Cataloging Data : Bastrt, Olivr: Nw Idas for Canonically Computing Graph Algbras; Tchn. Univ. Munchn, Fak. f. Math, Rport TUM M9803 (98) Mathmatics Su

Transcription

1 T U M Fakultat fur Mathmatik Nw Idas for Canonically Computing Graph Algbras Bastrt, Olivr ABCDE FGHIJ KLMNO TUM-M9803 Juni 98 Tchnisch Univrsitat Munchn

2 Cataloging Data : Bastrt, Olivr: Nw Idas for Canonically Computing Graph Algbras; Tchn. Univ. Munchn, Fak. f. Math, Rport TUM M9803 (98) Mathmatics Subjct Classication : 68R10;05C85;05E99 Editor : H. Wahling (wahling@mathmatik.tu-munchn.d), Fakultat fur Mathmatik dr Tchnischn Univrsitat Munchn, D Munchn, Grmany. For th Elctronic Vrsion s : Copyright c XCVIII Fakultat fur Mathmatik und Fakultat fur Informatik dr Tchnischn Univrsitat Munchn. D Munchn. All rights rsrvd. Printd in Grmany.

3 Nw Idas for Canonically Computing Graph Algbras Olivr Bastrt Zntrum Mathmatik Tchnisch Univrsitat Munchn, Grmany Th work rportd in this papr has bn partially supportd by th Grman Israli Foundation for Scintic Rsarch and Dvlopmnt undr contract # I /93. 1

4 Abstract In this papr, w discuss som nw idas on computing cohrnt algbras. A spac optimal algorithm which computs a canonical cohrnt coloring and fullls th bst known tim bound of O(n 3 log n) is givn. Morovr, w prsnt computational rsults obtaind by an implmntation of th algorithm. 2

5 1 Introduction Cohrnt algbras hav bn introducd and studid by B. J. Wisfilr and A. A. Lhman in [21]. A combinatorially quivalnt notion is th on of cohrnt congurations (s Higman [11]). A short history of both of ths and som rlatd notions ar givn in [14]. Nowadays, cohrnt algbras ar wll studid objcts in algbraic graph thory and hav many applications in various aras of graph thory. Rcntly, rsults on rcombination spacs [17] and rcognition of circulant graphs [15] hav bn obtaind by using cohrnt algbras. In addition, cohrnt algbras play an important rol in th thory of graph isomorphism problms: a ncssary condition for th xistnc of an isomorphism btwn two graphs is that th corrsponding cohrnt algbras ar isomorphic. In som cass,.g. for algbraic forsts which wr introducd in [7], this condition is vn sucint. Furthrmor, cohrnt algbras turn out to b usful in a polyhdral approach to th graph isomorphism problm. Namly, thy ar th foundation in th thory of wakly compact graphs (s [6] and [7]) which wr studid in continuation of th work on compact graphs (s [18], [19] and [10]). Wisfilr and Lhman wr th rst to show that cohrnt algbras of graphs (i.., thir standard bass) can b computd in polynomial tim [20]. If an algorithm computs cohrnt algbras in a crtain \canonical" way, thn it is straightforward to tst whthr th cohrnt algbras gnratd by two graphs ar isomorphic or not. Morovr, such an implmntation can asily b mbddd into a framwork for tsting isomorphism of arbitrary graphs (s [3] for dtails). Sinc computing cohrnt algbras is a spcial partitioning problm, it can b solvd in tim O(n 3 log n) (s [1], [2] and [13] for dtails). Th crucial point is that a straightforward algorithm nds O(n 3 ) spac. Th main rsults of this papr ar that w ar abl to rduc th spac rquirmnts to O(n 2 ) and that an implmntation basd on this algorithm provs to b vry cint in practic. W start with som basic dnitions and facts and giv a short ovrviw of diffrnt algorithms for computing cohrnt algbras and of thir implmntations that w know. Thn w prov that cohrnt algbras can b computd in tim O(n 3 log n) and spac O(n 2 ). In th last sction, an implmntation of th algorithm prsntd in this papr is discussd and computational rsults ar givn. 2 Dnitions and Basics Lt G = (V; E; f) b a complt dirctd colord graph, i.. G consists of a vrtx st V := f1; 2; : : : ; ng, an dg st E := V V and a coloring f : V V?! f1; 2; : : : ; n 2 g. By u, v, w, w always dnot vrtics, always dnots an dg, b, c and d dnot colors and w assum w.l.o.g. that f maps onto F := f1; 2; : : : ; n f g, n f := jf(e)j. A st of thr dgs of th form f(u; v); (u; w); (w; v)g is calld a triangl and will b dnotd by (u; v; w). Th dg (u; v) is th basis dg and th dgs (u; w), (w; v) ar th non basis dgs of th triangl. Furthrmor, w idntify a loop (u; u) with its vrtx u. W assum f(u) 6= f() for all vrtics u and dgs. 3

6 Edgs with th sam color c ar collctd in a color class C(c) := f j f() = cg and w dn C() := C(f()). For an dg st F, w dn F t := f(v; u) j (u; v) 2 F g. Instad of fg t, w writ t. W assum w.l.o.g. that th coloring f fullls th condition 8c9c : C(c) t = C(c): (1) If f dos not hav this proprty, w rn f appropriatly. Th intgrs p c;d := jfw 2 V j f((u; w)) = c; f((w; v)) = d and = (u; v)gj ar calld th structur valus of G. p c;d is th numbr of triangls with basis dg whos non basis dgs ar colord with th colors c and d. Lt L() := f(c; d; p c;d ) j p c;d 6= 0g b th structur list of and L(c; d) := f(; p c;d ) j p c;d 6= 0g b th structur list of c and d. In L(), th numbrs of triangls distinguishd by th colors of th non basis dgs which contain as basis dg ar collctd. L(c; d) collcts dgs and thir numbr of triangls whos non basis dgs ar colord with th colors c and d. Now, w ar abl giv a graph thortical formulation of th Wisfilr-Lhman algorithm. Algorithm 1 Wisfilr-Lhman Input: G = (V; E; f) Output: A stabl coloring f of G 1: rpat 2: comput L(); 8 2 E 3: rcolor, i.., f() = f( 0 ) :, L() = L( 0 ); ; 0 2 E 4: until n f did not chang On itration of Algorithm 1 is calld a stp (lins 2,3) and will b dnotd by hstpi. Wisfilr and Lhman stat thir algorithm in trms of matrix multiplication. Instad of a colord graph, thy tak a matrix containing non commutativ variabls as input. Thus a hstpi is nothing but multiplying th matrix by itslf and thn assigning dirnt variabls to dirnt ntris in th rsulting matrix. Furthrmor, thy rn th coloring in ach stp according to condition (1). This is not ncssary and thus lft out in our formulation of Algorithm 1. Lmma 2.1 If condition (1) is tru at som point of Algorithm 1 thn it is tru throughout th rst of th algorithm. 4

7 Proof. Assum that condition (1) is tru bfor hstpi s. Lt L() = f(c 1 ; d 1 ; p c 1;d1 ); (c 2 ; d 2 ; p c 2;d2 ); : : : ; (c l ; d l ; p c l ;d l )g b th structur list of th dg 2 C(b). Sinc condition (1) holds th structur list of t 2 C(b) looks lik th following. L( t ) = f(c 1 ; d 1 ; p c 1;d1 ); (c 2 ; d 2 ; p c 2;d2 ); : : : ; (c l ; d l ; p c l ;d l )g Thus, two dgs ; 0 2 C(b) gt th sam color i t ; 0t 2 C(b) gt th sam color and hnc condition (1) is still tru aftr hstpi s. A coloring f of G which dos not chang if you apply Algorithm 1 to it is calld a stabl or cohrnt coloring. On proprty of stabl colorings is that, givn colors b; c; d, th condition 9k8 : f() = b ) p c;d = k holds. Hnc, w can dn p c;d f () := pc;d. Th bst known bound on th numbr of stps of Algorithm 1 is th trivial on of n 2 { in th worst cas only on color is split into two nw ons in ach stp. Furr [5] constructd xampls which nd (n) stps. Obviously, two dgs = (u; v) and 0 = (u 0 ; v 0 ) which got dirnt colors in som stp will rmain dirntly colord. This is bcaus th triangls (u; v; v) and (u 0 ; v 0 ; v 0 ) ar rprsntd in th structur lists L() and L( 0 ). Lt a graph G = (V; E; f) and a stabl coloring f ~ of G b givn. f ~ dns th linar bas of an algbraic structur, calld cohrnt or cllular algbra, in th following way (s [8] and [12] for dtails). Lt M n b th algbra of th complx valud n n matrics, A th conjugat transpos of A, I th idntity matrix, J th all 1 matrix and A B th Hadamard product (A B) ij := (a ij b ij ) of A = (a ij ) and B=(b ij ). A cohrnt algbra A is a subalgbra of M n which is closd undr conjugat transposition and Hadamard multiplication and contains I and J. Dn B(f) = fe c j c 2 Fg whr E c is th n n matrix with (E c ) ij = 1 if f(i; j) = c and 0 othrwis. Th st B( f) ~ = f Ec ~ j c 2 Fg ~ has th following proprtis: P (i) J = c2 ~ F (ii) I = P c2i ~E c ~E c for som I ~ F (iii) 8c9d : Ec ~ = E ~ P d (iv) 8c; d : Ec ~ Ed ~ = b2 ~ F p c;d b ~E b Thus, th span of B( f) ~ is a cohrnt algbra. W call it th cohrnt algbra gnratd by (V; E; f) and dnot it by A(f). Whil a cohrnt algbra A lik any matrix algbra has dirnt linar bass, it has xactly on bas consisting only of 0? 1 matrics which is calld th standard 5

8 bas of A. Obviously, B( f) ~ is th standard bas of A(f). Th numbrs p c;d ar calld b structur constants of A(f). Two graphs H = (W; F ) and H 0 = (W 0 ; F 0 ) ar calld isomorphic i thr xists a bijction : W! W 0 such that (u; v) 2 F, ((u); (v)) 2 F 0 holds for all u; v 2 W. is calld an isomorphism btwn H and H 0. If an uncolord graph is givn, an initial coloring can b dnd by f() = 0; 8 = (u; u), f() = 1; 8 2 E and f() = 2 othrwis. In this way, any graph H is a colord graph and gnrats a cohrnt algbra H. A ncssary condition for th xistnc of an isomorphism btwn H and H 0 is that th corrsponding cohrnt algbras H and H 0 rspctivly ar isomorphic. Dnot by B = ff 1 ; F 2 ; : : : F l g and B 0 = ff 0 1 ; F 0 2 ; : : : F 0 g th uniqu basis of th l cohrnt algbras of H and H 0 rspctivly, obtaind by a crtain algorithm for computing stabl colorings. This algorithm is calld canonical if th function : B! B 0 ; F c 7! F 0 c inducs an algbra isomorphism btwn H and H 0 whnvr H and H 0 ar isomorphic. In othr words, whnvr thr xists an isomorphism btwn H and H 0 which maps 2 F to 0 2 F 0 thn a canonical algorithm will assign to and 0 th sam colors. 3 Prvious Art Bfor w startd our work, thr algorithms (stabil [4], stabcol [3] and CC [16] 1 ) for computing cohrnt colorings wr known. In this sction, w will discuss ths algorithms briy and introduc thos idas which wr usful in our approach. A straightforward implmntation of Algorithm 1 which just computs all structur lists in ach stp would hav a running tim of at last O(n 5 ) and O(n 3 ) spac would b ndd. By running tim, w always man th worst cas running tim. Th only possibility to rduc th running tim is to comput lss than n 3 triangls in ach stp. W will show how this can b don without changing th rsult. First, it is ncssary to xplain how w will do th rcoloring. During a rcoloring, a color class C(c 0 ) will b split into l color classs. Th largst class will kp th old color c 0 and th othr ons will gt nw colors c 1 ; c 2 ; : : : ; c l?1. W rfr to this proprty as th LCOC condition (Largst Class Old Color). Th following lmma is a strongr formulation of lmmas givn by Aho t al. [1] and Babl [2]. Lmma 3.1 It sucs to comput only thos ntris in ach structur list which contain at last on non basis dg colord with a nw color in th prcding stp. Proof. W ssntially follow Babl [2]. Lt th coloring f b th rsulting coloring aftr th sth rcoloring. Assum that th colors c 0 and d 0 hav bn rnd during th sth rcoloring to c 0 ; c 1 ; : : : c lc?1 and 1 Th programs ar availabl at 6

9 d 0 ; d 1 ; : : : d ld?1. Lt ; 0 2 C(b) aftr th sth rcoloring. Of cours lcx l dx i=0 j=0 p c i;d j = lcx l dx i=0 j=0 p c i;d j 0 (2) holds. If w comput only thos ntris in ach structur list which contain at last on non basis dg colord with a nw color, th valus p c 0;d0 and p c 0;d0 would hav not 0 bn computd. Assum that p c 0;d0 6= p c 0;d0. Du to (2), thr is a pair (i; j) 6= (0; 0) 0 such that p c i;d j 6= p c i;d j. For that rason, dirnt colors ar assignd to and 0 in 0 th (s + 1)th rcoloring. Although this lmma rducs th numbr of triangls which ar considrd in ach stp, th worst cas bound on th numbr of triangls is still n 3. Du to th lmma and th way of rcoloring, ach dg gts at most log n tims a nw color. As a consqunc, th ovrall numbr of triangls which nd to b considrd is boundd by O(n 3 log n). Thus, th up to now bst known tim bound of O(n 3 log n) follows (s [2] for dtails). Th part of a structur list which contains only structur valus blonging to triangls with at last on rcntly { during th prcding stp { colord dg is calld rducd structur list. A canonical coloring can b obtaind by sorting th lmnts of th structur lists L() lxicographically and thn assigning nw colors according to th lxicographic ordr of th structur lists. This can b combind with th rcoloring dscribd abov to gt th canonical coloring algorithm prsntd by Babl in [2]. Th idas of Babl wr implmntd in stabcol. Th disadvantags of stabcol ar th spac rquirmnt of O(n 3 ) and th bad practical running tim. Anothr algorithm for computing cohrnt algbras is implmntd in th program stabil. This implmntation is vry cint in practic, has a thortical tim bound of O(n 7 ) and nds O(n 2 ) spac. Th authors of stabil trid to rduc th numbr of structur lists thy hav to stor simultanously. Th ida is to comput only th dirnt structur lists of on color class at a tim. Sinc this can b still up to n 2, stabil rns by considring only th th rst O(n) dirnt structur lists of a color class. This approach has two disadvantags. First, stabil dos not color canonically and scond, stabil might nd mor stps than th Wisfilr-Lhman algorithm. A rasonabl canonical implmntation of th Wisfilr-Lhman algorithm is includd in th packag CC but no information on th algorithm and no thortical running tim ar givn. 4 Th Algorithm In th following, w will show that canonical cohrnt colorings can b computd in O(n 2 ) spac and O(n 3 log n) tim. To rduc th spac rquirmnts, w hav to avoid to comput all n 2 structur lists at a tim. A straightforward approach would b to comput th (rducd) 7

10 structur lists of a fw dgs only. To gt a canonical algorithm, it is thn ncssary to comput th structur lists of all dgs of on color class or to comput som of thm at last twic. Howvr, sinc color classs can b larg, th rquird spac would b still (n 3 ) or th tim bound would not b as good as rquird. Th main ida of our approach is to comput only parts of th structur lists of th dgs but for all dgs at a tim. I.., w comput th lists L(c; d) instad of L(). So hstpi can b rformulatd as follows. Procdur 2 stp 1: ~ f f 2: for all c; d 2 F do 3: splitcolor(c; d), i.., comput L(c; d) and split th colors in th following way: ~f() = f( ~ 0 ) :, f() ~ = f( ~ 0 ) and p c;d = p c;d ; 8; 2 E : nd for 5: rcolor, i.., f f ~ Th ntris of lists (or sts) ar always visitd according to th currnt (natural) ordring of th list (st). In th formr procdur, th ntris ar visitd in incrasing ordr of c and d. hsplitcolor(c; d)i stors a psudo color ~ f at ach dg. This guarants that th information of th prviously computd structur valus will b mmorizd. Finally, hrcolori ovrwrits th color of ach dg with its currnt psudo color. Obviously, th rsult of this rnmnt procdur is th sam as th on of lins 2,3 in Algorithm 1. Sinc vry list L(c; d) has lngth up to n 2 and thr ar up to n 4 such lists, this approach dos not sm to b vry promising at rst sight. But at last it is only ncssary to stor th list of on pair (c; d) at a tim and still gt a canonical coloring (for dtails s Procdur 4). Thus, this approach maks it possibl to work in O(n 2 ) spac only. From now on, w will try to rduc th running tim to O(n 3 log n) and kp th good spac bound. To rduc th numbr of pairs (c; d) to b considrd, w can apply Lmma 3.1 (s Procdur 3). Th nw colors ar dnotd by N and th old colors by O := F n N. To mak a running tim analysis of Procdur 3, w nd to go into th dtails of th usd data structur. W stor th colord graph G as a colord matrix M, i.., M uv = f(u; v), 8u; v 2 V, and th color classs in an array of lngth n 2. With ach color class C(c), w associat doubly linkd lists of a rowwis and a columnwis ncoding of th dgs in that color class (ths corrspond to spars matrix rprsntations of E c ). W dnot ths lists by rowwis(c) and columnwis(c) rspctivly. Furthrmor, w stor doubly linkd lists of th rst dgs in th rows and columns of ach color class calld rows(c) and columns(c). Th lists L(c; d) ar stord as doubly linkd lists also. W always associat a list ntry with an dg and thus ar abl to hav accss to ach list ntry in constant tim by knowing th rfrring dg. It taks O(n 2 ) tim 8

11 Procdur 3 stp 1: ~ f f 2: for all c 2 N do 3: for all d 2 F do 4: splitcolor(c; d) 5: nd for 6: nd for 7: for all c 2 O do 8: for d 2 N do 9: splitcolor(c; d) 10: nd for 11: nd for 12: rcolor to initializ this data structur. In th following, w do not dscrib how and whn occurring lists and variabls ar dltd and rst rspctivly. It always should b clar from th contxt how and whn this is don. For an dg st W, rowindics(w ) and columnindics(w ) dnot th sts of row indics and column indics of W with rspct to M. If W = fg, w writ rowindx() and columnindx() instad of rowindics(fg) and columnindics(fg) rspctivly. To achiv th claimd tim bound, w also nd to xplain th implmntations of hsplitcolor(c,d)i and hrcolori in mor dtail. P It is asy to implmnt ths functions in tim O(n + #triangls) and O(n + triangls). By #triangls, w man th numbr of triangls which ar considrd for computing L(c; d), i.., #triangls = P2L(c;d) pc;d P, and by triangls th numbr of triangls which ar considrd in th currnt hstpi. Th rst lin of Procdur 4 is implmntd in Procdur 5. It is a spcial spars matrix multiplication and can b don in th rquird tim of O(n + #triangls). To comput L(c; d), th matrix product E := E c E d has to b computd which is nothing but a matrix rprsntation of L(c; d). Obsrv that th numbr of itrations of th loop in lin 1 of Procdur 5 which quals jcolumnindics(c(c)) \ rowindics(c(d))j is boundd by #triangls and th computing of columnindics(c(c)) \ rowindics(c(d)) can b don by scanning through th lists columns(c) and rows(d) whos lngths ar boundd by n. Th innr loop nds an ovrall tim of O(#triangls) bcaus togthr with th indx w, w gt th rst lmnts with column indx w and row indx w of th color classs C(c) and C(d) rspctivly, and hav accss in tim O(1) to th succssors of th lmnts. An dg is calld hit by (c; d) if p c;d > 0, a color class C(b) is calld hit by (c; d) if som 2 C(b) is hit by (c; d). C(b):hit dnots th numbr of hit lmnts of C(b). This numbr is ndd in hsplitcolori and can b asily computd in Procdur 5 (which is don in lins 3-5). In hsplitcolori, th psudo rcoloring will b don in th following way. Nw psudo colors ar assignd according to an incrasing ordring of th f() ~ and p c;d. 9

12 Procdur 4 splitcolor(c; d) 1: comput L(c; d) 2: sort L(c; d) by th valus ~ f() 3: sort L(c; d) by th valus p c;d 4: for all with ( rst dg in L(c; d) with color f()) ~ do 5: b = f() ~ 6: if C(b):hit < C(b):siz thn 7: C(b):p := 0 8: ls 9: C(b):p := p c;d 10: nd if 11: C(b):LastColor := b 12: C(b):hit := 0 13: nd for 14: for all 2 L(c; d) do 15: if C( f()):p ~ 6= p c;d thn 16: C( f()):p ~ := p c;d 17: C( f()):lastcolor ~ := n ~f : nd if 19: C( f()):siz ~?? 20: f() ~ := C( f()):lastcolor ~ 21: C( f()):siz ~ : nd for Procdur 5 comput L(c; d) 1: for all w 2 columnindics(c(c)) \ rowindics(c(d)) do 2: for all = (u; v) with (u; w) 2 C(c) and (w; v) 2 C(d) do 3: if p c;d == 0 thn 4: C( f()):hit ~ + + 5: nd if 6: p c;d + + 7: nd for 8: nd for 10

13 In Procdur 4, w dtrmin th smallst p c;d of ach color class C(b) hit by (c; d) (stord in C(b):p) bcaus th dgs with th smallst p c;d kp thir old (psudo) color and th othr ons gt nw (psudo) colors. It is not possibl to do this by scanning through all lmnts of C(b) bcaus C(b) or at last th sum of th sizs of all hit color classs might b too larg. On possibl solution for computing th smallst p c;d is shown in lins 4? 13. That is why w nd to updat th sizs of th color classs immdiatly which is don in lins 19,21. In lins 14? 22, th nw psudo colors ar assignd as dscribd bfor. To analyz th running tims of lins 2,3, w nd to xplain th sorting procdur w usd. Lt a list L of lngth m b givn. Assum that ach lmnt of L consists of at most k numbrs out of th intrval f1; 2; : : : ; ng (or of on natural numbr boundd by n k ), thn L can b sortd using buckt sort in tim O(k (n + m)) and spac O(m + n) [1]. W us an in situ implmntation of buckt sort. It follows that th sorting in lin 3 in Procdur 4 can b don with buckt sort in tim O(#triangls) sinc th p c;d in L(c; d) ar boundd by #triangls. Lin 2 nds tim O(n + #triangls) and thus, this procdur has a running tim of O(n + #triangls). Sinc hsplitcolor(c; d)i is calld in lxicographical ordr of (c; d) and th assignmnt of th nw colors dpnds only on th structur valus and th prvious coloring, th psudo coloring is again canonical. Procdur 6 rcolor 1: Lt L b th list of all dgs which got a nw psudo color. 2: for all 2 L do 3: dlt from its color class C(f()) 4: appnd to C( ~ f()) 5: nd for 6: sort L by th tupls (rowindx(); columindx()) and initializ with th hlp of this ordring th row ncodings of th nw color classs 7: sort L by th tupls (columnindx(); rowindx()) and initializ with th hlp of this ordring th column ncodings of th nw color classs 8: for all c = f(); 2 L do 9: comput d with jc(d)j = maxfjc( ~ f())j j f() = cg 10: if jc(d)j > jc(c)j thn 11: xchang th colors of th color classs C(c) and C(d). 12: nd if 13: nd for 14: updat f To nish th stp w hav to transform th psudo colors distributd by hsplitcolori into a nw coloring according to th LCOC condition. hrcolori (s Procdur 6) nsurs that th largst color classs gt th old colors and dos th updating of th color classs and colors. In ordr to updat our data structurs, th dgs hav to b movd from thir old color class to thir nw on. In our data structurs, dlting an lmnt and 11

14 appnding an lmnt to a nw list { without any furthr updating of th data structur { taks tim O(1). Th sorting in Procdur 6 (lins 6,7) can b don in tim O(n + P triangls) using buckt sort. Bcaus L is sortd proprly, th initialization of th row and column ncodings is nothing but an appnding procdur and thus can b don in tim O( P triangls). Lins 8-13 tak tim O( P triangls) sinc two colors will b xchangd only if th nw color is largr than th old color. W conclud that all statmnts of Procdur 6 can b xcutd in tim O(n + P triangls). If on dos an amortizd cost analysis of th algorithm dscribd so far, on will com up with a worst cas running tim of O(n 5 ). Th rason for th still bad running tim of Procdur 3 is th fact that many mpty structur lists ar computd and that th running tim of hsplitcolori contains th trm n. To achiv th claimd running tim, w nd to nsur that only triangls which appar in th graph will b computd, th multiplication will tak only O(#triangls) tim and th sorting in lin 2 of Procdur 4 will b omittd. Thrfor, w chang th procdur hstpi again. In Procdur 7, w comput lists of paths of lngth 2 which appar in th graph and thn comput th corrsponding triangls. Procdur 7 stp 1: fexamin triangls whos rst non basis dg was rcntly colordg 2: for all c 2 N do 3: for all (u; w) 2 columns(c(c)) do 4: for (v = 1; v vrtics; v + +) do 5: if (w; v) is rst in som row of its color class thn 6: appnd ((u; w); (w; v)) to th list rf 7: nd if 8: nd for 9: nd for 10: sort rf by th colors of th scond dgs; 11: for all d 2 rf do 12: splitcolor(c; d) 13: nd for 14: dlt rf; 15: nd for 16: - 29: fth sam has to b don for th cas whn th scond color is nwg 30: rcolor Th list rf consists of all dirctd paths of lngth 2 whrby th rst dg has color c. Thrfor, rf has at most n jcolumns(c)j lmnts. Thus, th sorting of rf can b implmntd in tim proportional to n jcolumns(c)j. Obsrv that no supruous but all ncssary triangls ar computd in this procdur and again th ordring allows a canonical coloring. 12

15 Th sorting in lin 11 can b don in tim proportional to O(jrf j + n) = O( Pc triangls) bcaus in vry call of this sorting procdur, at last n triangls ar considrd. Pc triangls dnots th numbr of triangls with th rst dg colord by th color c to b considrd in this stp. With th information givn in rf, it is asy to implmnt th multiplication in hsplitcolori in tim proportional to th numbr of triangls bcaus th sts columnindics(c) \ rowindics(d) for all ncssary pairs (c; d) ar stord in rf. To omit th sorting in lin 2 of Procdur 4, w nd to introduc mor data structurs. With ach dg, w stor its parnt color, i.., th (psudo) color class to which blongd to bfor its color was changd th last tim, and with ach color class C(c) a list of childrn, i.., a list of dgs which wr in C(c) bfor. If a child of a color class C(d) is rcolord, it is dltd from th list of childrn of C(d). During th initialization, ach dg will b dnd as child of C(). For dtails s procdur Procdur 8. Procdur 8 splitcolor(c; d) 1: comput L(c; d) 2: sort L(c; d) by th valus p c;d 3: for all with rst dg in L(c; d) with color f() ~ do 4: b := f() ~ 5: if C(b):hit < C(b):siz thn 6: C(b):p := 0 7: ls 8: C(b):p := p c;d 9: nd if 10: C(b):LastColor := b 11: C(b):hit := 0 12: nd for 13: for all 2 L(c; d) do 14: if C( f()):p ~ 6= p c;d thn 15: C( f()):p ~ := p c;d 16: C( f()):lastcolor ~ := n ~f : nd if 18: C( f()):siz ~?? 19: dlt from its parnt's childrn list 20: appnd to th childrn list of C( f()) ~ 21: 22: assign C( f()) ~ as parnt of f() ~ := C( f()):lastcolor ~ 23: C( f()):siz ~ : nd for Using this data structur, it is possibl to insrt som lins in hrcolori to gt a canonical coloring (s Procdur 9). Obviously, all rcntly dnd changs do not xcd th xpctd running tim. 13

16 Procdur 9 insrt in Procdur 6 btwn lin 1 and 2 1: sort L by th old colors f 2: introduc a dummy root r 3: for all 2 L; f(parnt()) n f do 4: assign as child of r 5: assign r as parnt of 6: nd for 7: fth parnt childrn rlationship dns a tr T on th color classs in L.g 8: assign nw colors n f + 1; : : : ; n ~f to th color classs in T and to th dgs in L by walking through T in post ordr (or som othr wll dnd ordr). Thus, hsplitcolori can b implmntd in tim O(#triangls) and th asymptotic running tim of hrcolori did not chang. Obsrv that th prsntd data structurs can still b stord in O(n 2 ) spac. Th algorithm nds at most n 2 calls of hstpi to comput a stabl coloring. Howvr, th ovrall running tim of all calls of hsplitcolori (including Procdur 5) is proportional to th numbr of all triangls which ar considrd. This numbr is boundd by O(n 3 log n) bcaus ach triangl is considrd at most O(log n) tims. Th sam amount of tim is ndd for all xcutions of th rmaining part of Procdur 7 up to lin 29. P Sinc on xcution of hrcolori taks tim O(n+ triangls), th ovrall running tim for this part of hstpi is also boundd by O(n 3 + n 3 log n) = O(n 3 log n). Hnc, w gt a running tim of O(n 3 log n). Thorm 4.1 Givn a colord graph G, a stabl coloring of G can b computd in tim O(n 3 log n) and spac O(n 2 ). 5 Th Implmntation Th abov algorithms ar implmntd in th program qwil 2 which was writtn in C++. W lik to introduc som idas which dcras th running tim of th qwil implmntation considrably. Instad of rning th coloring by considring all ncssary pairs (c; d) and thn rcoloring onc, w rn th coloring by considring on pair of colors only and thn rcolor immdiatly. This rsults in much mor rcolorings but rducs th sizs of th color classs vry quickly. Obsrv that this mthod still colors canonically. Du to th abov considrations, it dos not mak sns to maintain th list rf all th tim. rf may chang aftr vry rcoloring. In our implmntation, w just stor colors in rf and do not do any updating with rf. This may caus th calculation of triangls which do not xist and thus nlarg th thortical running tim but 2 Th program is availabl at 14

17 it provs to b cint in practic. For a bttr undrstanding, th implmntd procdur is givn blow (s Procdur 10). Procdur 10 stp 1: fexamin triangls whos rst non basis dg was rcntly colordg 2: for all c 2 N do 3: for all (u; w) 2 colums(c(c)) do 4: for (v = 1; v vrtics; v + +) do 5: appnd f(w,v) to th list rf 6: nd for 7: nd for 8: sort rf; 9: for all d 2 rf do 10: splitcolor(c; d) 11: rcolor 12: nd for 13: dlt rf; 14: nd for 15: - 28: fth sam has to b don for th cas whn th scond color is nw g On can obsrv that many structur lists will not yild a rnmnt of th coloring. It is asy to chck whthr a color class C(b) will b split by a structur list L(c; d) or not. If X j 2 C(b)g jc(b)j = maxfp c;d 2C(b) p c;d ; thn th color class C(b) will not b split, othrwis it will. A structur list which rns th coloring is calld usful. If w chck this condition aftr computing th structur list and thn rduc th structur list if possibl, th running tim dcrass. Furthrmor, w us Procdur 4 instad of Procdur 8. W tstd our implmntation on typical tst instancs and compard it to th implmntations stabil, CC and stabcol. Du to th mmory rstrictions, w wr not abl to tst stabcol on instancs with mor than 150 nods. Th instancs bnzn, mobius and dynkin ar takn from [4] and dscribd thr in dtail. stp dnots th instancs givn by Furr [9] that rquir a numbr of stps proportional to n. path dnots th instancs which ar paths. In th tabls, w giv th nam of th instanc, th numbr of vrtics of th instanc, th numbr of colors of th rsulting cohrnt coloring and th running tims of th dirnt algorithms. Th tims ar givn in sconds and th bold ntris mark th fastst algorithm for ach instanc. Th rsults hav bn obtaind on a Sun Sparc Ultra 170, Sun OS 5.6, 128 Mb mmory using th GNU C/C++ compilr, vrsion Tabl 1 shows that qwil is by far th fastst canonical implmntation. On all instancs, including th small ons, qwil is fastr than CC and stabcol. By ts, w dnot th tim and spac optimal algorithm dscribd in th prvious sction. W 15

18 addd to ts th dltion of uslss structur lists. Tabl 1 shows that ts is fastr than stabcol and thus th fastst implmntation of an algorithm for computing cohrnt algbras with a thortical tim bound of O(n 3 log n). W hav implmntd a non canonical vrsion of qwil by just skipping th sorting at crtain points in th algorithm. W dnot this variant by :canonical. Both, qwil and :canonical, ar on th small instancs oftn slowr than stabil. Howvr, on th larg instancs :canonical and vn qwil prov to b much fastr than stabil (s Tabl 2 and Tabl 5 for dtails). On may obsrv that th dirnc in th running tim of qwil and stabil incrass with incrasing valus of # colors. n In Tabl 3, w hav listd som mor variants of qwil. By :ir, w dnot th algorithm which dos no immdiat rcoloring, by :usful th algorithm which dos not dlt uslss structur lists. Tabl 3 shows that not only th good thortical bounds but also th idas prsntd in this sction ar rsponsibl for th good practical running tim of qwil. Obsrv that if w omit th immdiat rcoloring, qwil computs in ach stp th sam numbr of nw colors as th Wisfilr-Lhman algorithm and thus nds th sam numbr of stps. 6 Concluding Rmarks W hav prsntd a nw canonical algorithm for computing cohrnt colorings which satiss th bst known thortical bounds. Furthrmor, w hav shown that our approach yilds by far th fastst known implmntation for computing cohrnt colorings. In Tabl 4, w compar th numbr of computd structur lists with th numbr of usful structur lists. It turns out that only vry fw structur lists nd to b computd. This obsrvation givs ris to an algorithm which computs only a fw structur lists in ach stp. Sinc w do not know which structur lists ar usful, w implmntd a randomizd vrsion of our algorithm which computs only fw randomly chosn structur lists. W hav mad som rst practical xprincs in this dirction and th rsults sm to b vry promising. Howvr, th major problm is that, up to now, w ar not abl to chck rasonably fast whthr th rsulting coloring is stabl or not. Furthrmor, it is asy to gnraliz our approach and obtain a k-dimnsional Wisfilr-Lhman algorithm which nds only O(kn 2 ) spac and fullls th bst known tim bound of O(k 2 n k+1 log n) [13]. 7 Acknowldgmnts W would lik to thank L. Babl and G. Tinhofr for hlpful discussions and commnts. 16

19 Rfrncs [1] A. V. Aho, J. E. Hopcroft, and J. D. Ullman, Th Dsign and Analysis of Computr Algorithms, Addison-Wsly, [2] L. Babl, Computing cohrnt algbras. to b publishd, [3] L. Babl, S. Baumann, M. Ludck, and G. Tinhofr, STABCOL: Graph isomorphism tsting basd on th Wisfilr-Lman Algorithm, Tch. Rp. TUM-M9702, Tchnisch Univrsitat Munchn, [4] L. Babl, I. V. Chuvava, M. Klin, and D. V. Paschnik, Algbraic combinatorics in mathmatical chmistry. Mthods and algorithms. II. Program implmntation of th Wisfilr-Lman Algorithm, Tch. Rp. TUM-M9701, Tchnisch Univrsitat Munchn, [5] J.-Y. Cai, M. Furr, and N. Immrman, An optimal lowr bound on th numbr of variabls for graph idntication, Combinatorica, 12 (1992), pp. 389{ 410. [6] S. Evdokimov, M. Karpinski, and I. Ponomarnko, Compact cllular algbras and prmutation groups, Tch. Rp CS, Univrsitat Bonn, [7] S. Evdokimov, I. Ponomarnko, and G. Tinhofr, On a nw class of wakly compact graphs, Tch. Rp. TUM-M9715, Tchnisch Univrsitat Munchn, [8] S. Fridland, Cohrnt algbras and th graph isomorphism problm, Discrt Applid Mathmatics, 25 (1989), pp. 73{98. [9] M. Furr, Privat communication, [10] C. D. Godsil, Compact graphs and quitabl partitions, Linar Algbra and its Applications, 255 (1997), pp. 209{239. [11] D. G. Higman, Cohrnt congurations, I. Rnd. Sm. Mat. Univ. Padova, 44 (1972), pp. 1{25. [12], Cohrnt algbras, Linar Algbra and its Applications, 93 (1987), pp. 209{239. [13] N. Immrman and E. Landr, Dscribing graphs: A rst-ordr approach to graph canonization, in Complxity Thory Rtrospctiv, Springr-Vrlag, 1990, pp. 59{81. [14] M. Klin, C. Ruckr, G. Ruckr, and G. Tinhofr, Algbraic combinatorics in mathmatical chmistry. Mthods and algorithms. I. Prmutation groups and cohrnt (cllular) algbras., Tch. Rp. TUM-M9510, Tchnisch Univrsitat Munchn, [15] M. E. Muzychuk and G. Tinhofr, Rcognizing circulant graphs of prim ordr in polynomial tim, Elctronic Journal of Combinatorics, 5 (1998), p. 28. [16] I. Ponomarnko, CC Usr's Guid, Fb [17] P. F. Stadlr and G. P. Wagnr, Algbraic thory of rcombination spacs, Evolutionary Computation, 5 (1998), pp. 241{

20 [18] G. Tinhofr, Graph isomorphism and thorms of Birkho typ, Computing, 36 (1986), pp. 285{300. [19], A not on compact graphs, Discrt Applid Mathmatics, 30 (1991), pp. 253{264. [20] B. J. Wisfilr, On Construction and Idntication of Graphs, Springr, Brlin, [21] B. J. Wisfilr and A. A. Lman, Rduction of a graph to a canonical form and an algbra arising during this rduction, Naucho - Tchnicksagy Informatsia, 9 (1968), pp. 12{16. Russian. 18

21 Nam Vrtics Colors qwil CC stabcol ts tim tim ratio tim ratio tim ratio bnzn bnzn bnzn bnzn bnzn bnzn bnzn bnzn bnzn bnzn dynkin dynkin dynkin dynkin dynkin dynkin dynkin dynkin dynkin dynkin mobius mobius mobius mobius mobius mobius mobius mobius mobius mobius path path path path path path path path path path stp stp stp stp stp stp stp stp stp stp Tabl 1: Canonical Implmntations 19

22 Nam Vrtics Colors :canonical stabil qwil tim tim ratio tim ratio bnzn bnzn bnzn bnzn bnzn bnzn bnzn bnzn bnzn bnzn dynkin dynkin dynkin dynkin dynkin dynkin dynkin dynkin dynkin dynkin mobius mobius mobius mobius mobius mobius mobius mobius mobius mobius path path path path path path path path path path stp stp stp stp stp stp stp stp stp stp Tabl 2: Non Canonical Implmntations 20

23 Nam Vrtics Colors qwil :usful :ir :ir :usful tim tim ratio tim ratio tim ratio bnzn bnzn bnzn bnzn bnzn bnzn bnzn bnzn bnzn bnzn dynkin dynkin dynkin dynkin dynkin dynkin dynkin dynkin dynkin dynkin mobius mobius mobius mobius mobius mobius mobius mobius mobius mobius path path path path path path path path path path stp stp stp stp stp stp stp stp stp stp Tabl 3: Variants of qwil 21

24 Nam Vrtics Colors Structur Lists ovrall usful % bnzn bnzn bnzn bnzn bnzn bnzn dynkin dynkin dynkin dynkin dynkin dynkin mobius mobius mobius mobius mobius mobius path path path path path path stp stp stp stp stp stp Tabl 4: Structur Lists Statistics Nam Vrtics Colors :canonical stabil qwil tim tim ratio tim ratio bnzn bnzn dynkin dynkin mobius mobius path path stp stp Tabl 5: Larg Instancs 22