Project: ArangoDB (App Definition and Development / Database)
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E/#0Ecien t Algorithms for Optim um Cycle Mean and Optim um Cost to Time RatioProblemsAli Dasdan Sandy S/. Irani and Ra jesh K/. GuptaDept/. of Computer Science Dept/. of Information and Computer ScienceUniv ersit y of Illinois/, Urbana/, IL /6/1/8/0/1 Univ ersit y of California/, Irvine/, CA /9/2/6/9/7dasdan/@cs/.uiuc/.edu f irani/,rgupta g /@ics/.uci/.ed uAbstractThe goal of this pap er is to iden tify the most e/#0Ecien t al/-gorithms for the optim um mean cycle and optim um costto time ratio problems and compare them with the p opu/-lar ones in the CAD comm unit y /. These problems ha v en u/-merous imp ortan t application s in CAD/, graph theory /, dis/-crete ev en t system theory /, and man ufacturing systems/. Inparticular/, they are fundamen tal to the p erformance anal/-ysis of digital systems suc h as sync hronous/, async hronous/,data/#0Do w/, and em b edded real/-time systems/. F or instance/,algorithms for these problems are used to compute the cyclep erio d of an y cyclic digital system/. Without loss of gen/-eralit y /,w e discuss these algorithms in the con text of theminim um mean cycle problem /#28MCMP/#29/. W e p erformed acomprehensiv e exp erimen tal study of ten leading algorithmsfor MCMP /.W e programmed these algorithms uniformly ande/#0Ecien tly /.W e systematically compared them on a test suitecomp osed of random graphs as w ell as b enc hmark circuits/.Ab o v e all/, our results pro vide imp ortan t insigh ti n to the p er/-formance of these algorithms in practice/. One of the mostsurprising results of this pap er is that Ho w ard/'s algorithm/,kno wn primarily in the sto c hastic con trol comm unit y /,i s b yfar the fastest algorithm on our test suite although the onlykno wn b ound on its running time is exp onen tial/. W e pro videt w o stronger b ounds on its running time/./1 Intro ductionConsider a digraph G /=/#28 V/; E /#29 with n no des and m arcs/.Asso ciate with eac h arc e in E t w on um b ers/: a w eigh t /#28orcost/#29 w /#28 e /#29 and a transit time t /#28 e /#29/. The w eigh t and tran/-sit time of a path in G is equal to the sum of the w eigh tsand transit times of the arcs on the path/, resp ectiv ely /. Thelength of a path is equal to the n um b er of arcs on the path/.Let w /#28 C /#29/, t /#28 C /#29/, and j C j denote the w eigh t/, transit time/, andlength of a cycle C in G /.The /#28cycle/#29 r atio /#1A /#28 C /#29 and the /#28cycle/#29 me an /#15 /#28 C /#29 of cycleC are de/#0Cned as/#1A /#28 C /#29/=
w /#28 C /#29t /#28 C /#29
/;t /#28 C /#29 /#3E /0 /; and /#15 /#28 C /#29/=
w /#28 C /#29j C j
/;Published in Pro c/. /3/6th Design Automation Conf/./#28D A C/#29/, pp/. /3/7/-/4/2/, Jun/. /1/9/9/9/.
resp ectiv ely /. Note that /#1A /#28 C /#29 giv es the a v erage w eigh tp e rtransit time/, and /#15 /#28 C /#29 giv es the a v erage arc w eigh to n C /.The cycle ratio is historically called the cost to time ratio/.The mean of C is a sp ecial case of its ratio in that its meanis obtained from its ratio b y setting the transit time of ev eryarc on C to unit y /.The minimum cycle r atio /#28MCR/#29 /#1A
/#03and the minim umcycle mean /#28MCM/#29 /#15
/#03of G are de/#0Cned as/#1A
/#03/= minC /2 G
f /#1A /#28 C /#29 g and /#15
/#03/= minC /2 G
f /#15 /#28 C /#29 g /;resp ectiv ely /. In b oth cases/, C ranges o v er all the cycles inG /. The resp ectiv e problems are called the minimum cycler atio pr oblem /#28MCRP/#29 and the minimum me an cycle pr ob/-lem /#28MCMP/#29
/1/. The de/#0Cnitions of the maxim um v ersions ofb oth of these problems are analogous/./1/./1 Applications in CADThe applicatio ns of b oth MCRP and MCMP are imp ortan tand n umerous/. See /#5B/1/2/#5D for the applications in graph the/-ory /. Our fo cus is on the applications in the CAD of digitalsystems/. These problems ha v e fundamental imp ortance tothe p erformance analysis of discrete ev en t systems /#5B/3/#5D/, whic hcan also mo del digital systems/. These problems are appli/-cable to the p erformance analysis of suc h digital systems assync hronous /#5B/2/3/#5D/, async hronous /#5B/4/#5D/, DSP /#5B/1/4/#5D/, or em b eddedreal/-time /#5B/1/8/#5D/. Simply put/, the algorithms for these problemsare essen tial to ols to /#0Cnd the cycle p erio d of a giv en cyclicdiscrete ev en t system/. Once determined/, the cycle p erio dis used to describ e the b eha vior of the system analyticall yo v er an in/#0Cnite time p erio d/. F or instance/, the algorithmsfor these problems are used to compute the iteration b oundof a data/#0Do w graph /#5B/1/4/#5D/, the time separation b et w een ev en to ccurrences in a cyclic ev en t graph /#5B/1/3/#5D/, and the optimalclo c ks c hedules for circuits /#5B/2/2 /#5D/./1/./2 Related w o rkThere are man y algorithms prop osed for b oth MCRP andMCMP /.W e giv e a comprehensiv e classi/#0Ccation of the fastestand the most common ones in T able /1/. References to a fewold algorithms can b e found in /#5B/1/2/#5D/. Note that as MCMP isa sp ecial case of MCRP /,a n y algorithm for the latter prob/-lem can b e used to solv e the former problem/. Con v ersely /,it is also p ossible to solv e MCRP using an algorithm for/1W ei n ten tionally use MCMP to refer to /#5Cthe minim um mean cycleproblem/"/./1T able /1 /: Minim um mean cycle and minim um cost to time ratio algorithms for a graph G with n no des and marcs/. /#28 W /, the maxim um arc w eigh t/; T /, the total transit time of G /; N is the pro duct of the out/-degrees of allthe no des in G /./#29Minim um mean cycle algorithmsName Source Y ear Running time Result Complexit y/1 DG Dasdan /& Gupta /#5B/8/#5D /1/9/9/7 O /#28 nm /#29 Exact P olynomial/2 HO Hartmann /& Orlin /#5B/1/2 /#5D /1/9/9/3 O /#28 nm /#29 Exact P olynomial/3 Karp/'s Karp /#5B/1/5/#5D /1/9/7/8 /#02/#28 nm /#29 Exact P olynomial/4 Hartmann /& Orlin /#5B/1/2 /#5D /1/9/9/3 O /#28 nm /+ n
/2lg n /#29 Exact P olynomial/5 YTO Y oung/, T arjan/, /& Orlin /#5B/2/5 /#5D /1/9/9/1 O /#28 nm /+ n
/2lg n /#29 Exact P olynomial/6 Karp /& Orlin /#5B/1/6 /#5D /1/9/8/1 /#02/#28 n
/3/#29 Exact P olynomial/7 K O Karp /& Orlin /#5B/1/6 /#5D /1/9/8/1 O /#28 nm lg n /#29 Exact P olynomial/8 O A/1 Orlin /& Ah uja /#5B/2/1 /#5D /1/9/9/2 O /#28
pn m lg /#28 nW /#29/#29 Appro ximate Pseudop oly /./9 O A/2 Orlin /& Ah uja /#5B/2/1 /#5D /1/9/9/2 O /#28
pn m lg
/2/#28 nW /#29/#29 Appro ximate Pseudop oly /./1/0 Cuninghame/-Green /& Yixun /#5B/7/#5D /1/9/9/6 O /#28 n
/4/#29 Exact P olynomialMinim um cost to time ratio algorithmsName Source Y ear Running time Result Complexit y/1/1 Burns/' Burns /#5B/4/#5D /1/9/9/1 O /#28 n
/2m /#29 Exact P olynomial/1/2 Megiddo /#5B/1/9 /#5D /1/9/7/9 O /#28 n
/2m lg n /#29 Exact P olynomial/1/3 Hartmann /& Orlin /#5B/1/2 /#5D /1/9/9/3 O /#28 Tm /#29 Exact Pseudop oly /./1/4 La wler/'s La wler /#5B/1/7/#5D /1/9/7/6 O /#28 nm lg/#28 nW /#29/#29 Appro ximate Pseudop oly /./1/5 I t o/&P arhi /#5B/1/4/#5D /1/9/9/5 O /#28 Tm /+ T
/3/#29 Exact Pseudop oly /./1/6 Gerez et al/. /#5B/1/0/#5D /1/9/9/2 O /#28 Tm /+ T
/3lg T /#29 Appro ximate Pseudop oly /./1/7 Gerez et al/. /#5B/1/0/#5D /1/9/9/2 O /#28 Tm /+ T
/4/#29 Exact Pseudop oly /./1/8 Ho w ard/'s Co c het/-T errasson et al/. /#5B/6/#5D /1/9/9/7 O /#28 Nm /#29 Exact Pseudop oly /.MCMP /#5B/1/1/#5D/. Th us/, w e will fo cus only on MCMP in thispap er/.In T able /1/, the p olynomial and pseudop olyno mia l algo/-rithms are resp ectiv ely ordered according to their w orst/-caserunning times/. Those with the same running time are pre/-sen ted in alphab etical order of their in v en tors/' names/. Somereferences are cited more than once b ecause they con tainmore than one algorithm/. Some algorithms cannot pro ducethe exact MCM or MCR/, in whic h case they are said toreturn appro ximate results/. The amoun t of error that canb e tolerated in the appro ximate results can usually b e con/-trolled/. This amoun t of error is denoted b y /#0F /, and called theprecision of the algorithm/./1/./3 Motivations/, contributions/, and metho dologyOur main goal in this pap er is to iden tify the most e/#0Ecien talgorithms for MCMP and MCRP and compare them withthe curren t practice in the CAD comm unit y for solving sim/-ilar problems/. W eh a v e realized this goal with the follo wingmotiv ations and con tributions/:/#28/1/#29 Despite the imp ortance of MCMP and MCRP /, mostof the w ork in the CAD comm unit y that deals with theseproblems are not a w are of most of the algorithms in T a/-ble /1/. The most p opular algorithms in the CAD comm u/-nit y are Karp/'s algorithm /#5B/1/5/#5D/, La wler/'s algorithm /#5B/1/7/#5D/, andto some exten t/, Burns/' algorithm /#5B/4/#5D/. W e sho w that thereare far more e/#0Ecien t algorithms than these p opular ones/.In particular/, w e sho w that Ho w ard/'s algorithm is signi/#0C/-can tly faster than all the others/. In addition/, w e pro vide animpro v ed v ersion of this algorithm as w ell as t w o strongerb ounds on its running time/./#28/2/#29 Most of the earlier w ork do es not presen ta n y exp er/-imen tal analysis of ev en the algorithms that they in tro duce/.There has not b een a clear understanding of the p erformanceof an y of the algorithms studied in this pap er although the/-oretical b ounds on their running times ha v e b een pro v en/.Finding more e/#0Ecien t implemen tation of these algorithmsis v ery imp ortan t b ecause their applicati ons require that
they b e run man y times/, e/.g/./, see /#5B/8/#5D/. This pap er is the /#0Crststudy that systematically compares their p erformance and
pro vides a great deal of insigh ti n to their b eha vior/. W e alsopro vide some implemen tational impro v emen ts for most ofthe algorithms/.In this study /,w e fo cus on the ten leading MCM algo/-rithms and the MCM v ersions of the MCR algorithms fromT able /1/, all of whic h are named in the table/. The remain/-ing algorithms in this table are not included in our studyb ecause they are v ery similar to the c hosen ones/. W e imple/-men ted eac h algorithm in a uniform and e/#0Ecien t manner/.W e tested them on a series of random graphs/, obtained us/-ing one generator from /#5B/5/#5D/, and real b enc hmark circuits/, ob/-tained from logic syn thesis b enc hmarks/. The running timeas w ell as represen tativ e op eration coun ts/, as adv o cated in/#5B/2/#5D/, are measured and compared/. W en o w giv e a review ofthe these algorithms and then presen t the exp erimen tal re/-sults and our observ ations/./2 Minimum Mean Cycle Algo rithmsW e /#0Crst giv e a di/#0Beren t form ulation of MCMP that is moreuseful to explain the b eha vior of the minim um mean cyclealgorithms/.The minim um cycle mean /#15
/#03of a graph G /=/#28 V/; E /#29 canb e de/#0Cned as the optim um v alue of /#15 in the follo wing linearprogram/:max /#15 s/.t/. d /#28 v /#29 /, d /#28 u /#29 /#14 w /#28 u/; v /#29 /, /#15/; /8 /#28 u/; v /#29 /2 E/; /#28/1/#29where d /#28 v /#29 is called the distanc e /#28or the no de p oten tial/#29 ofv /. The maxim um is c hosen o v er all v alues for d /#28 /#01 /#29/. Whenthe inequaliti es are all satis/#0Ced/, d /#28 v /#29 is equal to the w eigh tof the shortest path from s to v in G when /#15
/#03is subtractedfrom ev ery arc w eigh t/. The no de s is arbitrarily c hosen asthe sour c e in adv ance/. Let G/#15
denote the graph obtainedfrom G b y subtracting /#15 from the w eigh to f e v ery arc/. Theminim um cycle mean /#15
/#03is the largest v alue of /#15 for whic hG/#15
has no negativ e cycles/./2W es a y that an arc /#28 u/; v /#29 /2 E is critic al if d /#28 v /#29 /, d /#28 u /#29/=w /#28 u/; v /#29 /, /#15 /, whic hw e refer to as the critic ality criterion /.W es a y that a no de is critic al if it is adjacen t to a criticalarc/, and that a graph is critic al if all of its arcs are critical/.The critical subgraph of G/#15
/#03con tains all the minim um meancycles of G /, as implied b y Equation /1/. The critical subgraphis imp ortan t to compute b ecause the critical subgraph of agraph G con tains all the arcs and no des that determine thep erformance of the system mo deled b y G /. After running anMCM or MCR algorithm on G /, the critical subgraph of Gcan easily b e computed using its de/#0Cnition/. As a result/, w epresen t eac h algorithm in the con text of computing /#15
/#03only /.W e assume that the input graph G to the algorithm incon text is cyclic and strongly connected/. This assumptionsimpli/#0Ces most of the algorithms and generally impro v estheir running times in practice/. Note that if G is not stronglyconnected/, its minim um cycle mean can b e found easily/: /#0Crstpartition G in to its strongly connected comp onen ts/, run thealgorithm on eac h strongly connected comp onen t/, and thentak e as the minim um cycle mean of G the minim um of thecycle means returned b y the algorithm/. This is the w a yw eimplemen ted all of the algorithms/.W en o w review the minim um mean cycle algorithms inour study /. More detailed discussion of these algorithms to/-gether with their pseudo co de is giv en in /#5B/9 /#5D/./2/./1 Burns/' algo rithmBurns/' algorithm /#5B/4/#5D is actually the minim um mean cyclev ersion of the original Burns/' algorithm for MCRP /.W eh a v edisco v ered that the algorithm in /#5B/7/#5D is iden tical to Burns/'algorithm/. Burns/' algorithm is based on linear program/-ming/. It is an iterativ e algorithm constructed b y apply/-ing the primal/-dual metho d/. It solv es the ab o v e linear pro/-gram /#28Equation /1/#29 and its dual sim ultaneousl y /. In essence/,the b eha vior of Burns/' algorithm is v ery similar to that ofthe parametric shortest path algorithms b elo w suc ha st h eK O algorithm/: The K O algorithm impro v es up on an ini/-tial acyclic critical subgraph of G un til the critical subgraphb ecomes cyclic/, at whic h p oin t the minim um cycle mean isfound/. Burns/' algorithm also op erates on the critical sub/-graph and terminates when it b ecomes cyclic/. It di/#0Bers fromthe K O algorithm in that at ev ery iteration/, it reconstructsthe critical subgraph from scratc h/./2/./2 Ka rp/'s algo rithm and its va riantsDe/#0Cne Dk
/#28 v /#29 to b e the w eigh t of the shortest path of length kfrom s /, the source/, to v /;i f n os u c h path exists/, then Dk
/#28 v /#29/=/+ /1 /. Karp/'s algorithm /#5B/1/5/#5D is based on his observ ation that/#15
/#03/= minv /2 V
max/0 /#14 k /#14 n /, /1
Dn
/#28 v /#29 /, Dk
/#28 v /#29n /, k
/;whic h is called Karp/'s theorem/. Karp/'s algorithm computeseac h Dk
/#28 v /#29b y the recurrenceDk
/#28 v /#29/= min/#28 u/;v /#29 /2 E
f Dk /, /1
/#28 u /#29/+ w /#28 u/; v /#29 g /;k /=/1 /; /2 /;/:/:/:/;n /;D/0
/#28 s /#29/= /0 /;D/0
/#28 v /#29/= /+ /1 /;v /6/= s/:Note that d /#28 v /#29 and D /#28 v /#29 are related to eac h other b y theequation d /#28 v /#29 /= min/0 /#14 k /#14 n /, /1
f Dk
/#28 v /#29 /, k/#15 g /. As observ ed in/#5B/8/, /1/2 /, /2/5/#5D/, this recurrence/, whic h is not computed recur/-siv ely /, mak es the b est and w orst cases of Karp/'s algorithmthe same/, whic hi sw h y it runs in /#02/#28 nm /#29/.W eh a v e three impro v emen ts on Karp/'s algorithm/: theDG algorithm/, the HO algorithm/, and the Karp/2 algorithm/.
The DG algorithm /#5B/8/#5D impro v es up on Karp/'s algorithm b yeliminatin g unnecessary w ork in tro duced b y the ab o v e re/-currence/. It w orks in a breadth/-/#0Crst manner in that startingfrom the source/, it visits the successors of no des rather thantheir predecessors/, as done in the recurrence/. This pro cesscreates an unfolding of G /, and when the algorithm is imple/-men ted using link ed lists/, its running time b ecomes equal tothe size of the /#5Cunfolded/" graph/. Dep ending on the struc/-ture of G /, the running time ranges from /#02/#28 m /#29t o O /#28 mn /#29/.The HO algorithm /#5B/1/2/#5D also impro v es up on Karp/'s algo/-rithm/. It helps to terminate Karp/'s algorithm early with/-out c hanging its structure/, i/.e/./, it still uses the ab o v e re/-currence/. It is based on the observ ation that man y of theshortest paths computed b y Karp/'s algorithm will con taincycles/. If one of these cycles is critical/, then the minim umcycle mean is found/, whic h su/#0Eces to terminate the algo/-rithm/. The HO algorithm essen tially c hec ks the critical/-it y of eac h cycle on the shortest paths computed/. If theearly termination is not p ossible/, this algorithm can add ano v erhead of O /#28 n
/2/+ m lg n /#29 in total to the running time ofKarp/'s algorithm although it do es not c hange the runningtime asymptotically /.The Karp/2 algorithm is a space e/#0Ecien tv ersion of Karp/'salgorithm
/2/. Karp/'s algorithm tak es up /#02/#28 n
/2/#29 space in orderto store the D /-v alues/. The Karp/2 algorithm reduces thisspace requiremen tt o /#02 /#28 n /#29/. The Karp/2 algorithm p erformst w o passes/. In the /#0Crst pass/, it computes Dn
/#28 v /#29 for eac hno de v without storing Dk
/#28 v /#29 for k/#3C n /. In the second pass/,it computes the fraction in Karp/'s theorem as it computeseac h Dk
/#28 v /#29/, k/#3Cn /. The DG and HO algorithms also su/#0Berfrom this large space complexit y problem/. F ortunately /, thetec hnique used in the Karp/2 algorithm is also applicable tothese v arian ts/./2/./3 P a rametric sho rtest path algo rithmsThe K O algorithm /#5B/1/6/#5D and the YTO algorithm /#5B/2/5/#5D are inthe category of parametric shortest path algorithms/. TheYTO algorithm is essen tially an e/#0Ecien t implemen tation ofthe K O algorithm/. These algorithms are based on the ob/-serv ation that the minim um cycle mean /#15
/#03is the largest /#15suc h that G/#15
do es not ha v ea n y negativ e cycles/. Th us/, thesealgorithms start with /#15 /= /,/1 and alw a ys main tain a tree ofshortest paths to a source no de s /. These algorithms c hange/#15 incremen tally so that the shortest path tree c hanges b yone arc in eac h iteration/. When a cycle of w eigh t zero isdetected in G/#15
/, that cycle is the cycle with the minim ummean/./2/./4 La wler/'s algo rithmLa wler/'s algorithm /#5B/1/7/#5D is based on the same observ ation asthe parametric shortest path algorithms/. It also uses the fact
that /#15
/#03of G lies b et w een the minim um and the maxim umarc w eigh ts in G /.L a wler/'s algorithm do es a binary searc ho v er the p ossible v alues of /#15
/#03and c hec ks for a negativ e cyclein G/#15
ev ery iteration/. If one is found/, then the c hosen /#15 isto o large so it is decreased/; if not/, it is to o small so it isincreased/. La wler/'s algorithm terminates when the in terv alfor the p ossible v alues of /#15
/#03b ecomes to o small/. The size ofthat in terv al/, /#0F /, determines the precision of the algorithm/./2Suggested b y S/. Gaub ert of INRIA/, F rance/./3Input /: A strongly connected digraph G /=/#28 V/; E /#29/.Output /: The minim um cycle mean /#15
/#03of G /./1 for eac hn o d e u /2 V do d /#28 u /#29 /#20 /+ /1/2 for eac h arc /#28 u/; v /#29 /2 E do/3 if /#28 w /#28 u/; v /#29 /#3Cd /#28 u /#29/#29 then/4 d /#28 u /#29 /#20 w /#28 u/; v /#29/; /#19 /#28 u /#29 /#20 v /#2F/* /#19 is the p olicy /*/#2F/5 while /#28true/#29 do /#2F/* Main lo op /- Iterate /*/#2F/6 E/#19
/#20f /#28 u/; /#19 /#28 u /#29/#29 /2 E g /#2F/* Find the set E/#19
of p olicy arcs /*/#2F/#2F/* Compute /#15 in the p olicy graph G/#19
/*/#2F/7 Examine ev ery cycle in G/#19
/=/#28 V/; E/#19
/#29/./8 Let C b e the cycle with the smallest mean in G/#19
/./9 Let /#15 /#20 w /#28 C /#29 /= j C j/1/0 Select an arbitrary no de s /2 C /./#2F/* Compute the no de distances using the rev erse BFS /*/#2F/1/1 if /#28there is a path from v to s in G/#19
/#29 then/1/2 d /#28 v /#29 /#20 d /#28 /#19 /#28 v /#29/#29 /+ w /#28 v/; /#19 /#28 v /#29/#29 /, /#15/#2F/* Impro v e the no de distances /*/#2F/1/3 impr ov ed /#20 f alse/1/4 for eac h arc /#28 u/; v /#29 /2 E do/1/5 /#0E /#28 u /#29 /#20 d /#28 u /#29 /, /#28 d /#28 v /#29/+ w /#28 u/; v /#29 /, /#15 /#29/1/6 if /#28 /#0E /#28 u /#29 /#3E /0/#29 then/1/7 if /#28 /#0E /#28 u /#29 /#3E/#0F /#29 then impr ov ed /#20 tr ue/1/8 d /#28 u /#29 /#20 d /#28 v /#29/+ w /#28 u/; v /#29 /, /#15 /; /#19 /#28 u /#29 /#20 v/#2F/* If not m uc h impro v emen t in the no de distances/, exit /*/#2F/1/9 if /#28 N O T impr ov ed /#29 then return /#15Figure /1 /: An impro v ed v ersion of Ho w ard/'s mini/-m um mean cycle algorithm/./2/./5 Ho w a rd/'s Algo rithmAn impro v ed v ersion of Ho w ard/'s algorithm /#5B/6/#5D is giv en inFigure /1/. It is similar to the st yle of the parametric short/-est path algorithms except that it starts with a large /#15 anddecreases /#15 un til the shortest paths in G/#15
are w ell de/#0Cned/.It computes /#15 on the p olicy gr aph whic h is simply a sub/-graph of G suc h that the out/-degree of eac h no de is exactlyone/. Note that the p olicy graph has n arcs/. F or a giv en /#15 /,the algorithm attempts to /#0Cnd the shortest paths from ev/-ery no de to an c hosen no de s using the breadth/-/#0Crst searc h/#28BFS/#29 algorithm/. In doing so/, it either disco v ers that theshortest paths are w ell de/#0Cned in whic h case the correct /#15has b een found or it disco v ers a negativ e cycle in G/#15
/. In thelatter case/, the negativ e cycle has a smaller mean w eigh tthan the curren t /#15 /. In this case/, /#15 can b e up dated to themean w eigh t of the new cycle and the pro cess con tin ues/.The b eaut yo fH o w ard/'s algorithm is that eac h iterationis extremely simple and requires only /#02/#28 m /#29 time/. Mean/-while/, although it ensures that the v alue of /#15 is non/-increasingfrom one iteration to another/, it usually manages to mak esigni/#0Ccan t progress in decreasing the v alue of /#15 in a v ery fewn um b er of iterations/. In /#5B/9/#5D/, w eh a v e pro v ed that /#15 decreasesb y at least /#0F/=n at least ev ery n iterations of the main lo opof Ho w ard/'s algorithm/, where /#0F is the precision of the algo/-rithm/. This result leads to t w o stronger b ounds on the run/-ning time of Ho w ard/'s algorithm/: /#28/1/#29 its running time is atmost O /#28 nm/#0B /#29/, where /#0B is the n um b er of simple cycles in G /,or /#28/2/#29 its running time is at most O /#28 n
/2m /#28 wmax
/, wmin
/#29 /=/#0F /#29/,where wmax
and wmin
are the maxim um and minim um arcw eigh ts in G /.
/2/./6 Scaling algo rithmsThe O A/1 and O A/2 algorithms /#5B/2/1/#5D are in this category /. Theyassume that the arc w eigh ts are in tegers b ounded b y W /.I fW is p olynomial in n /, then these algorithms are asymp/-totically the fastest algorithms/. The O A/2 algorithm appliesscaling to a h ybrid v ersion of an assignmen t algorithm/, calledthe auction algorithm/, and the successiv e shortest path algo/-rithm/. It uses an appro ximate binary searc h tec hnique/. TheO A/1 algorithm is the same as the O A/2 algorithm except thatit do es not use the successiv e shortest path algorithm/./3 Exp erimental F ramew o rkW e programmed the algorithms in C/+/+ using the LED Alibrary v ersion /3/./4/./1/. This library is a template library fore/#0Ecien t data t yp es and algorithms /#5B/2/0/#5D/. In order to ensureuniformit y of implemen tation/, all the algorithms w ere im/-plemen ted in the same st yle b y one of us/. W e also /#0Dattenedeac h algorithm in that w e man ually inlined all the functionsother than the functions needed b y the LED A data t yp es/.This eliminated the o v erhead of function in v o cations/. Thetotal size of the programs is appro ximately /2/7/0/0 lines ofC/+/+ co de/.W e compiled and link ed eac h program using the SunC/+/+ compiler CC v ersion /3/./0/./1 under the O/4 optimizationoption/. W e conducted the exp erimen ts on a Sun Sparc /2/0Mo del /5/1/2 with t w o CPUs/, /6/4 MB of main memory /, and/1/0/5 MB of sw ap space/. The op erating system w as SunOSv ersion /5/./5/./1/.W e did t w o sets of exp erimen ts/: one to measure the run/-ning time of eac h algorithm and another to coun t the k eyop erations of eac h algorithm/, as adv o cated in /#5B/2/#5D/. Our testsuite con tained random graphs/, generated using SPRAND /#5B/5/#5D/,and cyclic sequen tial m ulti/-lev el logic b enc hmark circuits/,obtained from the /1/9/9/1 Logic Syn thesis and OptimizationBenc hmarks /#5B/2/4/#5D/. SPRAND pro duces a graph with n no desand m arcs b y /#0Crst building a Hamiltonian cycle on theno des and then adding m /, n arcs at random/. This cyclemak es the graph strongly connected/. W e generated /1/0 v er/-sions of eac h random graph/. The exp erimen tal data rep ortedfor these graphs in this pap er are the a v erage o v er these /1/0runs/. The arc w eigh ts in the random graphs w ere uniformlydistributed in /#5B/1/, /1/0/0/0/0/#5D/, whic h is the default w eigh ti n ter/-v al in SPRAND/. Due to lac k of space/, w e do not presen t theexp erimen tal results for the b enc hmark circuits/: they canb e found in /#5B/9/#5D/.The prop erties of the random graphs in our test suiteare giv en together with the running times in T ables /2/. W eused sparse random graphs in our test suite b ecause realcircuits are sparse and w ew an ted our random graphs torepresen t them as closely as p ossible/. W e did more exp eri/-men ts than w ere rep orted in this pap er/. Ho w ev er/, since thetrend for the dep endence of the p erformance on the graphparameters is eviden t from the results that w e included inthis pap er/, w e did not see an y need to include more exp er/-imen tal results/. When doing our exp erimen ts/, w e tried tofollo w the guideline s in /#5B/1/#5D/. When comparing the algorithmsusing their op eration coun ts/, w e compared only the relev an tones b ecause all the algorithms do not ha v e the same kindof op erations/. F or instance/, w e compared only the K O andYTO algorithms for the n um b er of heap op erations/./4/4 Exp erimental Results and Observations/4/./1 The minimum cycle mean and the graph pa rametersF or the random graphs/, the minim um cycle mean is almostindep enden t of the n um b er of no des/, and it c hanges in v erselywith the densit y of the graph/. This observ ation is exp ectedb ecause as the densit y of a graph increases/, the graph con/-tains more cycles and the critical cycles get smaller/. Thissimple observ ation will b e used to explain the b eha vior ofsome of the algorithms/./4/./2 K O versus YTOIn our implemen tation/, b oth algorithms use Fib onacci heaps/,whic h is the default heap data structure in LED A/. Since theYTO algorithm is essen tially an implemen tation of the K Oalgorithm using Fib onacci heaps/, their use in the YTO algo/-rithm w as a natural c hoice/. Their use in the K O algorithmw as preferred to mak e these t w o algorithms comparable/.F rom the exp erimen tal results rep orted in /#5B/9/#5D/, w e can seethat b oth the algorithms p erform almost the same n um be rof iterations on eac h test case/; ho w ev er/, the YTO algorithmpro vides sa vings in the n um b er of heap op erations/, esp e/-cially in the n um b er of insertions/. The sa vings are morepronounced on the random graphs/, and they get b etter asthe densit y increases b ecause the rate of increase in thesen um b ers in the K O algorithm is larger/. Their running timesare comparable but the YTO algorithm p erforms a bit fasterwhen the densit y increases/. This is exp ected b ecause theYTO algorithm p erforms few er heap op erations/./4/./3 Numb er of iterationsBurns/'/, K O/, YTO/, and Ho w ard/'s algorithms p erform a n um/-b er of iterations b efore they con v erge/. An upp er b ound onthe n um b er of iterations for the /#0Crst three algorithms is n
/2/.An upp er b ound for Ho w ard/'s algorithm is the pro duct ofthe out/-degrees of all the no des/. W e also measured the v alueof k when the HO algorithm terminates/. W e refer to thisv alue as /#5Cthe n um b er of iterations/" of the HO algorithm al/-though it is not one in the sense of the other algorithms/. Itis alw a ys less than n /.F rom the exp erimen tal results rep orted in /#5B/9/#5D/, the n um/-b er of iterations is alw a ys less than the n um be r o f n od e sfor eac h algorithm although there are a few anomalies withHo w ard/'s algorithm/. It seems that unless n /= m /, the n um/-b er of iterations for the /#0Crst three algorithms is around n/= /2on the random graphs/, eac h of whic h is strongly connected/.Moreo v er/, Burns/' algorithm p erforms few er n um b er of iter/-ations than the K O algorithm/, and the K O and YTO algo/-rithms p erform the same n um b er of iterations/. The n um be rof iterations of the Ho w ard/'s algorithm is drastically smallcompared to the other algorithms/. In /#5B/6/#5D/, it is conjecturedthat the n um b er of iterations is O /#28lg n /#29 on the a v erage/, andit is O /#28 m /#29 in the w orst case/. Our exp erimen ts supp ort thew orst case conjecture/. They also sho w that the n um b er of it/-erations for Ho w ard/'s algorithm and the HO algorithm getssmaller as the densit y of the graph increases although someanomalies exist/. This can b e explained b y the /#0Crst observ a/-tion/./4/./4 Ka rp/'s algo rithm and its va riantsF rom the exp erimen tal results rep orted in /#5B/9/#5D/, it seems thatthe impro v emen ta c hiev ed b y the DG algorithm in the n um/-b er of arcs visited during the computation of Dk
/#28 v /#29 for eac h
v is v ery small on the random graphs/, indicating that it isnot e/#0Bectiv e for dense graphs/. The impro v emen t on the cir/-cuits is far b etter/, whic h explains the b etter p erformance ofthe DG algorithm o v er Karp/'s algorithm/.The space e/#0Ecien tv ersion of Karp/'s algorithm/, the Karp/2algorithm/, roughly doubles its running time/, as exp ected/.The space e/#0Eciency of the Karp/2 algorithm is directly ap/-plicable to the DG and HO algorithms/. The most e/#0Bectiv eimpro v emen t on the Karp/'s algorithm is the HO algorithm/.Its running time is ev en b etter than those algorithms whic hare asymptotically faster than it/. Extrap olating from theKarp/2 algorithm/, w e can sa y that the space e/#0Ecien tv ersionof the HO algorithm will double its running time/, whic h stillmain tains its sup eriorit y to most of the other algorithms/./4/./5 Running timesThe running time comparisons are giv en in T ables /2/. Theresults sho w the follo wing/: Ho w ard/'s algorithm is the fastestb y a great margin/. The HO algorithm ranks second/, whic hindicates that the early termination sc heme in the HO al/-gorithm is v ery e/#0Bectiv e/. The slo w est algorithm is La wler/'salgorithm/.The go o d p erformance of Karp/'s algorithm/, esp eciallyon small test cases/, is mostly due to its simplici t y/; it con/-tains three simple nested lo ops/. Its simplicit y facilitates itsoptimization b y a compiler/, e/.g/./, when compiled without op/-timization/, the DG algorithm almost alw a ys b eats it/. Ho w/-ev er/, as the n um b er of no des gets larger/, its p erformancedegrades more rapidly /.Burns/' algorithm is slo w er than the K O and YTO algo/-rithms although it p erforms few er n um b er of iterations andit do es not p erform exp ensiv e op erations suc h as heap op er/-ations/. W e attribute this b eha vior to the fact that it is notincremen tal/; ev ery iteration builds from the scratc h/.The O A/1 and O A/2 algorithms are not as fast as their run/-ning time implies/. They are in general slo w er than Karp/'salgorithm/. W e attribute m uc h of this to their complexit y/;they are more di/#0Ecult to optimize than the other algorithms/./5 Conclusions and F uture W o rkW eh a v e presen ted e/#0Ecien t algorithms for the minim ummean cycle problem/. This pap er is the /#0Crst study that bringsthem to the atten tion of the CAD comm unit y /.W eh a v e sys/-tematically compared these algorithms on random graphs asw ell as b enc hmark circuits and pro vide imp ortan t insigh tsin to their individua l p erformance as w ell as relativ e p erfor/-mance in practice/. One of the most surprising results of thisstudy is that Ho w ard/'s algorithm is b y far the fastest al/-gorithm on the graphs tested in this study /. Unfortunately /,the kno wn b ounds on the running time of this algorithm/,including our b ounds/, are exp onen tial/. W e are w orking onimpro ving these algorithms based on the insigh t that w eha v e obtained from this study /. So far/, w eh a v e dev elop edimpro v ed v ersions of Ho w ard/'s algorithm and La wler/'s algo/-rithm/.Ackno wledgmentsThe authors w ould lik et o a c kno wledge supp ort from the follo winga w ards/: NSF MIP /9/5/-/0/1/6/1/5 /#28CAREER/#29/, NSF CCR/-/9/8/0/6/8/9/8/, NSFCCR/-/9/6/2/5/8/4/4/, D ARP AD ABT/6/3/-/9/8/-C/-/0/0/4/5/, the Univ ersit y of Cali/-fornia MICR O program/, the In terstate Electronics F ello wship/, andthe D A C Design Automation Graduate Sc holarship/./5T able /2 /: The running time comparisons of Burns/'/, K O/, YTO/, Ho w ard/'s/, HO/, Karp/'s/, DG/, La wler/'s/, Karp/2/,and O A/1 algorithms on the random graphs with n no des and m arcs/. F or the cases mark ed with N/#2FA/, eitherw e could not get a result in a da y /,o r w e ran out of memory b ecause of the quadratic space complexit y of thealgorithm in con text/.n m Burns K O YTO Ho w ard HO Karp DG La wler Karp/2 O A/1/5/1/2 /5/1/2 /3/./4/8 /1/./5/1 /1/./6/7 /0/./0/1 /1/./0/0 /0/./7/9 /0/./0/6 /1/1/./0/9 /1/./4/1 /3/2/8/./8/8/5/1/2 /7/6/8 /2/./3/4 /1/./0/4 /1/./1/2 /0/./1/6 /0/./3/2 /0/./9/8 /1/./0/3 /6/./5/1 /1/./8/3 /5/./8/0/5/1/2 /1/0/2/4 /2/./7/2 /1/./2/1 /1/./2/1 /6/./7/5 /0/./2/9 /1/./1/7 /1/./2/6 /9/./2/6 /2/./2/5 /5/./6/6/5/1/2 /1/2/8/0 /4/./1/1 /1/./8/2 /1/./7/3 /0/./1/7 /0/./3/1 /1/./3/7 /1/./4/7 /1/0/./6/2 /2/./7/1 /6/./9/8/5/1/2 /1/5/3/6 /3/./5/2 /1/./5/9 /1/./5/2 /0/./1/3 /0/./2/7 /1/./5/7 /1/./6/9 /1/0/./9/8 /2/./8/7 /6/./5/1/1/0/2/4 /1/0/2/4 /1/3/./9/8 /5/./8/7 /6/./5/0 /0/./0/2 /4/./0/3 /3/./3/6 /0/./2/5 /4/4/./8/2 /6/./7/2 /2/7/9/0/./1/2/1/0/2/4 /1/5/3/6 /1/0/./1/7 /4/./4/1 /4/./6/1 /0/./3/4 /1/./0/7 /4/./1/7 /4/./6/6 /3/4/./6/7 /7/./8/7 /1/2/./3/4/1/0/2/4 /2/0/4/8 /1/1/./3/2 /4/./9/8 /4/./9/9 /0/./2/1 /0/./8/4 /5/./0/5 /5/./6/4 /3/0/./3/3 /9/./0/4 /1/3/./7/8/1/0/2/4 /2/5/6/0 /1/5/./1/6 /6/./7/4 /6/./6/2 /0/./2/3 /0/./9/4 /5/./9/1 /6/./6/3 /5/4/./7/7 /1/0/./8/2 /2/3/./6/7/1/0/2/4 /3/0/7/2 /1/3/./9/1 /6/./2/5 /5/./9/0 /0/./2/2 /0/./8/7 /6/./7/7 /7/./5/6 /5/1/./9/1 /1/4/./6/0 /1/7/./1/3/2/0/4/8 /2/0/4/8 /5/5/./8/8 /2/3/./1/3 /2/5/./4/6 /0/./0/4 /1/6/./4/5 /1/3/./4/8 /1/./0/2 /1/8/6/./3/5 /2/1/./8/0 /2/0/1/1/0/./2/8/2/0/4/8 /3/0/7/2 /4/4/./5/5 /2/0/./3/7 /2/2/./1/9 /0/./6/4 /4/./2/6 /1/7/./1/4 /1/9/./4/5 /1/7/8/./8/6 /2/9/./6/5 /6/2/./8/1/2/0/4/8 /4/0/9/6 /4/2/./8/8 /2/0/./5/9 /2/0/./3/1 /0/./8/8 /3/./1/4 /2/1/./8/7 /2/4/./9/6 /1/6/5/./6/1 /4/2/./2/5 /3/7/./0/4/2/0/4/8 /5/1/2/0 /6/3/./2/2 /3/0/./9/5 /2/9/./9/5 /0/./7/6 /3/./5/6 /2/7/./1/0 /3/0/./8/3 /2/2/1/./9/0 /5/3/./3/0 /8/0/./9/7/2/0/4/8 /6/1/4/4 /7/3/./9/2 /3/6/./5/6 /3/4/./6/1 /0/./8/0 /3/./5/3 /3/2/./8/6 /3/7/./0/5 /2/4/4/./0/5 /6/4/./8/9 /8/5/./8/7/4/0/9/6 /4/0/9/6 /2/1/8/./3/1 /9/1/./5/0 /1/0/0/./4/0 /0/./0/7 N/#2FA /5/5/./7/6 /4/./5/6 /6/5/9/./7/4 /8/9/./5/9 N/#2FA/4/0/9/6 /6/1/4/4 /1/6/1/./0/7 /7/9/./0/9 /8/1/./0/5 /7/./0/0 N/#2FA /7/6/./8/2 /8/6/./6/4 /7/3/6/./1/6 /1/3/5/./4/6 N/#2FA/4/0/9/6 /8/1/9/2 /1/6/7/./6/3 /8/8/./8/6 /8/8/./0/1 /1/./4/7 N/#2FA /1/0/3/./1/3 /1/1/5/./8/1 /7/8/1/./8/4 /1/9/5/./3/5 N/#2FA/4/0/9/6 /1/0/2/4/0 /2/4/2/./7/5 /1/3/2/./2/6 /1/3/0/./0/1 /1/./6/2 N/#2FA /1/2/9/./0/3 /1/4/4/./7/5 /1/3/0/5/./4/7 /2/5/9/./1/9 N/#2FA/4/0/9/6 /1/2/2/8/8 /2/3/6/./7/1 /1/3/9/./2/2 /1/3/7/./8/7 /1/3/./8/4 N/#2FA /1/5/6/./7/0 /1/7/3/./9/4 /1/1/3/2/./5/7 /3/1/3/./0/6 N/#2FA/8/1/9/2 /8/1/9/2 /8/2/6/./0/8 /3/6/3/./4/5 /3/9/8/./1/1 /0/./1/4 N/#2FA N/#2FA N/#2FA /2/8/1/9/./3/0 /3/5/5/./0/0 N/#2FA/8/1/9/2 /1/2/2/8/8 /5/5/9/./8/8 /3/0/6/./5/2 /3/2/9/./7/3 /4/./0/9 N/#2FA N/#2FA N/#2FA /2/9/4/9/./2/5 /5/9/5/./0/2 N/#2FA/8/1/9/2 /1/6/3/8/4 /6/2/6/./6/5 /3/8/2/./8/2 /3/8/0/./5/8 /4/./5/3 N/#2FA N/#2FA N/#2FA /3/7/0/8/./9/8 /8/5/8/./0/1 N/#2FA/8/1/9/2 /2/0/4/8/0 /8/4/0/./5/0 /5/3/6/./5/0 /5/2/4/./8/2 /4/./7/3 N/#2FA N/#2FA N/#2FA /5/1/1/2/./6/7 /1/1/1/0/./8/4 N/#2FA/8/1/9/2 /2/4/5/7/6 /8/7/4/./2/1 /6/0/9/./6/0 /5/8/7/./5/1 /5/./5/7 N/#2FA N/#2FA N/#2FA /5/4/1/7/./5/8 /1/9/0/5/./2/0 N/#2FAReferences/#5B/1/#5D Ah uja/, R/. 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