ebook img

Complexity of the path avoiding forbidden pairs problem revisited PDF

0.41 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Complexity of the path avoiding forbidden pairs problem revisited

Complexity of the path avoiding forbidden pairs problem revisited JakubKova´cˇ DepartmentofComputerScience,ComeniusUniversity,Mlynska´Dolina, 84248Bratislava,Slovakia Abstract LetG =(V,E)beadirectedacyclicgraphwithtwodistinguishedverticess,t,andletF beasetofforbiddenpairsof 1 vertices. WesaythatapathinG issafe,ifitcontainsatmostonevertexfromeachpair{u,v} ∈ F. GivenG and F, 1 thepathavoidingforbiddenpairs(PAFP)problemistofindasafes–tpathinG. 0 2 We systematically study the complexity of different special cases of the PAFP problem defined by the mutual positionsoffobiddenpairs. Fixonetopologicalordering≺ofvertices;wesaythatpairs{u,v}and{x,y}aredisjoint, c e ifu≺v≺ x≺y,nested,ifu≺ x≺y≺v,andhalving,ifu≺ x≺v≺y. D The PAFP problem is known to be NP-hard in general or if no two pairs are disjoint; we prove that it remains NP-hard even when no two forbidden pairs are nested. On the other hand, if no two pairs are halving, the problem 5 isknowntobesolvableincubictime. WesimplifyandimprovethisresultbyshowinganO(M(n))timealgorithm, 2 whereM(n)isthetimetomultiplytwon×nbooleanmatrices. ] M Keywords: path,forbiddenpairs,NP-hard,dynamicprogramming D . s c 1. Introduction [ 2 LetG =(V,E)beadirectedgraphwithtwodistinguishedverticess,t∈V andletF ⊆V×V beasetofforbidden v pairs of vertices. We say that a path π is safe, if it does not contain any forbidden pair, i.e., π contains at most one 6 vertexfromeachpair{u,v} ∈ F. GivenGandF,thepathavoidingforbiddenpairsproblem(henceforthPAFP)isto 9 findasafe s–tpathinG. Inthispaper,westudythecomplexityofdifferentspecialcasesoftheproblemondirected 9 3 acyclicgraphs. . 1 1.1. Motivation 1 1 ThePAFPproblemwasfirststudiedbyKrauseetal.[1]andSrimaniandSinha[2]motivatedbydesigningtest 1 cases for automatic software testing and validation. We can represent a program as a directed graph where vertices : v representsegmentsofcodeandedgesrepresenttheflowofcontrolfromonecodesegmentintoanother. Thegoalisto i coverthisgraphwith s–tpathscorrespondingtodifferenttestcases. However,notallpathscorrespondtoexecutable X sequencesintheprogram.ThereforeKrauseetal.[1]introducedforbiddenpairswhichidentifythemutuallyexclusive r a codesegmentsandformulatedthePAFPproblem. Unfortunatelly,asshownbyGabowetal.[3],theproblemisNP- hardevenfordirectedacyclicgraphs. Adifferentmotivationcamefrombioinformaticsandtheproblemofpeptidesequencingviatandemmassspec- trometry. Peptidesarepolymerswhichcanbethoughofasstringsovera20characteralphabetofaminoacidsand thesequencingproblemistodeterminetheaminoacidsequenceofagivenpeptide. Tothisend,manycopiesofthe peptidearefragmentedandthemassofthefragmentsismeasured(veryprecisely)bymassspectrometer. Theresult oftheexperimentisamassspectrumwhereeachpeakcorrespondstomassofsomeprefixorsomesuffixoftheamino acidsequence,orisanoise. Thespectrumisthencomparedagainstadatabaseofknownfragmentweights. Emailaddress:[email protected](JakubKova´cˇ) PreprintsubmittedtoDiscreteAppliedMathematics December30,2011 Chenetal.[4]suggestedthefollowingformulationofthepeptidesequencingproblem: Letuscreateaspectrum graphwithtwovertices p ands foreachpeakw withweightsw(p)=w −1andw(s)=W−w +1,whereW isthe i i i i i i i weightofthewholepeptide. Weaddanedgefromxtoyifthedifferencebetweenweightsw(y)−w(x)equalsthetotal massofsomeknownsequenceofaminoacids. Thus,pathsinthisgraphcorrespondtoaminoacidsequences. Paths goingthrough p correspondtow beingaweightofsomeprefixandsimilarly,pathsgoingthrough s correspondto i i i w beingaweightofsomesuffix. (Pathsgoingthroughneither p nors correspondtow beinganoise.) However,w i i i i i cannotbeapreffixweightandasuffixweightatthesametime,so{p,s}willformaforbiddenpairforeachi. Thisis i i averyspecialcaseofthePAFPproblemindirectedacyclicgraphswherealltheforbiddenpairsarenestedandChen etal.[4]showedthatitispolynomiallysolvable. ThePAFPproblemondirectedacyclicgraphsalsoaroseinacompletelydifferentapplicationinbioinformatics– genefindingusingRT-PCRtests[5]. Inthisapplication,wehaveasocalledsplicinggraphwhereverticesrepresent non-overlapping segments of the DNA sequence, length of a vertex is the number of nucleotides in this segment, andedge(u,v)indicatesthatsegmentvimmediatelyfollowssegmentuinsomegenetranscript. Thus, pathsinthis splicinggraphcorrespondtoputativegenes. Theproblemistoidentifythetruegeneswithahelpofinformationfrom RT-PCRexperiments. Without going into biology details, let us define a (simplified) result of an RT-PCR experiment as a triple t = (u,v,(cid:96)),whereu,v∈V aretwoverticesand(cid:96)isthelengthofaproduct. Letπbeapathgoingthroughuandvinthe splicinggraph;ifthelengthoftheu–vsubpathisequalto(cid:96),wesaythatπexplainstestt,otherwise,itisinconsistent withtestt. WecandefineascoreofapathπwithrespecttoasetoftestsT asasumofthescoresofallofitsvertices andedges,plusabonusBforeachexplainedtestfromT,andminusapenaltyPforeachinconsistenttest. Thegene findingwithRT-PCRtestsproblemistofindan s–t pathwiththehighestscoreinthegivensplicinggraphG witha setofRT-PCRtestsT. Notethatifwesetalllengthstoanunattainablevalue,say−1,andwesetahigh(infinite)penaltyPforinconsistent tests,webasicallygetthePAFPproblem. Thus,thePAFPproblemisatthecoreofgenefindingwithRT-PCRtests and the latter problem inherits all NP-hardness results for the PAFP problem. On the positive side, we have shown in our previous work [5] that some polynomial solutions for special cases of the PAFP problem can be extended to pseudo-polynomialalgorithmsforthegenefindingproblem. 1.2. Previousresults AsshownbyGabowetal.[3],thePAFPproblemisNP-hardingeneral,butseveralspecialcasesarepolynomially solvable. Yinnone[6]studiedthePAFPproblemunderskewsymmetryconditionswhereforeachtwoforbiddenpairs {u,u(cid:48)},{v,v(cid:48)} ∈ F, if there is an edge from u to v, there is also an edge from v(cid:48) to u(cid:48). He proved that under such conditions, the problem is polynomially equivalent to finding an augmenting path with respect to a given matching andthuspolynomiallysolvable. Fordirectedacyclicgraphs,wehavealreadymentionedthatthenestedcaseissolvableinpolynomialtime[4];Kol- manandPangra´c[7]wereabletodeviseapolynomialalgorithmifthesetofforbiddenpairshasawell-parenthesized orahalvingstructure(seePreliminaries). Recently,approximabilityandparameterizedcomplexityofthePAFPproblemhavebeenstudied:Weadd1tothe objectivefunctiontodisallowazerocostsolutions–otherwisetheproblemistriviallyinapproximable. Hajiaghayi etal.[8]showedthateventhenthereisaconstantc > 0suchthatminimizing1+thenumberofforbiddenpairson an s–t pathisnotc·n-approximable. Bodlaenderetal.[9]studiedthePAFPproblemonundirectedgraphs. When parameterized by the vertex cover ofG = (V,E), the problem is W[1]-hard (the proof also carries over to directed acyclicgraphs). Ontheotherhand,whenparameterizedbythevertexcoverofH =(V,F)(whereedgesareforbidden pairs), the problem is fixed parameter tractable (FPT), but has no polynomial kernel unless NP ⊆ coNP/poly. The problemisalsoFPTwhenparameterizedbythetreewidthofG∪H. 1.3. Contributionsandroadmap Inthispaper,wesystematicallystudydifferentspecialcasesofthePAFPproblemondirectedacyclicgraphs. In thenextsection,weintroducethedifferentspecialcasesbasedonmutualpositionsofforbiddenpairs. InSection3, weprovethatthePAFPproblemisNP-hardevenifthesetofforbiddenpairshasorderedstructureandinSections4 and 5, we improve upon the results of Chen et al. [4] and Kolman and Pangra´c [7] for the nested, halving, and well-parenthesizedforbiddenpairs. 2 Table1:ComplexityofthePAFPproblemforitsdifferentspecialcases;nandmdenotethenumberofverticesandedgesofG,respectively;O(nω) isthecomplexityofbooleanmatrixmultiplication,ω<2.3727[10,11]. AllowedForbiddenPairs Problem Complexity Example disjoint nested halving generalproblem (cid:88) (cid:88) (cid:88) NP-hard[3] overlappingstructure × (cid:88) (cid:88) NP-hard[7] ordered (cid:88) × (cid:88) NP-hard[new] well-parenthesized (cid:88) (cid:88) × O(n3)[7],O(nω)[new] halving × × (cid:88) O(n5)[7],O(nω+1)[new] nested × (cid:88) × O(nm)[4],O(nω)[new] disjoint (cid:88) × × O(n+m)[trivial] 2. Preliminaries LetG =(V,E)beadirectedacyclicgraphandletF bethesetofforbiddenpairs. AsalreadynoticedbyYinnone [6]andKolmanandPangra´c[7],wemayassumethateveryvertexexceptforsandtbelongstoexactlyoneforbidden pair,i.e.,(cid:83)F =V −{s,t}. Thisissimplybecauseifvertexvdoesnotbelongtoanyforbiddenpair,wecanremoveit andreplaceall2-edgepathsu,v,wbyadirectedge(u,w). Ontheotherhand,ifvbelongstok > 1forbiddenpairs, wecanreplaceitbyadirectedpathoflengthkandmovetheendsofforbiddenpairstodifferentverticesonthispath. Todefinespecialcasesofinterest, wefixonetopologicalorderingofvertices. Wesaythatvertexuisbeforeor precedesv,u≺v,ifuprecedesvinthislinearorder. Letusdenotetheforbiddenpairs{f, f(cid:48)}fori=1,...,k,where i i f ≺ f(cid:48)and f ≺ f ≺···≺ f ,i.e.,weorderthembypositionoftheleftmemberofthepair. i i 1 2 k Werecognizethreepossibletypesofmutualpositionofpairs{u,v}and{x,y}(withoutlossofgenerality,letu≺v, x≺y,andu≺ x): disjoint(u,v≺ x,y;seeFig.1(a)),nested(u≺ x,y≺v;seeFig.1(b)),andhalving(u≺ x≺v≺y; see Fig. 1(c)). All the special cases are obtained by restricting the set of forbidden pairs F to only certain types of mutualpositions(seeTable1). Thisgivesus23 =8cases,fromwhichthese6classesarenon-trivialandinteresting: (a) disjointpairs (b) nestedpairs (c) halvingpairs Figure1:Differentmutualpositionsoftwoforbiddenpairs. 1. generalcase–therearenoconstraintsonthepositionsofpairs; 2. overlappingstructure1–everytwoforbiddenpairsoverlap(theymaybenestedorhalving,butnotdisjoint);as aconsequence, f ≺ f ≺···≺ f ≺ f(cid:48) ≺ f(cid:48) ≺···≺ f(cid:48) forsomepermutationσ; 1 2 k σ(1) σ(2) σ(k) 3. ordered – there may be disjoint and halving pairs, but no two forbidden pairs are nested; as a consequence f ≺ f ≺···≺ f and f(cid:48) ≺ f(cid:48) ≺···≺ f(cid:48); 1 2 k 1 2 k 4. well-parenthesized–theremaybedisjointandnestedpairs,butnotwopairsarehalving;thiscasedeservesits namesinceifwewrite( and) forthei-thpair,wegetawell-parenthesizedsequence; i i 5. halving–everytwopairshalveeachother; f ≺ f ≺···≺ f ≺ f(cid:48) ≺ f(cid:48) ≺···≺ f(cid:48); 1 2 k 1 2 k 6. nested–thereareonlynestedpairsi.e.,theverticesinforbiddenpairsareordered f ≺ f ≺ ... ≺ f ≺ f(cid:48) ≺ 1 2 k k ···≺ f(cid:48) ≺ f(cid:48);thisisaspecialcaseofthewell-parenthesizedcase. 2 1 1notethatthisspecialcaseisreferedtoashalvingstructurebyKolmanandPangra´c[7];wereservetheterm“halving”forsetswhereeverytwo pairshalveeachother 3 ThepreviousworkandourownresultsaresummarizedinTable1. Forcompletenessandasawarm-up,weincludeourownproofofNP-hardnessofthePAFPprobleminthegeneral andoverlappingcase. ThisproofisalsosimplerthantheonegivenbyKolmanandPangra´c[7]. Theorem1. ThePAFPproblemisNP-hard,evenwhenthesetofforbiddenpairshasoverlappingstructure. Proof. By reduction from 3-SAT: Let φ = (cid:86) φ be a formula over m variables x ,...,x , with n clauses φ = 1≤i≤n i 1 m i ((cid:96) ∨(cid:96) ∨(cid:96) ),whereeachliteral(cid:96) iseither x or¬x . WewillconstructgraphG andasetofforbiddenpairs F i,1 i,2 i,3 i,j k k suchthatthereisans–tpathavoidingpairsinF ifandonlyifφissatisfiable. x1 x2 x3 xm ··· ··· s t ··· ··· ¬x1 ¬x2 ¬x3 ··· ¬xm φ1 φ2 φ3 ··· φn Figure2:InputforthePAFPproblemfortheformulaφ1∧φ2∧···∧φn.Alledgesaredirectedfromlefttoright. G consistsoftwoparts: Thefirstpartcontainsavertexforeachvariable x anditsnegation¬x (seeFig.2). A k k path traversing this first part corresponds to a truth assignment of variables where the visited vertices are true. The secondpartcontainsavertexforeachliteral(cid:96) (seeFig.2). Forbiddenpairsconnectingeveryliteralfromthefirst i,j parttoeveryoccurenceofitsnegationinthesecondpartofGwillensurethatwecanonlygothrough“true”vertices. Thusan s–tpathavoidingF existsifandonlyifeveryclauseissatisfied. Sinceeveryforbiddenpairstartsinthefirst partandendsinthesecondpart,allpairsoverlap. 3. Orderedforbiddenpairs In this section, we turn to a seemingly more restricted version of the PAFP problem, allowing only disjoint and halvingforbiddenpairs. Thisspecialcasehasnotbeenstudiedbefore. Theorem2. ThePAFPproblemisNP-hard,evenwhenthesetofforbiddenpairsisordered. Proof. We will prove the claim by reduction from 3-SAT. Let φ be a logical formula over m variables x ,...,x , 1 m whichisaconjunctionofnclausesφ ∧···∧φ ,whereφ = ((cid:96) ∨(cid:96) ∨(cid:96) )andeachliteral(cid:96) iseither x or¬x . 1 n i i,1 i,2 i,3 i,j k k WewillconstructgraphGwithalinearorder≺onitsverticesandanorderedsetofforbiddenpairsF suchthatthere isans–tpathavoidingpairsinF ifandonlyifφissatisfiable. Graph G consists of several blocks B and B of 2m vertices shown in Fig. 3(a), (b). The blocks are connected (cid:96) togetherasoutlinedinFig.3(c).Anyleft-to-rightpaththroughtheblockBnaturallycorrespondstoatruthassignment ofthevariablesand,sinceB hasanisolatedvertex¬(cid:96),apaththroughblockB correspondstoanassignmentwhere (cid:96) (cid:96) (cid:96) is true. A clause gadget consists of three such blocks, each corresponding to one literal. Any s–t path must pass throughoneofthethreeblocks,andthuschooseanassignmentthatsatisfiestheclause. TheforbiddenpairsinF will enforcethattheassignmentofthevariablesisthesameinallblocks. Thisisdonebyaddingaforbiddenpairbetween allliterals(cid:96)(cid:48)intheB -blockswiththeircounterparts¬(cid:96)(cid:48)inthepreviousandthefollowingB-block. (cid:96) TheorderofliteralsinaB-blockis¬x ≺ x ≺¬x ≺···≺ x ,whiletheorderinaB -blockis x ≺¬x ≺ x ≺ 1 1 2 m (cid:96) 1 1 2 ··· ≺ ¬x . Letvi ≺ vi ≺ vi ≺ ··· betheorderofverticesingraphGi. AzippingoperationtakesgraphsG1,G2,G3 m 1 2 3 andproducesanewgraphG1∪G2∪G3withverticesorderedv1 ≺v2 ≺v3 ≺v1 ≺v2 ≺v3 ≺···. Theclausegadgets 1 1 1 2 2 2 areproducedbyzippingthethreeblockscorrespondingtotheirliterals. Ifwedonotallowmultipleforbiddenpairs startingorendinginthesamevertex,wecansubstituteverticesinGforshortpathsasinFig.3(d). Itiseasytocheck thatundersuchlinearorder,notwopairsinF arenested. 4 ¬x1 ¬x2 ¬x3 ··· ¬xm x1 x2 x3 ··· ‘ ··· xm x1 x2 x3 ··· xm ¬x1 ¬x2 ¬x3 ··· ¬‘ ··· ¬xm (a) Block B–verticesofthisgraphcorrespondto (b) BlockB(cid:96)issimilartoaB-block,exceptthattheorderof positive and negative literals; a path through this verticesisdifferentandvertex¬(cid:96)isisolated. Thus,apath blockcorrespondstoatruthassignmentofthevari- throughB(cid:96)correspondstoanassignmentwhere(cid:96)istrue. ables. φ φ φ 1 2 n B B B ‘1,1 ‘2,1 ‘n,1 s B B B B B t ‘1,2 ‘2,2 · · · ‘n,2 B B B ‘1,3 ‘2,3 ‘n,3 (c) ConstructionofGfromtheblocksandzippedblockscorrespondingtotheclauses.Forbiddenpairsenforcethat theassignmentofvariablesisthesameinallblocks. forbiddenpairs xk xk+1 ··· B‘k,1 ··· ¬xk ¬xk+1 ¬xk ¬xk+1 ··· xk xk+1 ··· B‘k,2 ··· ··· xk xk+1 ¬xk ¬xk+1 blockB xk xk+1 ··· B‘k,3 ··· ¬xk ¬xk+1 clauseφk (d) AnenlargedviewofgraphGshowingblockB,thefollowingblocksforclauseφkandthewaythey areconnectedbyforbiddenpairs.Notethatnotwoforbiddenpairsarenested. Figure3:ConstructionofthegraphGfora3-SATformulaφ.Alledgesaredirectedfromlefttoright. 4. Well-parenthesizedforbiddenpairs ThefirstpolynomialalgorithmforthePAFPproblemwithwell-parenthesizedforbiddenpairswasgivenbyKol- manandPangra´c[7]. Theiralgorithmusesthreerulesforreducingtheinputgraph: 1. contractionofavertex–ifvdoesnotappearinanyforbiddenpair,removeitandaddadirectedge(u,w)for everypairofedges(u,v),(v,w); 2. removalofanedge–ifedgee∈ E∩F joinstwoverticesthatmakeupaforbiddenpair,removeefromE; 3. removal of a forbidden pair – if (u,v) ∈ F is a forbidden pair, but there is no path from u to v, remove (u,v) fromF. Thesethreerulesarealternatelyappliedtotheinputgraphuntilweendupwithvertices sandtonly–eitherjoined byanedgeordisconnected–whichisatrivialproblem. AsimpleimplementationofthisapproachgivesanO(n2m)algorithm. Usingfastmatrixmultiplication,thetime complexitycanbereducedtoO(nω+1)≈O(n3.373)andusingadynamicdatastructurefor“findingpathsanddeleting edgesindirectedacyclicgraphs”byItaliano[12],itcanbereducedstilltoO(n3). HerewedescribeourownO(n3)algorithm,itsadvantagesbeingsimplicity,extensibility,andimprovability: The algorithmdoesnotuseanyadvanceddatastructures. Itcanbeeasilyextendedtosolveproblemssuchas • findans–tpathpassingtheminimumnumberofforbiddenpairsor 5 • givenagraphwherealledgeshavescoresandtherearebonusesorpenaltiesforsome(well-parenthesized)pairs ofvertices,findans–tpathwithmaximumscore(thisproblemwasconsideredbyKova´cˇ etal.[5]). Itseemsunlikelythattheseproblemscanbesolvedusingtheformerapproach(becauseofrule2). Furthermore,our algorithm can be improved using the Valiant’s technique and fast matrix multiplication algorithms [13, 14] or the Four-Russianstechnique[15]. Notethatthereductiontomatrixmultiplicationisnotonlyoftheoreticalinterest,since therearefastandpracticalhardware-basedsolutionsformultiplyingtwomatrices[16,17]. Theorem3. ThePAFPproblemwithwell-parenthesizedforbiddenpairscanbesolvedinO(n3)time. Proof. Wemodifytheinputgraphsothatnotwoforbiddenpairsstartorendinthesamevertex. LetP[u,v]betrueif asafeu–vpathexistsandlet J[u,v]betrueifthereisaforbiddenpair(q,v) ∈ F,u ≺ q ≺ v,andthereisasafeu–v pathsuchthatthefirstedgejumpsoverq. ThevaluesofPand J canbefoundbydynamicprogramming: Itiseasytocompute J[u,v](ifwealreadyknow P[w,v]forallu ≺ w (cid:22) v)byinspectingtheneighboursofuandconversely,wecanalsocompute P[u,v]efficiently usingthetable J: Ifnoforbiddenpairendsinvorvertexuis“inside”theforbiddenpair(q,v) ∈ F, wejustsearch theneighboursofvforavertexthatcouldbepenultimateontheu–vpath. Otherwise, let(q,v) ∈ F beaforbidden pairsuchthatu ≺ q ≺ v. Supposethatasafeu–vpathexistsandletwbethelastvertexonthispathbeforeq. Then P[u,w] and J[w,v] are both true. Conversely, if P[u,w] and J[w,v] are true for some w ≺ q, by concatenating the correspondingpaths,wegetasafeu–vpath: Thepathobviouslyavoidsallforbiddenpairsbeforeorafterq(fromthe definitionofP[u,w]andJ[w,v])andtherearenoforbiddenpairshalving(q,v). Thus,P[s,t]canbecomputedincubictimeusingthefollowingtworecurrences: (cid:40) (cid:87) P[w,v] ifu≺qand(q,v)∈ F isaforbiddenpair (1) J[u,v]= (u,w)∈E,q≺w undefined otherwise  true ifu=v P[u,v]=  (cid:87)(cid:87)falus(cid:22)ew≺v,((Pw[,vu)∈,EwP]∧[u,Jw[w],v]) iiifff((nuqo,,vvfo))ri∈bsiFaddfieosnrabpifdaodirrebenidnpddaesinri,npuvai≺orrq(q≺,vv)∈ F forq≺u ((23)) u(cid:22)w≺q This algorithm can be further improved to O(nω) time by using fast boolean matrix multiplication. The proof is actually simple thanks to the work of Zakov et al. [14] that simplified and generalized the Valiant’s technique [13]. TheyintroduceagenericproblemcalledInsideVectorMultiplicationTemplate(VMT)whichcanbesolvedin subcubictime. AproblemisconsideredanInsideVMTproblemifitfulfillsthefollowingrequirements: 1. Thegoaloftheproblemistocomputeforeveryi, jaseriesofinsidepropertiesβ1 ,β2 ,...,βK. i,j i,j i,j (cid:76) (cid:16) (cid:17) 2. Let 1 ≤ k ≤ K, and let µk be a result of a vector multiplication of the form µk = βk(cid:48) ⊗βk(cid:48)(cid:48) , for i,j i,j q∈(i,j) i,q q,j some1 ≤ k(cid:48),k(cid:48)(cid:48) ≤ K. Assumethatthefollowingvaluesareavailable: µk ,allvaluesβk(cid:48) for1 ≤ k(cid:48) ≤ K and i,j i(cid:48),j(cid:48) (i(cid:48), j(cid:48))(cid:40)(i, j)andallvaluesβk(cid:48) for1≤k(cid:48) <k. Then,βk canbecomputedino(n)time. i,j i,j 3. In the multiplication variant that is used for computing µk , the ⊕ operation is associative, and the domain of i,j elementscontainsazeroelement. Inaddition,thereisamatrixmultiplicationalgorithmforthismultiplication variant,whoserunningtimeM(n)overtwon×nmatricessatisfiesM(n)=o(n3). Theorem 4 (Zakov et al. [14]). For every Inside VMT problem there is an algorithm whose running time is o(n3). Inparticular, let M(n)bethecomplexityofthematrixmultiplicationusedandsupposethatβk canbecomputedin i,j Θ(1) time in item 2 of the definition above. Then the time complexity is Θ(M(n)logn), if M(n) = O(n2logkn); and Θ(M(n)),ifM(n)=Ω(n2+ε)forε>2and4M(n/2)≤d·M(n)forsomed <1andsufficientlylargen. Corollary1. ThePAFPproblemwithwell-parenthesizedforbiddenpairscanbesolvedinO(nω)time,where2<ω< 2.3727istheexponentinthecomplexityofthebooleanmatrixmultiplication. 6 Proof. WeformulateoursolutionfromTheorem3asanInsideVMTproblem. Thegoalistocomputeinsideproper- tiesA,J,α,β,P,P(cid:100)•,andP•(cid:100). Properties J andP correspondtothedynamicprogrammingtablesfromtheproof u,v u,v ofTheorem3,otherpropertiesareauxilliary.PropertyAistheadjacencymatrixofgraphGanditisconstant(A =1 u,v ifandonlyif(u,v)∈ E). Propertiesα,βareusedtostorethepartialresultsfromcases(2)and(3)inthecomputation of P[u,v]. Finally,theauxiliaryproperties P(cid:100)• and P•(cid:100) canbecomputedfrom Pinconstanttimeandaredefinedas follows: P(cid:100)• = P ∧(q≺w) if(q,v)∈ F,elsefalse P•(cid:100) = P ∧(w≺q) if(q,v)∈ F,elsefalse w,v w,v w,v w,v NowwecanrewritethecomputationofJ andP usingbooleanvectormultiplicationasfollows: u,v u,v (cid:87) P[w,v] (cid:32) J =(cid:76) (A ⊗P(cid:100)•) (1’) (u,w)∈E,q≺w u,v w∈(u,v) u,w w,v (cid:87) P[u,w] (cid:32) α =(cid:76) (P ⊗A ) (2’) u(cid:22)w(cid:22)v,(w,v)∈E u,v w∈(u,v) u,w w,v (cid:87) (P[u,w]∧J[w,v]) (cid:32) β =(cid:76) (P•(cid:100) ⊗J ) (3’) u(cid:22)w≺q u,v w∈(u,v) w,v w,v PropertyP canbecomputedfromα andβ inconstanttime. u,v u,v u,v 5. Theothercasesandconcludingremarks NotethattheO(nω)algorithmforwell-parenthesizedforbiddenpairsalsoimprovesupontheresultbyChenetal. [4]forthenestedcase. Itremainsanopenproblemwhetherthereisamoreefficientalgorithmforthenestedcase. AnO(nω+1)timealgorithmforhalvingforbiddenpairsisachievedbyarefinedversionofthealgorithmgivenby KolmanandPangra´c[7].Recallthatinthiscase,theinputgraphGconsistsoftwoparts:alltheforbiddenpairsstartin thefirstpart,andendinthesecondpartinthesameorder. Letusdenotetheverticesinthefirstparts≺ x ≺···≺ x 1 n andverticesinthesecondparty ≺ ··· ≺ y ≺ t,where{x,y}areforbiddenpairs. Wemayassumethatallvertices 1 n i i areaccessiblefromsandthattisaccessiblefromeveryvertex. Ifthereisadirectedgefromstothesecondpartorifthereisanedgefromthefirstparttot,asafes–tpathexists trivially. Otherwise,wereducethehalvingcasetoninstancesofthenestedcase. Therewillbeasafes–tpathinGif andonlyifthereisasafes–t(cid:48)pathinatleastoneoftheproducedinstances. First,removeallthe(x,y )edges,addanewterminalvertext(cid:48),andreversethedirectionofalledgesinthesecond i j part ofG. Note that in this new order, s ≺ x ≺ ··· ≺ x ≺ t ≺ y ≺ ··· ≺ y ≺ t(cid:48), the forbidden pairs are nested. 1 n n 1 The k-th instance is obtained by adding edges (x ,t) and (y ,t(cid:48)) for each edge (x ,y ), so there is a safe s–t path k (cid:96) k (cid:96) s,...,x ,y ,...,tintheoriginalgraphGifandonlyifthereisasafes–t(cid:48)paths,...x ,t,...,y ,t(cid:48)inthenewgraph. k (cid:96) k (cid:96) Itremainsanopenproblemwhetheramoreefficientalgorithmexists. Acknowledgements. The autor would like to thank Bronˇa Brejova´ for many constructive comments. The research of Jakub Kova´cˇ is supported by APVV grant SK-CN-0007-09 Marie Curie Fellowship IRG-231025 to Dr. Bronˇa Brejova´, Come- niusUniversitygrantUK/121/2011,andbyNationalScholarshipProgramme(SAIA),SlovakRepublic. Preliminary versionofthisworkappearedinKova´cˇ etal.[5]. References [1] K.Krause, R.Smith, M.Goodwin, Optimalsoftwaretestplanningthroughautomatednetworkanalysis, in: Proc.1973IEEESymp.on ComputerSoftwareReliability,18–22,1973. [2] P.K.Srimani,B.P.Sinha,Impossiblepairconstrainedtestpathgenerationinaprogram,Inf.Sci.28(2)(1982)87–103. [3] H.N.Gabow,S.N.Maheswari,L.J.Osterweil,OnTwoProblemsintheGenerationofProgramTestPaths,IEEETrans.SoftwareEng.2(3) (1976)227–231. [4] T.Chen,M.-Y.Kao,M.Tepel,J.Rush,G.M.Church,ADynamicProgrammingApproachtoDeNovoPeptideSequencingviaTandem MassSpectrometry,J.Comput.Biol.8(3)(2001)325–337. [5] J.Kova´cˇ,T.Vinaˇr,B.Brejova´,PredictingGeneStructuresfromMultipleRT-PCRTests,in:S.Salzberg,T.Warnow(Eds.),WABI,vol.5724 ofLectureNotesinComputerScience,Springer,ISBN978-3-642-04240-9,181–193,2009. [6] H.Yinnone,OnPathsAvoidingForbiddenPairsofVerticesinaGraph,DiscreteAppl.Math.74(1)(1997)85–92. 7 [7] P.Kolman,O.Pangra´c,Onthecomplexityofpathsavoidingforbiddenpairs,DiscreteAppl.Math.157(13)(2009)2871–2876. [8] M.Hajiaghayi,R.Khandekar,G.Kortsarz,J.Mestre,TheCheckpointProblem,in:M.J.Serna,R.Shaltiel,K.Jansen,J.D.P.Rolim(Eds.), APPROX-RANDOM,vol.6302ofLectureNotesinComputerScience,Springer,ISBN978-3-642-15368-6,219–231,2010. [9] H.L.Bodlaender,B.M.P.Jansen,S.Kratsch,KernelBoundsforPathandCycleProblems,in:Proc.ofthe6thInternationalsymposiumon ParameterizedandExactComputation(IPEC),2011. [10] D.Coppersmith,S.Winograd,MatrixMultiplicationviaArithmeticProgressions,J.Symb.Comput.9(3)(1990)251–280. [11] V.Williams,BreakingtheCoppersmith-Winogradbarrier,2011. [12] G.F.Italiano,FindingPathsandDeletingEdgesinDirectedAcyclicGraphs,Inf.Process.Lett.28(1)(1988)5–11. [13] L.G.Valiant,GeneralContext-FreeRecognitioninLessthanCubicTime,J.Comput.Syst.Sci.10(2)(1975)308–315. [14] S.Zakov,D.Tsur,M.Ziv-Ukelson,ReducingtheworstcaserunningtimesofafamilyofRNAandCFGproblems,usingValiant’sapproach, AlgorithmsMol.Biol.6(1)(2011)20. [15] V.Arlazarov,E.Dinic,M.Kronrod,I.Faradzev,Oneconomicconstructionofthetransitiveclosureofadirectedgraph,in: SovietMath. Dokl.,vol.11,1209–1210,1970. [16] S.Ryoo,C.I.Rodrigues,S.S.Baghsorkhi,S.S.Stone,D.B.Kirk,W.meiW.Hwu,Optimizationprinciplesandapplicationperformance evaluationofamultithreadedGPUusingCUDA,in:S.Chatterjee,M.L.Scott(Eds.),PPOPP,ACM,ISBN978-1-59593-795-7,73–82,2008. [17] V.Volkov,J.Demmel,BenchmarkingGPUstotunedenselinearalgebra,in:SC,IEEE/ACM,ISBN978-1-4244-2835-9,31,2008. 8

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.