ♦r srtr ①trt♦♥ ♦♥ ♥♦r♠ ♥ ❱P ③♦ ♥

❬❪ ❯ts ♥ ♠r ♣♦♥t ♣r♦ss ♠♦ ♦r ♠t ♣♦♣ tt♦♥♥ ❱P ♦♦r♦ ♣r♥s ❯

❬❪ r♥ P rs ♠♣ r♦ ♥s ♦♥t r♦ ♦♠♣tt♦♥ ♥ ②s♥ ♠♦tr♠♥t♦♥ ♦♠tr

❬❪ st♥s ❲ ♦♥t r♦ s♠♣♥ s♥ r♦ ♥s ♥ tr ♣♣t♦♥s ♦♠tr

❬❪ ♥ ❩❲ ❩ ♥ ♠ s♠♥tt♦♥ ② ♥ t ♠♣s♦♥♠t♦ P

❬❪ rst r♥♥r ❯ ♥s♥ r ♠♣s♦♥ r♦ ♣r♦sss ♦♥♦rt♦♦♥ r♦♣s ♦r ♦t ♣♦s st♠t♦♥ ♦r♥ ♦ ttst P♥♥♥ ♥ ♥r♥

❬❪ ❩ ❩ ♠ ♠♥tt♦♥ ② tr♥ r♦ ♥ ♦♥t r♦ P

❬❪ r♥ss r♥ P Pr ♥s ② rt♦♥ ♥ rrs ♠♣ ♠♠ ♦r♦t r♦♥t♦♥ ♥ ❱ rst♦ ❯

❬❪ ②r rs r♦ ♥ t ♣rst♦♥ ♦ ♠♠s ♠ ♣r♦ss♥♥ ♥tr♥t♦♥ ②♠♣♦s♠ ♦♥ Pr ♥ strt Pr♦ss♥ t♥t ❯

❬❪ ♦♥③③ ♦ ❨ rtt♦♥ str♥ Pr s s♠♣♥ r♦♠ ♦♦rs t♦ t♥ ♥t♦♥ trs ♦r♥ ♦ ♥ r♥♥ sr

❬❪ ♦r② r♠②♥ ❩r r ♦rr t ♦♥t♦rs ❱

♥

❨♥♥ ❱ré ♦r♥t r

❬❪ ♠♣ts② ❱ ❩ssr♠♥ r♥♥ t♦ ♦♥t ♦ts ♥ ♠s ♥ P ❱♥♦r♥

❬❪ ♦s ♦r♦ P sr tt♥ ♥ ♣rs♥ rttr t t② s r♦♠r♥ t ♥ ❱P ♥ r♥s♦ ❯

❬❪ ③s ❩ rst♥ ❱sr ♦♠♦♦r♦ ❱ r ♣♣♥ ♦tr ♦♠♣rt st② ♦ ♥r② ♠♥♠③t♦♥ ♠t♦s ♦r ♠r♦ r♥♦♠s t s♠♦♦t♥sss ♣r♦rs P

❬❪ r♦ ♥♦♠ ♦♥ ♥ ♠ ♥②ss ♣r♥r

❬❪ ❲ss ❨ r♠♥ ❲ ♥ t ♦♣t♠t② ♦ s♦t♦♥s ♦ t ♠①♣r♦t ♣r♦♣t♦♥ ♦rt♠ ♥ rtrr② r♣s r♥s ♦♥ ♥♦r♠t♦♥ ♦r②

❬❪ ♦②♦ ❨ ❱sr ❩ st ♣♣r♦①♠t ♥r② ♠♥♠③t♦♥ r♣ tsP

❬❪ s ♥ t sttst ♥②ss ♦ rt② ♣trs ♦r♥ ♦ t ♦② ttst♦t②

❬❪ ❱♥t ❱ r②♥♥ P ♦♥ ♠t s s♥ ♥r♠♥t ♣①♣♥s♦♥♠♦ ♦♥ t P❯s ♥ ❱ ❳♥ ♥

❬❪ ♠ss♦ s♦r P ♥♠♠ tt♥♥ ❨r ♦♠♣tt♦♥r♠♦r ♦r s♠t♥ ♦rs♥ ♠r♦s♦♣ ♠s t ♣♦♣t♦♥s r♥s♦♥ ♠♥

❬❪ r ♠r ①tr r♣rs♥tt♦♥ ② ♦♠tr ♦ts s♥ ♠♣s♦♥ ♣r♦ss ♥ ❱ s ❯

♥r

♦rs t ♣r③t♦♥ ♦ rs ♠♣ r♦ ♥ ♦♥t r♦ ♦rt♠s

♦♥t♥ts

♥tr♦t♦♥ P♦♥t ♣r♦sss ♦r s♦♥ ♣r♦♠s ♦tt♦♥s ♦♥trt♦♥s

P♦♥t Pr♦ss r♦♥

s♠♣♥ ♣r♦r ♠t♥♦s ♠t♣ ♣rtrt♦♥s ♦♥♥♦r♠ ♣♦♥t strt♦♥s ♠♣r ♦r♠t♦♥

①♣r♠♥ts ♠♣♠♥tt♦♥ P♦♥t ♣r♦sss ♥ P♦♥t ♣r♦sss ♥

♦♥s♦♥

♣♣♥s

ts ♦♥ t ♣♦♣t♦♥ ♦♥t♥ ♠♦ r♦♠ ♠s

ts ♦♥ t ♥♥t♦r ①trt♦♥ ♠♦ r♦♠ ♠s

ts ♦♥ t tr r♦♥t♦♥ ♠♦ r♦♠ ♣♦♥t ♦s

Pr♦r♠♥ tsts ♦♥ ♦♥♥t♦♥ r♦ ♥♦♠ ♠♦s

t♦♥ rsts ♥ ♦♠♣rs♦♥s

♥

RESEARCH CENTRE

SOPHIA ANTIPOLIS – MÉDITERRANÉE

2004 route des Lucioles - BP 93

06902 Sophia Antipolis Cedex

Publisher

Inria

Domaine de Voluceau - Rocquencourt

BP 105 - 78153 Le Chesnay Cedex

inria.fr

ISSN 0249-6399