An Auction Algorithm for Shortest Paths PDF

6 Apr 2007 GOOD I. J. (1965) The Estimation of Probabilities: An Essay on Modern Bayesian Methods. Cambridge

Dimitri P. Bertsekas a and David A. Castanon b 1. Introduction

Assignment problem auction algorithm; synchronous and asynchronous Linear Network Optimization: Algorithms and Codes (MIT Press

A Distributed Algorithm for the Assignment Problem

This paper describes a new algorithm for solving the classical assignment in developing distributed algorithms for optimization and other problems.

Mathematical Equivalence of the Auction Algorithm for Assignment

2 Laboratory for Information and Decision Systems M.I.T

A FORWARD/REVERSE AUCTION ALGORITHM FOR

2 Department of Electrical Engineering and Computer Science M. I. T.

An Auction Algorithm for Shortest Paths

AN AUCTION ALGORITHM FOR SHORTEST PATHS*. DIMITRI P. BERTSEKAS'. Abstract. A new and simple algorithm for finding shortest paths in a directed graph is

Gaussian mixture models and the EM algorithm

Expectation-Maximization (EM) algorithm first for the specific case of GMMs

Auction Algorithms

Auction Algorithms. Dimitri P. Bertsekas bertsekas@lids.mit.edu. Laboratory for Information and Decision Systems. Massachusetts Institute of Technology.

D.P. Bertsekas 1. INTRODUf;rION Relaxation methods for optimal

The algorithm can also be inter- preted as a Jacobi -like relaxation method for solving a dual problem. Its. (sequential) worst -case complexity for a

Rollout Algorithms for Discrete Optimization: A Survey

dimitrib@mit.edu This chapter discusses rollout algorithms a sequential approach to ... A rollout algorithm starts from some given heuristic.

ANAUCTIONALGORITHMFORSHORTESTPATHS*

DIMITRIP.BERTSEKAS'

algorithmterminates. processusingsimpledatastructures.

425Downloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

426DIMITRIP.BERTSEKAS

algorithm'sperformance. whichisverysimilartotheoneprovidedhere. thepresentpaper.

7containscomputationalresults.

formforthesingleoriginandsingledestinationcase,andwedeferthediscussionofDownloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

AUCTIONFORSHORTESTPATHS427

ofnodes. ofitsarclengths. (1,i,i_,ik-). that (la)p,<-aij+pV(i,j), weimplicitlyassumethatPissimple.) thedefaultpair

P(1),p,=O

apair.Downloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

428DIMITRIP.BERTSEKAS

TYPICALITERATION

LetbetheterminalnodeofP.If

(2)pStep1:(Contractpath).Set

Pi:=min{ao+pj},i,j)4

(4) andif1,contractP.Gotothenextiteration.

Step2:(Extendpath).ExtendPbynodejiwhere

jiargmin{ai+p}.(i,j)e nextiteration. typicalwhentheinitialpricesareallzero. theshortestdistancefromtoj. min{aj+p},(5)p,, cf.condition(4)).

Supposenextthat

p/5imin{aij+p},(i,j)Downloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

AUCTIONFORSHORTESTPATHS429

P2=2 2pl

Oration

P3=2

Iteration

PathPPricevectorpTypeofaction

kP.Thiscompletestheinductionproof. lastassertionoftheproposition. fromtheorigintotheterminalnodeofP. algorithm, pl-pj<=DjVjcdf, whilebyProposition2,wehave

Pl--PDiforallinP.Downloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

430DIMITRIP.BERTSEKAS

Itfollowsthat

Di+pipt<-_Dj+pjptlPandj.

mayviewthequantity

Dj+p-pt

algorithm'spathonlyifD+p-p,isminimal. pationinashortestpathfrom1tot. nodes. (7)pOinitialpriceofnodei. (8)d,D,+p

Letusindextheiterationsby1,2,..,andlet

(b)Ifk2,whilethelengthofPisPl_pO. algorithmandkstaybounded.Wenextclaimthatpmuststayboundedforalli.Indeed,inordertoDownloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

AUCTIONFORSHORTESTPATHS431

toi,implyingthatp-->oo.U (10)#-{ild,<=d,}; pathsgeneratedbythealgorithm.Letusdenote (11)!numberofnodesin#, andletusalsodenotebyEtheproduct (13)E=I.G. runningtimeofthealgorithmisO(E(D,+p-p)).

1),itfollowsthat

(14)pi-PiandpriceincreasesisO(E(D,+pOplo)).Downloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

432DIMITRIP.BERTSEKAS

NotethatwehaveDt<-_hL,where

(15)Lmaxaij,(i,j) (17)O(EhL).

OriginDestination

roughlyequaltotwo)by n,-l+(2n,1),,q,i problemofFig.1,theaboveestimateisexact. andthedirectionofeacharc.

assumethatallarclengthsarenonnegative.WeintroduceaversionofthealgorithmDownloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

AUCTIONFORSHORTESTPATHS433

lengthsandstartingprices.Let (18)K=[logLJ+l andfork=1,...,K,define (19)ao(k)=2K_kV(i,j)e binaryrepresentationofa0.Define nextsection. (21)p(k+l)=2p*(k)VieW, p,.():ovi. (22)O<-ao(k+l)-2ao(k)<-IV(i,j)s,

CScondition

(23)p(k+l)<=py(k+l)+ao(k+l)V(i,j)s allthesubproblems.

Itcanbeseenusing(22)that

D,(k+1)=<2D,(k)+h(k),

andinviewof(21),weobtain k,k=k+l,.,K,is

(25)O(E(k)h(k)),Downloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

434DIMITRIP.BERTSEKAS

timeofthealgorithmforsubproblemkis (26)O(E(k)D,(k)), aij()<2 wehave (27)D,(/7)<2h(/7). thealgorithmis (28)02E(/)h(/)+E(k)h(k)k=/+! itfollowsthat

Piaiji+Pj,,

min{a+pa},aiji+PJii,j)e.g beingtheterminalnode,wecalculate min{a+p},i,j)a togetherwithanarc(i,j)suchthat

jiargmin{a+p}.i,j)eagDownloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

AUCTIONFORSHORTESTPATHS435

terminalnodeofthepath. computethe"bestneighbor" jiargmin{ai+p}i,j.l kargmin{a+pj},(i,j)4,jJi andthecorresponding"secondbestlevel"

Wiaitqh-Pkg.

theminimumintheexpression min(aj+p},i,j.4 neighbor," kargmin{a;+p;}(i,j)

P<--ai+ffiVi,j4,

thateachnodeexceptfortheoriginhasatleastoneincomingarc.Downloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

436DIMITRIP.BERTSEKAS

g(t),pi=OVi, ifallarclengthsarenonnegative.

TYPICALITERATIONOFTHEREVERSEALGORITHM

LetjbethestartingnodeofR.If

pj>max{Piaij},(i,j). gotoStep1;elsegotoStep2.

Step1:(Contractpath).Set

pg:=max{paig},i,j) iteration.

R,precedingj),where

/argmax{Pi-aij}.i,j),4 iteration.

COMBINEDALGORITHM

anincreaseoftheoriginpricePl.GotoStep2. ofthedestinationpricept.GotoStep1.

Step2mustcontainonlyafinitenumberofiterationsoftheforwardandthereverseDownloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

AUCTIONFORSHORTESTPATHS437

counterexample. p.Initially, ={(i,j)6g[a0Thetypicaliterationisasfollows:

TYPICALPREPROCESSINGITERATION

fromgandgotoStep2.

Step2:(Addaffectedarcsto).Ifp>ag+pg,set

Pi:=aij+p

Wehavethefollowingproposition.

withapricevectorpsatisfying (29)p<=ao+pgVi,j6gwithiCt. wehave

{(i,j)6lpi>aq+pj,iSt}.Downloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

438DIMITRIP.BERTSEKAS

distancefromktoifkt,Pk*=/,ifk=t, andletrbeasufficientlylargescalarsothat p,>-p*r (30)Pk>=Prlkt. have ai+p>ao+pj.*r>min{a,+p}rp*r,i,m),. theproof. ={(i,j)ClaoALTERNATIVEPREPROCESSIN6ITERATION from'andgotoStep2.

Pj:--Piai

thereforefollowsfromProposition6. ofiterationswithapricevectorpsatisfying

P,<=aij+piV(i,j)ewithjl.Downloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

AUCTIONFORSHORTESTPATHS439

[aij];(k)=/2K_k[V(i,j),,

1-- and (31)p(k+l)<=p)?(k+l)+ij(k+l)+ll(i,j), discussesparalleltwo-sidedalgorithms. handlethepathofanotherorigin.
experimentallyonasharedmemorymachine.Downloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

440DIMITRIP.BERTSEKAS

Pi:=min{au+pj};i,,sd
implementation. (32)Pin:=max{p,,p}Vn. pricevectorpsatisfiesthecondition
Pm= p<=a,,+p',,,p<--_a,,+pV(m,n).(33) Then, (34) and (35) and max{p'm,p}<--_a,,,.+max{p',,,p}V(m,n)M, min{pi,.,,,p}Proof.From(33),wehave
pPJmamn"k-max{pi,,,p}V(m,n)4.

time-consumingifnovectorprocessinghardwareisavailableattheprocessors.Downloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

AUCTIONFORSHORTESTPATHS441

condition min{cj+p}(36)Ciji-[-"pj(i,J) jargmin{ci+p};i,j), andthen"

PJi+Wil.)i

price), vimin{%+pj},i,j.s wi=min{ci+p}.i,j.sC,j#ji anobject.

difference;whiletheauctionalgorithmisguaranteedtoterminateinafinitenumberDownloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

442DIMITRIP.BERTSEKAS

teesoptimalityofthefinalassignment. t=4 assignmentcostsareshownnexttothearcs. psatisfyingtheCScondition(la),i.e., (37)pi<=aij+pl(i,j)g, andthepartialassignment (i',i)ti#l,t. assignedarcs(i',i)iszero. min{alj+p},(1,j)d

changingthepricep,.Downloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

AUCTIONFORSHORTESTPATHS443

thefollowingcanbeverifiedbyinduction: (38)(1',il),(i,i2),''',(i-1,ik), togetherwiththeadditionalarcs i',i)foril,ik,t, min{c./+p}p(/,j)

Piaiji"]"

theshortestpathalgorithm. (LNF)minimize,aoxo(i,j)

(40)0<--xoV(i,j),91,Downloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

444DIMITRIP.BERTSEKAS

where

S1,S=-1,

si=Ofi#l,t, andisthegivendestination. primalcostisequaltotheoptimaldualcost.

Xij__flifandjaresuccessivenodesinP,

[0otherwise. complementaryslacknessconditions p0pao+p.li,j. changeindualcost.

wheneachofthereversepathshavemettheforwardpath.Downloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

AUCTIONFORSHORTESTPATHS445

thanTWO_TREE_SHEAP,asshowninTable1. possible).

themeritsofouralgorithm.WealsonotethattheideasinthispaperarenewandDownloaded 10/30/14 to 128.31.7.81. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

446DIMITRIP.BERTSEKAS

1,1000].

NAAUCTION_SPSHEAPTWO_TREE_SHEAP

1,0004,0000.0330.2500.033

1,00010,0000.0500.2000.133

2,0008,0000.0170.0170.017

2,00020,0000.0670.8670.150

3,00012,0000.0670.9830.100

3,00030,0000.0331.1170.100

4,00016,0000.0671.2330.100

4,00040,0000.0330.3830.100

5,00020,0000.0501.3830.083

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[PDF] An Auction Algorithm for Shortest Paths