Cooperative Search and Nogood Recording

Transcription

1 Cooperative Searh and Nogood Reording Cyril Terrioux LSS - Equipe NCA (LM) 39 rue Joliot-Curie F-1343 Marseille Cedex 13 (Frane) terrioux@limuniv-mrsfr Abstrat Within the framework of onstraint satisfation problem we propose a new sheme of ooperative parallel searh The ooperation is realized by exhanging nogoods (instantiations whih an t be extended to a solution) We assoiate a proess with eah solver and we introdue a manager of nogoods in order to regulate exhanges of nogoods Eah solver runs the algorithm Forward-Cheking with Nogood Reording We add to algorithm a phase of interpretation whih limits the size of the searh tree aording to the reeived nogoods Solvers differ from eah other in ordering variables and/or values by using different heuristis The interest of our approah is shown experimentally n partiular we obtain linear or superlinear speed-up for onsistent problems like for inonsistent ones up to about ten solvers 1 ntrodution n onstraint satisfation problems one of main tasks onsists in determining whether there exists a solution ie an instantiation of all variables whih satisfies all onstraints This task is a NP-omplete problem n order to speed up the resolution of problems parallel searhes are used A basi one is independent parallel searh whih onsists in running several solvers (eah one using a different heuristi) instead of a single solver The aim is that at least one of the solvers gets an heuristi suitable for the problem whih we solve Tested on the graph oloring problem ([Hogg and Williams 1994]) this approah has better results than a lassial resolution with a single solver but the gains seem limited Hogg and Williams reommend then the use of a ooperative parallel searh A ooperative parallel searh is based on the same ideas as the independent searh with in addition an exhange of informations between solvers in order to guide solvers to a solution and then to speed up the resolution Experimental results on ryptarithmeti problems ([Clearwater et al 1991; Hogg and Huberman 1993]) and on graph oloring problem ([Hogg and Huberman 1993; Hogg and Williams 1993]) show a signifiant gain in time with respet to an independent searh n both ases the exhanged informations orrespond to partial onsistent instantiations n [Martinez and Verfaillie 1996] a ooperation based on exhanging nogoods (ie instantiations whih an t be extended to a solution) is proposed Exhanged nogoods permit solvers to prune their own searh tree So one an expet to find more quikly a solution Solvers run the algorithm Forward Cheking with Nogood Reording (noted FC-NR [Shiex and Verfaillie 1993]) The realized implementation gathers all solvers in a single proess whih simulates the parallelism t is turned to a monoproessor system Experimentations on random CSPs show that ooperative searh is better than independent one However the weak gain with report to a single solver gives a doubt about effiieny of a suh searh From the idea of Martinez and Verfaillie we define a new sheme of ooperation with exhange of nogoods turned to systems with one or several proessors We assoiate a proess with eah solver Eah solver runs FC-NR n order to avoid problems raised by the ost of ommuniations we introdue a manager of nogoods whose role is to regulate exhange of nogoods n addition to sending and reeipt of messages we add a phase of interpretation to FC-NR in order to limit the size of the searh tree aording to reeived nogoods Our seond main aim is to answer an open question ([Martinez and Verfaillie 1996]) about effiieny of a ooperative parallel searh with exhange of nogoods We show experimentally the interest of our approah The plan is as follows n setion we give basi notions about CSPs nogoods and FC-NR Then in setion 3 we present our sheme by desribing the manager of nogoods and the phase of interpretation Finally after providing experimental results in setion 4 we onlude in setion Definitions 1 Definitions about CSPs A onstraint satisfation problem (CSP) is defined by a quadruplet is a set of variables Eah variable takes its values in the domain from Variables are subjet to onstraints from Eah onstraint involves! #" a set of variables A relation (from ) is assoiated with eah onstraint % suh '&)(((*&+%#" that represents the set of allowed $ -uplets over A CSP is alled binary if eah onstraint involves two vari-

2 / E B N Q N < < B < J A B $ 0 Q p r P ables Let and - be two variables we note the orresponding onstraint Afterwards we onsider only binary CSPs However our ideas an be extended to n-ary CSPs Given /10 suh that / 34 an instantiation of variables from / is a $ 6 7 -uplet from &8(((9&: 7At is alled onsistent if ;=< 7 inonsistent otherwise We use indifferently the term assignment 6 7 instead of instantiation We note the in the more meaningful form CB DB A solution is a onsistent F6 > G > instantiation of all variables E Given an instane determine whether has a solution is a NP-omplete problem E H Given a CSP and an instantiation E K L> M 7 J 7 is the CSP indued by E from with a Forward Cheking filter suh that: PHQ QSRG> 9 - ;ON T >RVU > W PQ - ;ON RG6 7 H#"#Y < 4X ;3 < - ;ON N4Z Y Y\[ 98 Y6Y6[^]6 _&+ [ > H` ba is said FC-onsistent if ;-N Nogoods: definitions and properties n this part we give the main definitions and properties about nogoods and FC-NR ([Shiex and Verfaillie 1993]) A nogood orresponds to an assignment whih an t be extended to a solution More formally ([Shiex and Verfaillie 1993]) given an instantiation and a subset of onstraints ()0 6 > > ) is a nogood if the CSP doesn t have any solution whih d ontains is alled the nogood s justifiation (we note the variables subjet to onstraints from ) The arity of the nogood is the number of assigned variables in We fe? note / the restrition of to variables whih are both in and in / For instane every inonsistent assignment orresponds to a nogood The onverse doesn t hold To alulate justifiations of nogoods we use the notion of value-killer (introdued in [Shiex and Verfaillie 1993]) and we extend it in order to exploit it in our sheme Given an assignment E the CSP indued by and the set g Rkj of nogoods found by other solvers a onstraint (Nih $ ) is a value-killer of value from for if one of the following onditions holds: 1 is a value-killer of for lm br $ > n` #"#Y and and < 3! B B < g oa f a unique solver is used g (whih orresponds to the definition presented in [Shiex and Verfaillie 1993]) Assume that an inonsisteny is deteted beause a domain beomes empty The reasons of failure (ie justifiations) orrespond to the union of value-killers of The following theorem formalizes the reation of nogoods from dead-ends Theorem 1 Let be an assignment and be an unassigned variable Let p be the set of value-killers of f it doesn t remain any value in? fq A then p is a nogood 6l The two next theorems make it possible to reate new nogoods from existing nogoods The first theorem builds a new nogood from a single existing nogood Theorem (projetion [Shiex and Verfaillie 1993]) f is a nogood is a nogood then? %d A n other words we keep from instantiation the variables whih are involved in the inonsisteny Thus we produe a new nogood whose arity is limited to its strit minimum Theorem 3 we build a new nogood from a set of nogoods: Theorem 3 Let be an instantiation be an unassigned variable Let p be the set of value-killers of Let be the ir extension of by assigning the vu value u to (! sb u t ) f w T is a nogood >x are nogoods then A nogood an be used either to bakjump or to add a new onstraint or to tighten an existing onstraint n both ases it follows from the use of nogoods a pruning of the searh tree FC-NR explores the searh tree like Forward Cheking During the searh it takes advantage of dead-ends to reate and reord nogoods These nogoods are then used as desribed above to prune the searh tree The main drawbak of FC-NR is that the number of nogoods is potentially exponential So we limit the number of nogoods by reording nogoods whose arity is at most (ie unary or binary nogoods) aording to the proposition of Shiex and Verfaillie ([Shiex and Verfaillie 1993]) Nevertheless the ideas we present an be easily extended to n-ary nogoods 3 Desription of our multiple solver Our multiple solver onsists of y sequential solvers whih run independently FC-NR on the same CSP Eah solver differs from another one in ordering variables and/or values with different heuristis Thus eah one has a different searh tree The ooperation onsists in exhanging nogoods A solver an use nogoods produed by other solvers in order to prune a part of its searh tree whih should speed up the resolution During the searh solvers produe nogoods whih are ommuniated to other solvers Therefore when a solver finds a nogood it must send y{z messages to inform its partners Although the number of nogoods is bounded the ost of ommuniations an beome very important prohibitive even So we add a proess alled manager of nogoods whose role is to inform solvers of the existene of nogoods Aordingly when a solver finds a nogood it informs the manager whih ommuniates at one this new information to a part of other solvers n this way a solver sends only one message and gets bak more quikly to the resolution of the problem The next paragraph is devoted to the role and the ontribution of manager in our sheme 31 The manager of nogoods Role of the manager The manager s task is to update the base of nogoods and to ommuniate new nogoods to solvers Update the base of

3 P P g p ` nogoods onsists in adding onstraints to initial problem or in tightening the existing onstraints To a unary (respetively binary) nogood orresponds a unary (resp binary) onstraint Eah nogood ommuniated to manager is added to the base n order to limit the ost of ommuniations the manager must inform only solvers for whih nogoods may be useful A nogood is said useful for a solver if it allows this solver to limit the size of its searh tree The next theorem haraterizes the usefulness of a nogood aording to its arity and the urrent instantiation Theorem 4 (haraterization of the usefulness) (a) a unary nogood is always useful (b) a binary nogood! B~} B~ is useful if in the urrent instantiation and are assigned respetively to } and () a binary nogood! B~} - B~ is useful if in the urrent instantiation (resp - ) is assigned to } (resp ) - (resp ) isn t assigned and (resp } ) isn t removed yet Proof: see [Terrioux 001] From this theorem we expliite what solvers reeive some nogoods (aording to their usefulness): (a) every unary nogood is ommuniated to all solvers (exept the solver whih finds it) (b) binary nogood! B } - B is ommuniated to eah solver (exept the solver whih finds it) whose instantiation ontains B } or B n ase (b) we an t ertify that the nogood is useful beause the solver may have baktraked between the sending of the nogood by the manager and its reeipt by the solver With a view to limit the ost of ommuniations only the in- stantiation of nogood is onveyed Communiate the justifiation isn t neessary beause this nogood is added to the problem in the form of a onstraint Thanks to reeived information solvers an forbid with justifiation Contribution of the manager n this part we show the ontribution of the manager of nogoods to our sheme with respet to a version without manager The omparison is based on the total number of messages whih are exhanged during all searh Let g be the total number of nogoods whih are exhanged by all solvers We ount ƒ unary nogoods and binary ones Note that among these g nogoods doubles may exist n effet two solvers an find independently a same nogood n a sheme without manager eah solver ommuniates P! the nogoods it finds to y z other solvers So ƒ P! y z messages are sent for unary nogoods and y z for binary ones But in a sheme with manager nogoods are first sent to manager by solvers During all searh solvers onvey to manager ƒ messages for unary nogoods and for binary ones Then the manager sends only unary nogoods to ynz solvers These nogoods orrespond to ƒ nogoods minus the doubles Likewise for binary nogoods doubles aren t ommuniated Furthermore for the remaining binary nogoods the manager restrits the number of reipients Let be the P our sheme we exhange ƒb : ŷ z nogoods and 8 messages for binary ones number of messages sent by manager for binary nogoods n messages for unary n the worst ase the sheme with manager produes up to additional messages in omparison with the sheme without manager But in general and are little enough so that the sheme with manager produes fewer messages 3 Phase of interpretation The method whih we are going to desribe is applied whenever a nogood is reeived Solvers hek whether a message is reeived after developing a node and before filtering n the phase of interpretation solvers analyze reeived nogoods in order to limit the size of their searh tree by stopping branh whih an t lead to solution or by enforing additional filtering For unary nogoods this phase orresponds to a permanent deletion of a value and to a possible bakjump Method 1 details the phase for suh nogoods Method 1 (phase of interpretation for unary nogoods) Let be the urrent instantiation Let ŠB } reeived nogood We delete } from be the (a) f is assigned to the value } then we bakjump to f? %q A is empty we reord the nogood p with p the set of value-killers of (b) f is assigned to ( } ) we do nothing () f isn t assigned we hek whether is empty f? qna is empty we reord the nogood p with p the set of value-killers of Theorem The method 1 is orret Proof: see [Terrioux 001] For binary nogoods the phase orresponds to enfore an additional filtering and to a possible bakjump Method desribes the ations done during this phase for binary nogoods Method (phase of interpretation for binary nogoods) Let be the urrent instantiation and! B } - B be the reeived nogood (a) f and are assigned in to } and respetively then we bakjump to the deepest variable among and O f - (resp ) is this variable we delete by filtering (resp } ) from (resp ) (b) f (resp O ) is assigned to } (resp ) and - (resp ) isn t be assigned we delete by filtering (resp } ) from (resp ) f? qna (resp ) beomes empty we reord the nogood with p the set of value-killers of (resp ) Theorem 6 The method is orret Proof: see [Terrioux 001] Unlike the phase of interpretation for unary nogoods here the deletion isn t permanent Whereas the phase of interpretation is orret its addition to FC-NR may in some ases ompromise a basi property of FC-NR The next paragraph is devoted to this problem

4 Œ Œ Œ 0 Q 33 Maintening FC-onsisteny We remind first a basi property of FC-NR (from FC): Property 1 Every instantiation built by FC-NR is FConsistent After a baktrak to the variable FC-NR (like FC) anels the filtering whih follows the assignment of So it restores eah domain in its previous state n partiular if a domain is wiped out after filtering it isn t empty after restoring t ensues the preserve of the property 1 With the addition of ommuniations and of the phase of interpretation this property may be ompromised For example we onsider the searh tree MŒn explored by> a solver whih ooperates with other ones Let } This solver assignes first } to then to J Enfore FC-onsisteny after assigning J removes } Œ from The filtering whih follows the assignment of! to Ž deletes and Œ from The solver assignes to and then it visits the orresponding subtree Assume that this subtree ontains only failures and that the solver reeives unary nogoods whih forbid assigning values and to So the solver baktraks and t baktraks reords a unary nogood whih forbids for again (to Œ Ž ) and assignes J to Ž whih raises a problem namely is empty (due to permanent removals of and from Œ by unary nogoods) So the urrent instantiation isn t FC-onsistent and the property 1 doesn t hold The next theorem haraterizes the problem: Theorem 7 Let B l B l B be a FConsistent instantiation We onsider the exploration by FC- NR of subtree rooted in O B Let g be the set of values of whih remain forbidden by nogoods at the end of the exploration suh that these nogoods are reorded or reeived during this exploration and none of them doesn t involve - f all values of are removed during the exploration after anelling the filtering no value is restored in l following the assignment of to if and only if g Proof: see [Terrioux 001] t ensues that in some ases the union of reeipts and reations of nogoods with the filtering indues the existene of empty domains and thus property 1 doesn t hold A solution onsists in heking whether a domain is wiped out after anelling the last filtering and if there exists suh domain in baktraking until the domain isn t empty t isn t neessary to hek the emptiness of every domain thanks to the following lemma whih determines potential origins of this problem Lemma 1 Only reording or reeipt of a nogood may indue the loss of property 1 Proof: see [Terrioux 001] This lemma enables to hek only the emptiness of domains whih beome empty after reording or reeiving a nogood f emptiness is indued by reeiving a nogood we introdue an additional phase of bakjump This phase follows every detetion of empty domain after reeiving a nogood and makes it possible to keep on with the searh from a FConsistent instantiation l Method 3 (bakjump s phase) f is empty due to a reeived nogood we bakjump: - until isn t empty or until the urrent instantiation is empty if the nogood is unary - until isn t empty if the nogood is binary Note that we baktrak to empty instantiation only for inonsistent problems Theorem 8 The method 3 is orret Proof: see [Terrioux 001] f emptiness is indued by reording a nogood a solution onsists in reording only nogoods whih don t wipe out a domain However we ommuniate all nogoods to manager This solution is easy to implement but it doesn t take advantage of all found nogoods 4 Experimental results 41 Experimental protool We work on random instanes produed by random generator written by D Frost C Bessière R Dehter and J-C Régin This generator takes 4 parameters g and t builds a CSP of lass g with g variables whih have domains of size and binary onstraints ( Q= Q ' lm š Q: J ) in whih tuples are forbidden ( J ) \ Experimental Gœ -P!œ žo results we give afterwards onern lasses with whih varies between 433 and 447 Considered lasses are near Ÿ ž to the satisfiability s threshold whih orresponds to However Ÿ4ž for FC-NR the diffiulty s shape is observed for Every problem we onsider has a onneted graph of onstraints Given results are the averages of results obtained on 100 problems per lass Eah problem is solved 1 times in order to redue the impat of non-determinism of solvers on results Results of a problem are then the averages of results of 1 resolutions For a given resolution the results we onsider are ones of the solver whih solves the problem first Experimentations are realized on a Linux-based PC with an ntel Pentium 0 MHz proessor and 6 Mb of memory 4 Heuristis n order to guarantee distint searh trees eah solver orders variables and/or values with different heuristis As there exist few effiient heuristis from an effiient heuristi hoosing variables we produe several different orders by hoosing differently the first variable and then applying We use the heuristi 4 t ª«([Bessière and Régin 1996]) for whih the next variable to assign is one whih minimizes the ratio s (where is the urrent domain of and is the set of variables whih are onneted to by a binary onstraint) for This heuristi is onsidered better in general than other lassial heuristis That s why we hoose it n our implementation only the size of domains varies for eah instantiation The degree X X is updated when a new onstraint is added thanks to a nogood As regards the hoie of next value to assign we onsider values in appearane order or in reverse order n following results (unless otherwise speified) half solvers use the appearane order to order domains other half reverse order

5 ± ² # onsistent Problems Table 1: Effiieny (in %) for onsistent and inonsistent problems for lasses ³ œ OP œtžo 43 Results Effiieny n this paragraph we assess the speed-up and the effiieny of our method Let be the run-time of a single solver for the resolution of a serie of problems and be the run-time for y solvers whih are run in parallel We define speed-up as the ratio µ and effiieny as the ratio µ The speed-up is µ µ alled linear with report to the number y of solvers if it equals to y superlinear if it is greater than y sublinear otherwise Table \ Gœ -P!œ žo 1 presents effiieny (in %) obtained for lasses with whih varies between 433 and 447 n table 1 up to 10 solvers we obtain linear or superlinear speed-up for all lasses (exept 3 lasses) Above 10 solvers some lasses have linear or superlinear speed-up but most lasses have sublinear speed-up We note also a derease of effiieny with the inrease of number of solvers Aording to onsisteny of problems we observe a better speed-up for onsistent problems (whih remains linear or superlinear) We note the same derease of effiieny with the number of solvers But effiieny for inonsistent problems is less important whereas it is greater than 1 up to 10 solvers t follows the appearane of sublinear speedup above 10 solvers This lak of effiieny for inonsistent problems infers a derease of effiieny for overall problems Explanations of obtained results First we take an interest in explaining observed gains Possible origns are multiple orders of variables and ooperation We define an independent version of our sheme (ie without exhange of nogoods) We ompare the two versions by alulating the ratio of the run-time for the independent version over one for ooperative version Figure 1 presents results obtained for the lass (013439) with a number of solvers between and 8 We observe similar results for other lasses We note first that ooperative version is always better than independent one Then we observe that the ratio is near 1 for onsistent problems (solid line) That means that the good quality of results for these problems results mostly from multiple orders of variables However the ratio remains Ratio p Consistent nonsistent Figure 1: Ratio independent searh / ooperative searh greater than 1 So exhange of nogoods partiipates in obtained gains too Finally for inonsistent problems (dashed line) ratio is more important than for onsistent ones t inreases with the number of solvers n other words obtained results for these problems are mostly due to ooperation and the ontribution of ooperation inreases with the number of solvers For inonsistent problems we must underline the predominant role of values heuristis For eah solver (exept one if the number of solvers is odd) there exists a solver whih uses the same variables heuristi as and the values heuristi whih is reverse with report to one of Without exhanging nogoods these two solvers visit similar searh trees With exhanging nogoods eah one explores only a part of its searh tree thanks to reeived nogoods t s the same for onsistent problems but this effet is less important beause the searh stops as soon as a solution is found We fous then on possible reasons of effiieny s derease With a sheme like our an usual reason of effiieny s lak is the importane of ost of ommuniations Our method doesn t make exeption But in our ase there is another reason whih explains the derease of performanes We ompare multiple solvers ¹º and ¹ º Z Both have 8 solvers For ordering values half solvers of ¹mº use the appearane or-

6 der other half the reverse order All solvers of ¹ º Z use the appearane» order We realise that ¹º has a better effiieny than ¹ º Z The number of messages for ¹ º Z is greater than for ¹ º But above all it s the same for the number of nodes So ¹ º Z explores more important trees ¹º and ¹ º Z differ in used heuristis for ordering values and variables The heuristis we use for ordering variables are near eah other Using two different orders of values adds diversity to resolution Thus º This differene of diversity per- º The lak of diversity is the main reason (with the inrease of number of ommuniations) of the effiieny s derease ¹º is more various than ¹9Z mits to explain the gap of effiieny between ¹º and ¹ Z Number of messages and real ontribution of manager n order to measure the real ontribution of manager we ompare the osts of ommuniations in a sheme with manager and one without manager n presented results we onsider that the ost of a message is independent of presene or not of the manager and that the ommuniation of a binary nogood is twie as expensive as one of a unary nogood (beause a binary nogood onsists of two pairs variable-value against a single pair for a unary nogood) Figure presents the ratio of ost of ommuniations for a sheme without manager over one for a sheme with manager Considered problems (onsistent and inonsistent) belong to lass (013439) For information we observe similar results for other lasses Ratio P Total Unary Nogoods Binary Nogoods Figure : Ratio between shemes with and without manager First we note an inrease of manager s ontribution with y Thus we an hope that the number of solvers above of whih the ost of ommuniations penalizes effiieny is greater in a sheme with manager than one without manager More preisely we note that ommuniations of unary nogoods is more expensive in the sheme with manager when y is less important This result was foreseeable ndeed in ase of unary nogoods the manager removes only doubles As the probability that two solvers find simultaneously the same nogoods is all smaller sine there are few solvers We explain thus that the sheme with manager beomes better than one without manager when the number of solvers inreases Regarding the ost of ommuniations for binary nogoods the sheme with manager is signifiantly heaper and this eonomy inreases with y This result is explained by the fat that binary nogoods are in general useful for few solvers On overall ommuniations the sheme with manager is the best due essentially to the number of binary nogoods whih is signifiantly greater than one of unary nogoods (with a fator between 30 and 100) n onlusion the manager does its job by limiting the number of exhanged messages t avoids solvers a lak of time due to management of messages (in partiular reeipt of useless nogoods) Conlusions and future works n this paper from an idea of Martinez and Verfaillie we define a new sheme of ooperative parallel searh by exhanging nogoods and we assess experimentally its effiieny We observe then linear or superlinear speed-up up to 10 solvers for inonsistent problems and up to 16 solvers for onsistent ones So exhange of nogoods is an effiient form of ooperation We note a derease of effiieny with the number of solvers due to the inreasing number of ommuniations and to a lak of diversity of solvers A first extension of this work onsists in finding several effiient and diverse heuristis in order to improve effiieny as well as inrease the number of solvers Then we an extend our sheme by applying any algorithm whih maintains some level of onsisteny by using different algorithms (whih would permit to ombine omplete searh methods and inomplete ones like in [Hogg and Williams 1993] and to improve the diversity of solvers) or by generalizing it to another form of ooperation with exhange of informations Referenes [Bessière and Régin 1996] C Bessière and J-C Régin MAC and Combined Heuristis : Two Reasons to Forsake FC (and CBJ?) on Hard Problems n Pro of CP 96 pages [Clearwater et al 1991] S Clearwater B Huberman and T Hogg Cooperative Solution of Constraint Satisfation Problems Siene 4: Nov 1991 [Hogg and Huberman 1993] T Hogg and B A Huberman Better Than the Best: The Power of Cooperation n L Nadel and D Stein editors 199 Letures in Complex Systems volume V of SF Studies in the Sienes of Complexity pages Addison-Wesley 1993 [Hogg and Williams 1993] T Hogg and CP Williams Solving the Really Hard Problems with Cooperative Searh n Pro of AAA 93 pages [Hogg and Williams 1994] T Hogg and CP Williams Expeted Gains from Parallelizing Constraint Solving for Hard Problems n Pro of AAA 94 pages Seattle WA 1994 [Martinez and Verfaillie 1996] D Martinez and G Verfaillie Ehange de Nogoods pour la résolution oopérative de problèmes de satisfation de ontraintes n Pro of CNPC 96 pages Dijon Frane 1996 n frenh [Shiex and Verfaillie 1993] T Shiex and G Verfaillie Nogood Reording for Stati and Dynami Constraint Satisfation Problems n Pro of the ¼6½ EEE CTA 1993 [Terrioux 001] C Terrioux Cooperative searh and nogood reording Researh report LM 001