An Inclusion-Exclusion Algorithm for Network Reliability with Minimal Cutsets

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1 Amercan Journal of ompuaonal Mahemacs, 202, 2, 6-20 hp://dxdoorg/0426/acm Publshed Onlne December 202 (hp://wwwscrporg/ournal/acm) An Incluson-Excluson Algorhm for ework Relably wh Mnmal uses Yan-Ru Sun, We-Y Zhou School of Scences, orheasern Unversy, Shenyang, hna 2 School of Elecronc Informaon, Wuhan Unversy, Wuhan, hna Emal: yanrusun@26com, wey0564@26com Receved Augus 0, 202; revsed Ocober 9, 202; acceped Ocober 2, 202 ABSTRAT The ncluson-excluson formula (IEF) s a fundamenal ool for evaluang nework relably wh known mnmal pahs or mnmal cus However, he formula conans many pars of erms whch cancel Usng he noon of comparable node parons some properes of cancelng erms n IEF are gven Wh hese properes he hough of dynamc programmng mehod, a smple effcen ncluson-excluson algorhm for evaluang he source-o-ermnal relably of a nework sarng wh cuses s presened The algorhm generaes all he non-cancelng erms n he unrelably expresson The compuaonal complexy of he algorhm s Onm M, where n m are he numbers of nodes mnmal cus of he gven nework respecvely, M s he number of erms n he fnal symbolc unrelably expresson ha generaed usng he presened algorhm Examples are shown o llusrae he effecveness of he algorhm Keywords: Incluson-Excluson Formula; ework Relably; Mnmal use; Dynamc Programmng Inroducon The relably of a nework s an mporan parameer n desgn operaon of neworks There are many mehods o compue he relably of neworks [,2] Several algorhms exs n he leraure for evaluang he relably of a dreced graph by ncluson-excluson formula (IEF) based on eher pah (k-ree) enumeraon or cuse enumeraon [-8] In fndng he k-ermnal relably by IEF here are wo approaches, one based on enumerang all k-rees he oher based on enumerang all k-ermnal cus If here are m mnmal pahs (or cus) n a graph, here are 2 m possble nersecon erms n IEF However, he number of non-cancellng erms n IEF s consderably less Sarng wh he se of pahs (or k-rees) of a dreced graph, Sayanarayana coworkers [5,6] developed mehods of denfyng non-cancellng erms n IEF They showed ha he non-cancelng erms of he sourceo-ermnal relably correspond one-o-one wh he p-acyclc subgraphs of he gven graph An algorhm was gven for generang all he p-acyclc subgraphs of a dreced graph [5] Buzaco [] gave a correspondng resul for he non-cancellng erms n IEF sarng wh he se of cus of a graph Snce each erm n he resulng formula s assocaed wh a paron of he se of nodes of he graph, was called he node paron formula Fnd all he node parons of a graph s a very edous work Usng a lemma (he Lemma 4 of []) of ncomparable node parons of [] characerscs of cancelng erms n IEF, by he hough of dynamc programmng mehod a smple effcen ncluson-excluson algorhm s gven n hs paper for evaluang he sourceo-ermnal relably of a graph based on mnmal cus The algorhm generaes only he non-cancelng erms of he relably expresson of he graph 2 omenclaure, oaon Assumpon A nework s modeled as a dreced graph, E (abbrevaed o ) whch consss of a se of nodes a se of edges (lnks) E A node s s he source of s s he ermnal of 2 omenclaure Source o ermnal (s ) relably: he probably ha he source s s conneced o he ermnal node by pahs workng edges s cu: a subse of edges whose removal dvdes he node se of he nework no wo pars such ha, s, e he edge se from o opyrgh 202 ScRes

2 Y-R SU, W-Y ZHOU 7 Mnmal s cu: s cu whch no longer remans a s cu f s any edge s removed dae chld se of : an ordered se,,, 2 r conssng of all he node ses such ha, 2,, r, 2 r Incomparable node ses: a par of node ses 2 such ha oaon : subse of such ha s : complemen of,, : cu, e he edge se from o : ) cu, ) even ha all he edges of cu fal : ) unon of all he edges of cus 2) nersecon of evens U : unon of cus ( ): cdae chld se of Pr : he probably of even Q s, : source-o-ermnal (s ) unrelably of 2 Assumpon ) has perfecly relable nodes s-ndependen 2-sae (good faled) edges, he relably of each edge has been gven 2) Le,, be mncus of, hen, s also a mn- cu of Prelmnares 2 Le,,, m be he se of mncus of a gven nework where corresponds one-o-one wh he node se, e, (, 2,, m) The s unrelably of, by IEF, can be expressed as Q s, Pr 2 Pr Pr m m m Pr k Pr 2m km m he summaons are over all mncus mncu combnaons In formula (), here exs 2 m possble erms Bu s possble ha U U for some,, Indeed he mos vexng problem n relably analyss usng () s he appearance of large numbers of pars of dencal erms wh oppose sgn, whch cancel Fnd he charac- () ersc of cancelng erms n () s he keysone of an effcen algorhm Buzaco gave a smple very useful lemma (Lemma 4 of Ref []) o denfy some cancelng erms n () Lemma ven any wo mncus,, of such ha are ncom- parable, all erms n IEF conanng boh, s also a mncu [] cancel f In formula (), assume ha 2 m Accordng o Lemma, () can be changed no: Q s, Pr 2 m Pr Pr m m k km Pr k k Pr 2 k 2 k m 2 he summaons are over all mncus mncu combnaons ha sasfy he gven condons The erms n (2) can correspond one-o-one he verex of he m rooed rees wh he followng properes ) The roo verex of each rooed ree s he verex correspondng o cu se, s wegh s, sgn s + 2) Sons of each verex n every rooed ree are all elemens n, each son s wegh s he unon of s faher s wegh he cu se correspondng o hs son verex, sgn s s faher s sgn mes For example, le, 2 4, k (2), 2,, 4 Fgure are four rooed rees In Fgure (a), ree ( ) has a only verex, he roo verex Is wegh s 4, sgn s In Fgure (c), ree ( 2 ) s roo verex s 2, s wegh s 2, sgn s 2 has wo sons: The son verex s wegh equals o 2, sgn s ; s wegh equals o 4, sgn s s son s, here s wegh equals o 2 4, sgn s The wegh wh s sgn of each node n he rooed rees one-o-one corresponds wh he erm n he expresson of formula (2) So we dscuss m rooed rees gener- ang The rooed ree whose roo s s denoed as ree In fac, f we generae he rooed rees n non-ncreasng order of roo s modulus generae he sons of each verex n non-decreasng order of he son s modulus, we can use he rees whch have already been generaed o generae he followng rees For example, Fgure are four rooed rees, where ree s a branch of ree 2, ree 2 ree are he branches of ree If ree s generaed frsly, we can use he resul when we generae ree And when we 2 opyrgh 202 ScRes

3 8 Y-R SU, W-Y ZHOU (a) 2 he generang processes of he rees wh above properes ) 2) Such ha rees verces correspond wh he non-cancelng erms of (2) 4 Algorhm (b) (c) Ths secon presens an algorhm for effcenly generang all he non-cancelng erms n (2) The algorhm has four pars, The man par s o generae all rees whose verces correspond wh he non-cancelng erms of (2) 2 (d) Fgure Rooed sub-rees (a) Sub-ree( ); (b) Sub-ree( ); (c) Sub-ree( 2 ); (d) Sub-ree( ) generae ree, we can use ree ree 2 drecly Ths s he hough of dynamc programmng By hs hough he process of generang rees s grealy smplfed In formula (2) here are sll many erms ha can cancel each oher The properes of cancelng erms are dscussed as follow Theorem Le,, 2, be hree mncus of a dreced graph, 2, 2 Then for any mncus, ha, ha of, , 2 4 4,, 2, be hree Theorem 2 Le mncus of a dreced graph wh 2 If 2 hen doesn exs mncu 0, 0 0 of, 0 such ha ; doesn exs 4, 4 of, 4 such ha Theorems 2 mply ha f we fnd ou he all pars of cancelng erms ha unon of wo cus hree cus, e fnd ou all U 2 U wh U2 U, he all cancelng erms n (2) can be deermned The followng lemma gves a condon ha U2 U n (2) Lemma 2 Le,, 2, be hree mnmal cus of a dreced graph 2 Then 2 f only f 2, 2 Accordng o Theorems 2 Lemma 2 we gve 4 Algorhm ) Fnd all he mncus of he gven dreced nework, E whch sasfes assumpons Le, 2,, m be he mncus correspondng o he node parons, 2,, m, respecvely Order he node parons as, 2,, m such ha 2 m 2) Fnd, 2,, m ) enerae m rooed rees by he followng Algorhm-Tree, e generae all he non-cancelng erms of Q s, 4) Sum up he weghs wh sgn of verces of all he rees o oban he symbolc expresson of Q s, Fnally, we ge he symbolc relably expresson of R s, Q s, 42 Algorhm Tree By heorems 2, all he pars of cancelng erms n IEF can be known f we fnd he cancelng erms whch unon of wo hree cu ses Usng hs propery an algorhm Algorhm Tree s gven I has wo pars: Trees eneraon Weghed Trees I generaes rooed rees n he non-ncrease order of he roo verex s modulus, e generaes ree ( m ), ree ( m- ),, ree ( ) successvely 42 Trees eneraon We shall gve an algorhm o generae all rees as follow Algorhm Trees eneraon Inpu: ) he node parons of :, 2,, m such ha 2 m he correspondng mncus, 2,, m 2) he cdae chld ses:,,,,, 2 k k k2 k,,,,,, m m m Oupu: all he rees Begn Sep enerae he frs ree m wh he only verex, e roo verex m Sep 2 enerae he second ree wh a roo k m opyrgh 202 ScRes

4 Y-R SU, W-Y ZHOU 9 verex m s only son verex m Sep Suppose ha rees,, 2,, k m m have been generaed enerae ree mk ) The roo of ree m k s mk 2) enerae sons (we call hem he frs-generaon offsprng) of :,,, (where m k mk, mk,2 mk, m k mk,,, 2,, mk, k mk m, m k m k, mk m); mk,, mk,2,,, m k m k m k m k ) Whle m k do Begn Denoe he elemen wh he mnmal modulus n m k as m k,, mk m k m k m k, m k wh ree m k, Subsue s son m k, Denoe he son se of hs verex Suppose mk,,, m k, mk, mk, 2 k, k non-decreasng order of her modulus) mk mk 0,, For = o Begn If k mk, mk, m m k m k m k mk mk m k mk mkm k, ; as m k,,,, m k,, hen,,, ; (n he u m k s son m k, m k, s son mk, wh s offsprng from curren ree ; mk, Else nex End If 0 m k, m k,, hen mark, m k, called sll node, denoed as sll node m k, End Sep 4 Repea sep unl all he rees, m, m,,2, have been generaed End 422 Weghed Trees Tree s wegh s defned he sum of roo s all verces weghs wh her sgn In he order of generang rees, sarng from each ree s roo verex gves each verex a wegh n dephfrs-search The roo s wegh s he all edges of he cu se correspondng o he roo verex, sgn s + Each non-sll verex s wegh equals o he unon of s fa- s Fgure 2 ework her s wegh all he edges of he cu se correspondng hs verex, s sgn s s faher s sgn mes Each sll node s wegh equals o he unon of s faher s wegh he ree s wegh whose roo s hs verex s sgn s s faher s sgn mes 5 ompuaonal omplexy The man par of he presened algorhm s Algorhm Sub-ree I has wo pars One s Sub-rees eneraon, he oher s Weghed Sub-rees The man work of algorhm Sub-rees eneraon s o deermne wheher here exs edges beween wo ses I a mos needs m m2mm 2 comparson operaons for each sub-ree Weghed sub-rees runs n OM, where M s he number of erms n he las symbolc un-relably expresson In fac, M s more smaller han 2 m For example,he nework n Fgure 2, m = 8, m , bu M = 5 For each of sub-rees, oher operaons a mos ake Om me I akes On me o fnd all mncus, where n s he number of nodes of a gven nework I akes Om me o fnd each cdae chld se So he compuaonal complex- y of he presened algorhm s O nm M 6 oncluson Ths paper presens an effcen algorhm for evaluang he relably of nework based mncus The algorhm generaes all he non-cancelng erms n he unrelably expresson By he hough of dynamc programmng mehod each verex a mos generaes wo generaons chldren n every sub-ree The number of verces of he generaed sub-rees are more smaller han he number of non-cancelng erms n Q s, s expresson The algorhm has smaller me complexy REFEREES [] J olbourn, The ombnaorcs of ework Relably, Oxford Unversy Press, ew York, Oxford, 987 [2] M O Ball, J olbourn J S Provan, ework Relably, Hbook of Operaons Research: ework Models, Elsever orh-holl, Amserdam, Vol 7, 995, pp opyrgh 202 ScRes

5 20 Y-R SU, W-Y ZHOU [] J A Buzaco, ode Paron Formula for Dreced raph Relably, eworks, Vol 7, o 2, 987, pp do:0002/ne [4] J A Buzaco S K hang, u Se Inersecons ode Paron, IEEE Transacons on Relably, Vol, o 4, 982, pp [5] A Sayanarayana A Prabhakar, ew Topologcal Formula Rapd Algorhm for Relably Analyss of omplex eworks, IEEE Transacons on Relably, Vol 27, o, 978, pp do:009/tr [6] A Sayanarayana J Hagsrom, A ew Algorhm for Relably Analyss of Mul-Termnal eworks, IEEE Transacons on Relably, Vol 0, o 4, 98, pp 25-4 do:009/tr [7] L Zhao F J Kong, A ew Formula an Algorhm for Relably Analyss of ework, Mcroelecron Relably, Vol 7, o 4, 997, pp 5-58 [8] W Yeh, A reedy Branch--Bound Incluson-Excluson Algorhm for alculang he Exac Mul-Sae ework Relably, IEEE Transacons on Relably, Vol 57, o, 2008, pp 88-9 do:009/tr opyrgh 202 ScRes