Mechtild Stoer * Frank Wagner** Abstract. fastest algorithm known. The runtime analysis is straightforward. In contrast to

Transcription

1 SERIE B INFORMATIK A Simple Min Cu Algorihm Mechild Soer * Frank Wagner** B 9{1 May 199 Abrac We preen an algorihm for nding he minimum cu of an edge-weighed graph. I i imple in every repec. I ha a hor and compac decripion, i eay o implemen and ha a urpriingly imple proof of correcne. I runime mache ha of he fae algorihm known. The runime analyi i raighforward. In conra o nearly all approache o far, he algorihm ue no ow echnique. Roughly poken he algorihm coni of abou jv j nearly idenical phae each of which i formally imilar o Prim' minimum panning ree algorihm. *Konrad-Zue-Zenrum fur Informaionechnik Berlin, Heilbronner Sr. 10, Berlin, Germany, e- mail: oer@zib-berlin.de. **Iniu fur Informaik, Fachbereich Mahemaik und Informaik, Freie Univeria Berlin, Takurae 9, 1195 Berlin-Dahlem, Germany, wagner@mah.fu-berlin.de.

2 1 Overview Graph conneciviy i one of he claical ubjec in graph heory, and ha many pracical applicaion, e. g. in chip and circui deign, reliabiliy of communicaion nework, ranporaion planning and cluer analyi. Finding he minimum cu of an edge-weighed graph i a fundamenal algorihmical problem. Preciely, i coni in nding a nonrivial pariion of he graph' verex e V ino wo par uch ha he um of he weigh of he edge connecing he wo par i minimum. The uual approach o olve hi problem i o ue i cloe relaionhip o he maximum ow problem. The famou Max-Flow-Min-Cu-Theorem by Ford and Fulkeron [FF5] howed he dualiy of he maximum ow and he o-called minimum - cu. There, and are wo verice which are he ource and he ink in he ow problem and have o be eparaed by he cu, i.e., hey have o lie in dieren par of he pariion. Unil recenly all cu algorihm were eenially ow algorihm uing hi dualiy. Finding a minimum cu wihou pecied verice o be eparaed can be done by nding minimum - cu for all O(jV j ) pair of verice and hen elecing he lighe one. Gomory and Hu [GH1] reduced hi o O(jV j) pair of verice and hu O(jV j) maximum ow compuaion. Recenly Hao and Orlin [HO9] howed how o ue he maximum ow algorihm by Goldberg and Tarjan [GT88], he fae o far, in order o olve he minimum cu problem a fa a he maximum ow problem, i.e., in ime O(jV jjej log(jv j =jej). In he ame year Nagamochi and Ibaraki [NI9a] publihed he r minimum cu algorihm ha i no baed on a ow algorihm, ha he lighly beer running ime of O(jV jjej + jv j log jv j) bu i ill raher complicaed. In he unweighed cae hey ue a fa earch echnique o decompoe a graph' edge e E ino ube E 1 ::: E uch ha he union of he r k E i ' i a k-edge-conneced panning ubgraph of he given graph and ha ize a mo kjv j. Theyimulae hi approach in he weighed cae. Their work i one of a number of paper reaing queion of graph conneciviy by non-ow-baed mehod [NP89, NI9a, M9]. In hi conex we preen in hi paper a remarkably imple minimum cu algorihm wih he opimal running ime eablihed in [NI9b]. We reduce he complexiy of he algorihm of Nagamochi and Ibaraki by avoiding he unneceary imulaed decompoiion of he edge e. Thi enable u o give a comparably raighforward proof of correcne avoiding e. g. he diincion beween he unweighed, ineger-, raional-, and real-weighed cae. The Algorihm Throughou he paper we deal wih an ordinary undireced graph G wih verex e V and edge e E. Every edge e ha poiive realweigh w(e). In order o decribe he idea of he algorihm we ar by reminding he reader of Prim' minimum panning ree algorihm: MinimumSpanningTree(G w a)

3 A Simple Min Cu Algorihm A fag T while A = V add o A he mo looely conneced verex add o T he connecing edge A ube A of he graph' verice grow aring wih an arbirary ingle verex unil A i equal o V. In each ep he verex ouide of A mo looely conneced wih A i added. Formally, we add a verex z A uch ha w(a z) =minfw(b y) j b A y A by Eg where w(b y) i he weigh ofedgeby and az i he connecing edge. A he end he e of all he connecing edge hen form a minimum panning ree. The imple minimum cu algorihm we decribe here coni of jv j;1 phae each of which i very imilar o Prim' algorihm: MinimumCuPhae(G w a) A fag while A = V add o A he mo ighly conneced verex ore he cu-of-he-phae and hrink G by merging he wo verice added la A ube A of he graph' verice grow aring wih an arbirary ingle verex unil A i equal o V. In each ep he verex ouide of A mo ighly conneced wih A i added. Formally, we add a verex z A uch ha w(a z) = maxfw(a y) j y Ag where w(a y) iheumofheweigh of all he edge beween A and y. A he end of each uch phae he wo verice added la are merged, i.e., he wo verice are replaced by anewverex, and any edge from he wo verice o a remaining verex are replaced by anedgeweighed by he um of he weigh of he previou wo edge. Edge joining he merged node are removed. The cu of V ha eparae he verex added la from he re of he graph i called he cu-of-he-phae. The lighe of hee cu-of-he-phae i he reul of he algorihm, he deired minimum cu. MinimumCu(G w a) while jv j > 1 MinimumCuPhae(G w a) if he cu-of-he-phae i ligher han he curren minimum cu hen ore he cu-of-he-phae a he curren minimum cu Noice ha he aring verex a ay he ame hroughou he whole algorihm. An Example

4 w(1 ) = Figure 1: A graph G =(V E) wih edge-weigh. 1 a b c f d 8 e Figure : The graph afer he r MinimumCuPhae(G w a), a =, and he induced ordering a b c d e f of he verice. The r cu-of-he-phae correpond o he pariion f1g f 5 7 8g of V wih weigh w =5. a d e b 7 1 c 8 Figure : The graph afer he econd MinimumCuPhae(G w a), and he induced ordering a b c d e of he verice. The econd cu-of-he-phae correpond o he pariion f8g f1 5 7g of V wih weigh w =5.

5 A Simple Min Cu Algorihm 5 a d b c Figure : Afer he hird MinimumCuPhae(G w a). The hird cu-of-he-phae correpond o he pariion f7 8g f1 5 g of V wih weigh w =7. a a b c b Figure 5: Afer he fourh and fh MinimumCuPhae(G w a), repecively. The fourh cu-of-he-phae correpond o he pariion f 7 8g f1 5 g. The fh cu-of-he-phae correpond o he pariion f 7 8g f1 5 g wih weigh w =.

6 a V n Figure : Afer he ixh and evenh MinimumCuPhae(G w a) repecively. The ixh cu-of-he-phae correpond o he pariion fg f 7 8g wih weigh w = 7. The la cu-of-he-phae correpond o he pariion fg V nfg i weigh i w = 9. The minimum cu of he graph G i he fh cu-of-he-phae and he weigh iw =. Correcne The core of he proof of correcne i he following omewha urpriing lemma. Lemma Each cu-of-he-phae i a minimum - cu in he curren graph, where and are he wo verice added la in he phae. Auming, ha he lemma hold, we can how by a imple cae diincion, ha he malle of hee cu-of-he-phae i indeed he minimum cu we are looking for. Thi i done by inducion on jv j. The cae jv j = i rivial. If jv j, look a he r phae: If G ha a minimum cu, ha i a he ame ime a minimum - cu, hen, according o he lemma, he cu-of-he phae i already a minimum cu. If no, hen G ha a minimum cu wih and on he ame ide. Therefore, a minimum cu of G 0, he inpu graph of phae, ha dier from G by he merging of and, i a minimum cu of G. Now, by inducion, he lighe of he cu-of-he-phae ojv j;1iuch a minimum cu of G 0. Noice ha he applicaion of phae o jv j;1og 0 i he ame a he applicaion of he complee algorihm o G 0. Finally, we how he claimed propery of he cu-of-he-phae. The run of a MinimumCuPhae order he verice of he curren graph linearly, aring wih a and ending wih and, according o heir order of addiion o A. Nowwelook a an arbirary - cu C of he curren graph and how, ha i i a lea a heavy a he cu-of-he-phae.

7 A Simple Min Cu Algorihm 7 We callaverex v = a acive (wih repec o C) when v and he verex added ju before v are in dieren parofc. Le w(c) be he weigh ofc, A v he e of all verice added before v (excluding v), C v he pariion of A v [fvg induced by C, and w(c v ) he weigh of he induced cu, i.e., he um of he weigh of he edge going from one par of he induced pariion o he oher. We how ha for every acive verex v w(a v v) w(c v ) by inducion on he e of acive verice: For he r acive verex he inequaliy i aied wih equaliy. Le he inequaliy be rue for all acive verice added up o he acive verex v, and le w be he nex acive verex ha i added. Then we have w(a w w)=w(a v w)+w(a w n A v w)=: Now, w(a v w) w(a v v)a v wa choen a he verex mo ighly conneced wih A v. By inducion w(a v v) w(c v ). All edge beween A w n A v and w connec he dieren parofc. Thu hey conribue o w(c w bu no o w(c v ). So w(c v )+w(a w n A v w) w(c w ) A i alway an acive verex wih repec o C we can conclude ha w(a ) w(c ) which ay exacly ha he cu-of-he-phae i a mo a heavy a C. 5 Running Time A he running ime of he algorihm MinimumCu i eenially equal o he added running ime of he jv j;1 run of MinimumCuPhae, which i called on graph wih decreaing number of verice and edge, i uce o how ha a ingle MinimumCuPhae need a mo O(jEj + jv j log jv j) ime yielding an overall running ime of O(jV jjej + jv j log jv j). The key o implemening a phae ecienly i o make i eay o elec he nex verex o be added o he e A, he mo ighly conneced verex. During execuion of a phae, all verice ha are no in A reide in a prioriy queue baed on a key eld. The key of a verex v i he um of he weigh of he edge connecing i o he curren A, i.e.,w(a v). Whenever a verex v i added o A we have o perform an updae of he queue. v ha o be deleed from he queue, and he key of every verex w no in A, conneced o v ha o be increaed by he weigh of he edge vw, if i exi. A hi i done exacly once for every edge, overall we have o perform jv j ExracMax and jej IncreaeKey operaion. Uing Fibonacci heap [FT87], we can perform an ExracMax operaion in O(log jv j) amorized ime and a IncreaeKey operaion in O(1) amorized ime. Thu he ime we need for hi key ep ha dominae he re of he phae, i O(jEj + jv j log jv j). Noice, ha hi runime analyi i very imilar o he analyi of Prim' minimum panning ree algorihm.

8 8 Acknowledgemen The auhor hank Dorohea Wagner for her helpful remark. Reference [FT87] M. L. Fredman and R. E. Tarjan, Fibonacci heap and heir ue in improved nework opimizaion algorihm, Journal of he ACM (1987) 59{ 15 [FF5] L. R. Ford, D. R. Fulkeron, Maximal ow hrough a nework, Canadian Journal on Mahemaic 8 (195) 99{0 [GT88] A. V. Goldberg and R. E. Tarjan, A new approach o he maximum ow problem, Journal of he ACM 5 (1988) 91{90 [GH1] R. E. Gomory, Muli-erminal nework ow, Journal of he SIAM 9 (191) 551{570 [HO9] X. Hao and J. B. Orlin, A faer algorihm for nding he minimum cu inagraph, rd ACM-SIAM Sympoium on Dicree Algorihm (199) 15{17 [M9] D. W. Maula A linear ime + approximaion algorihm for edge conneciviy, Proceeding of he h ACM-SIAM Sympoium on Dicree Mahemaic (199) 500{50 [NI9a] H. Nagamochi and T. Ibaraki, Linear ime algorihm for nding a pare k-conneced panning ubgraph of a k-conneced graph, Algorihmica 7 (199) 58{59 [NI9b] H. Nagamochi and T. Ibaraki, Compuing edge-conneciviy in muligraph and capaciaed graph, SIAM Journal on Dicree Mahemaic 5 (199) 5{ [NP89] T. Nihizeki and S. Poljak, Highly conneced facor wih a mall number of edge, Preprin (1989) [P57] R. C. Prim, Shore connecion nework and ome generalizaion, Bell Syem Technical Journal (1957) 189{101