(cid:3) DETERMINANT: OLD ALGORITHMS, NEW INSIGHTS MEENA MAHAJANy AND V VINAYz Abstra t. In this paper we approa h the problemof omputing the hara teristi polynomial of a matrix from the ombinatorial viewpoint. We present several ombinatorial hara terizations of the oeÆ ients of the hara teristi polynomial, in terms of walks and losed walks of di(cid:11)erent kinds in the underlying graph. We develop algorithms based on these hara terizations, and show that they tally with well-known algorithms arrived at independently from onsiderations in linear algebra. Key words. Determinant,Algorithms,Combinatori s,Graphs,Matri es AMS subje t lassi(cid:12) ations. 05A15, 68R05,05C50,68Q25 1. Introdu tion. Computingthedeterminant,orthe hara teristi polynomial, ofamatrixisaproblemwhi h hasbeen studied severalyearsagofromthe numeri al analysisviewpoint. In the mid 40's, a seriesof algorithmswhi h employed sequential iterative methods to ompute the polynomial were proposed, the most prominent beingduetoSamuelson[19℄,Krylov,Leverier;see,forinstan e,thepresentationin[8℄. Then, in the 80's, a series of parallel algorithms for the determinant were proposed, due to Csanky, Chistov, Berkowitz [6, 5, 1℄. This ulminated in the result, shown independently by several omplexity theoristsin luding Vinay, Damm, Toda,Valiant [26,7,24,25℄,that omputingthedeterminantofanintegermatrixis ompleteforthe omplexity lassGapL,andhen e omputationallyequivalentinapre ise omplexity- theoreti sense to iterated matrix multipli ation or matrix powering. In an attempt to unravel the ideas that went into designing eÆ ient parallel al- gorithmsforthe determinant, Valiantstudied Samuelson'salgorithmand interpreted the omputation ombinatorially[25℄. Hepresenteda ombinatorialtheorem on ern- ing losedwalksingraphs,the orre tnessofwhi hfollowedfromthatofSamuelson's algorithm. This was the (cid:12)rst attempt to view determinant omputations as graph- theoreti rather than linear algebrai manipulations. Inspired by this, and by the purely ombinatorial and extremely elegant proof of the Cayley-Hamilton Theorem duetoRutherford[18℄(andindependentlydis overedbyStraubing[22℄;see[2,27℄for ni e expositions and see [3℄ for related material), Mahajan and Vinay [15℄ des ribed a ombinatorialalgorithm for omputing the hara teristi polynomial. The proof of orre tness of this algorithm is also purely ombinatorial and does not rely on any linear algebra or polynomial arithmeti . In this paper, we follow up on the work presented in [25, 22, 15℄ and present a unifying ombinatorial framework in whi h to interpret and analyse a host of algorithms for omputing the determinant and the hara teristi polynomial. We (cid:12)rst des ribewhatthe oeÆ ientsofthe hara teristi polynomialofamatrixM represent as ombinatorial entities in the graph GM whose adja en y matrix is M. We then (cid:3)A preliminary version of this paper appeared in the Pro eedings of the Sixth S andinavian Workshop on AlgorithmTheory SWAT'98, Vol.1432in the Springer-Verlag LNCS series,pp 276{ 287. yInstituteofMathemati alS ien es,Chennai600113,India. (meenaims .ernet.in). Partofthis workwasdonewhenthisauthorwasvisitingtheDepartmentofComputerS ien eandAutomation, IIS ,Bangalore. z Department of ComputerS ien e and Automation, Indian Institute of S ien e, Bangalore 560 012,India. (vinay sa.iis .ernet.in). 1 2 M.MAHAJANANDVVINAY onsider various algorithms for evaluating the oeÆ ients, and in ea h ase we relate the intermediate steps of the omputation to manipulation of similar ombinatorial entities, giving ombinatorialproofs of orre tnessof these algorithms. Inparti ular,inthegraph-theoreti setting, omputingthedeterminantamounts to evaluating the signed weighted sum of y le overs. This sum involves far too many terms to allow evaluationof ea h, and we show howthe algorithmsof [19, 5, 6℄ essentiallyexpandthis sum toin lude moretermsi.e., generalizationsof y le overs, whi h eventually an el out but whi h allow easy evaluation. The algorithm in [15℄ uses low sequen es expli itly; Samuelson's method [19℄ impli itly uses pre(cid:12)x low sequen es;Chistov'smethod[5℄impli itlyusestablesoftoursequen es;andCsanky's algorithm [6℄ hinges around Leverier's Lemma (see, for instan e, [8℄), whi h an be interpreted using loops and partial y le overs. In ea h of these ases, we expli itly demonstrate the underlying ombinatorial stru tures, and give proofs of orre tness whi h are entirely ombinatorial in nature. In a sense, this paper parallels the work done by a host of ombinatorialists in proving the orre tness of matrix identities using the graph-theoreti setting. Foata [9℄ used tours and y le overs in graphs to prove the Ma Mohan master theorem; RutherfordandStraubing[18,22℄reprovedtheCayley-Hamiltontheoremusing ounting over walks and y le overs; Garsia [11℄, Orlin [17℄ and Tempereley [23℄ independently found ombinatorialproofs of the matrix-tree theorem and Chaiken [4℄ gener- alized the proof to the all-minor matrix-tree theorem; Foata [10℄ and then Zeilberger [27℄ gave new ombinatorial proofs of the Ja obi identity; Gessel [12℄ used transitive tournaments in graphs to prove Vandermonde's determinant identity. More re ently, Minoux[16℄showedanextensionofthematrix-treetheoremtosemirings,againusing ounting arguments over arbores en es in graphs. For beautiful surveys of some of these results, see Zeilberger's paper [27℄ and hapter 4 of Stanton and White's book on Constru tive Combinatori s [20℄. Zeilberger ends with a host of \exer ises" in proving many more matrix identities ombinatorially. Thus, using ombinatorialinterpretations and arguments to prove matrix identities has been around for a while. To our knowledge, however, a similar appli ation of ombinatorialideas to interpret, orprove orre tnessof, oreven develop new algorithms omputingmatrixfun tions, hasbeenattemptedonlytwi ebefore: byValiant [25℄ in 1992, and by the present authors in our earlier paper in 1997 [15℄. We build on our earlier work and pursue a new thread of ideas here. Thispaperisthusa olle tionofnewinterpretationsandproofsofknownresults. The paper is by and large self- ontained. 2. Matri es, Determinants and Graphs. Let A be a square matrix of dimension n. For onvenien e, we state our results for matri es overintegers, but they apply to matri es overany ommutative ring. Weasso iatematri esofdimensionnwith ompletedire tedgraphsonnverti es, withweightsontheedges. LetGAdenotethe ompletedire tedgraphasso iatedwith thematrixA. Iftheverti esofGA arenumberedf1;2;:::;ng,thentheweightofthe edge hi;ji is aij. We use the notation [n℄ to denote the set f1;2;:::;ng. The determinant of the matrix A, det(A), is de(cid:12)ned as the signed sum of all weighted permutations of Sn as follows: X Y det(A)= sgn((cid:27)) ai(cid:27)(i) (cid:27)2Sn i where sgn((cid:27)) is the number (modulo 2) of inversions in (cid:27) i.e. the ardinality of the DETERMINANT:OLDALGORITHMS,NEWINSIGHTS 3 set fhi;jiji(cid:27)(j)g modulo 2. Ea h (cid:27) 2 Sn has a y le de omposition, and(cid:18)it orresponds to a se(cid:19)t of y les in 1 2 3 4 5 GA. For instan e, with n = 5, the permutation has a y le 4 2 3 5 1 de omposition (145)(2)(3) whi h orresponds to 3 y les in GA. Su h y les of GA have an important property: they are all simple (non-interse ting), disjoint y les; whenput together,theytou hea hvertexexa tlyon e. Su hsetsof y lesare alled y le overs. Note that y le overs of GA and permutations of Sn are in bije tion with ea h other. We de(cid:12)ne weights of y le overs to orrespond to weights of permutations. The weightofa y le isthe produ t ofthe weightsof all edgesin the y le. The weightof a y le over is the produ t of the weights of all the y les in it. Thus, viewing the Q y le over C as a set of edges, w(C)= e2Cw(eQ). Sin e the weights of the edges are di tated by the matrix A, we an write w(C)= hi;ji2Caij. We an also de(cid:12)ne the sign of a y le over onsistent with the sign of the orre- spondingpermutation. A y le overiseven(resp.odd)ifit ontainsanevennumber (resp. odd) of even length y les. Equivalently, the y le over is even (resp. odd) if the number of y les plus the number of edges is even (resp. odd). De(cid:12)ne the sign of a y le over C to be +1 if C is even, and (cid:0)1 if C is odd. Cau hy showed that with this de(cid:12)nition, the sign of a permutation (based on inversions) and the sign of the asso iated y le over is the same. For our use, this de(cid:12)nition of sign based on y le overs will be more onvenient. Let C(GA) denote the set of all y le overs in the graph GA. Then we have X X Y det(A)= sgn(C)w(C)= sgn(C) aij C2C(GA) C2C(GA) hi;ji2C Consider the hara teristi polynomial of A, n n(cid:0)1 (cid:31)A((cid:21))=det((cid:21)In(cid:0)A)= 0(cid:21) + 1(cid:21) +:::+ n(cid:0)1(cid:21)+ n Tointerpretthese oeÆ ients, onsiderthe graphGA((cid:21)) whose edgesarelabeled a ording to the matrix (cid:21)In(cid:0)A. The oeÆ ient l olle ts part of the ontribution to det((cid:21)In (cid:0)A) from y le overs having at least (n(cid:0)l) self loops. (A self loop at vertexk now arriesweight(cid:21)(cid:0)akk.) Thisisbe ausea y le overwithiselfloopshas weight whi h is a polynomial of degree i in (cid:21). For instan e, with n=4, onsider the y le over h1;4i;h2;2i;h3;3i;h4;1i in GA((cid:21)). This has weight ((cid:0)a14)((cid:21)(cid:0)a22)((cid:21)(cid:0) a33)((cid:0)a41), ontributing a14a22a33a41 to 4, (cid:0)a14a41(a22 +a33) to 3, a14a41 to 2, and 0 to 1. Followingnotation from [22℄, we onsider partial permutations, orresponding to partial y le overs. A partial permutation (cid:27) is a permutation on a subset S (cid:18) [n℄. The set S is alled the domain of (cid:27), denoted dom((cid:27)). The ompletion of (cid:27), denoted (cid:27)^, is the permutation in Sn obtained by letting all elements outside dom((cid:27)) be (cid:12)xed points. This permutation (cid:27)^ orrespondsto a y le over C in GA, and (cid:27) orresponds to a subset of the y les in C. We all su h a subset a partial y le overPC, and we all C the ompletion of PC. A partial y le over is de(cid:12)ned to have the same parity andsignasits ompletion. Itiseasytoseethatthe ompletionneednotbeexpli itly a ountedforintheparity;apartial y le overPC iseven(resp.odd)i(cid:11)thenumber of y les in it, plus the number of edges in it, is even (resp. odd). Getting ba k to the hara teristi polynomial, observe that to olle t the ontri- butions to l, we must lookat all partial y le overswith l edges. The n(cid:0)l verti es 4 M.MAHAJANANDVVINAY left un overed by su h a partial y le over PC are the self-loops, from whose weight the(cid:21) termhasbeen pi kedup. Ofthel verti es overed,self-loops, if any, ontribute the (cid:0)akk term from their weight, not the (cid:21) term. And other edges, say hi;ji for i 6= j, ontribute weights (cid:0)aij. Thus the weights for PC evidently ome from the l graph G(cid:0)A. If we interpret weights over the graph GA, a fa tor of ((cid:0)1) must be a ounted for independently. Formally, Definition 2.1. A y le is an ordered sequen e of m edges C = he1;e2;:::;emi where ei = hui;ui+1i for i 2 [m(cid:0)1℄ and em = hum;u1i and u1 (cid:20) ui for i 2 [m℄ and all the ui's are distin t. u1 is alled the head of the y le, denoted h(C). The length of the Qm y le is jCj = m, and the weight of the y le is w(C) = i=1w(ei). The vertex set of the y le is V(C)=fu1;:::;umg. An l- y le over C is an ordered sequen e of y les C = hC1;:::;Cki su h that V(Ci)\V(Cj)=(cid:30) for i6=j, h(C1)<::: v.) Then Ck = g[k;n;1;n℄, and su h variables an be evaluated by the formula 8 X >>< g[q;r;u;w(cid:0)1℄(cid:1)g[p(cid:0)q;l(cid:0)r;w;v℄ if p>1 g[p;l;u;v℄= q(cid:20)r(cid:20)q+(l(cid:0)p) >>: uu,(cid:12)ndapathoflength l(cid:0)2 to some vertex w >u through verti es all greater than u, and then return to u. So 8 < P auu if l=1 g[l;u℄=: P v;w>uauv(cid:1)awu if l=2 v;w>uauv(cid:1) [l(cid:0)2;u;v;w℄(cid:1)awu otherwise The variable [l;u;v;w℄ sums the weights of all length l paths from v to w going through verti es greater than u. These variables an be evaluated as follows: [1;u;v;w℄ =avw P [l;u;v;w℄ = x>u [p;u;v;x℄(cid:1) [l(cid:0)p;u;x;w℄ if l>1, where p=dl=2e 3.2. Clow Sequen eswiththe Pre(cid:12)x Property: Gettingto Samuelson's Method. The generalization from y le overs to low sequen es has a ertain ex- travagan e. The reason for going to low sequen es is that evaluating their weighted sum is easy, and this sum equals the sum over y le overs. However, there are several low sequen es whi h we an drop from onsideration without sa ri(cid:12) ing ease of omputation. One su h set arises from the following onsideration: DETERMINANT:OLDALGORITHMS,NEWINSIGHTS 7 In a y le over,all verti esare overed exa tly on e. Suppose we enumerate the verti es in the order in whi h they are visited in the y le over (following the order imposedbythe y leheads). Ifvertexhbe omestheheadofa y le,thenallverti es in this and subsequent y les are larger than h. So all the lower numbered verti es must have been already visited. So at least h(cid:0)1 verti es, and hen e h(cid:0)1 edges, must have been overed. We an require our low sequen es also to satisfy this property. We formalize the pre(cid:12)x property: a low sequen e C = hC1;:::;Cki has the pre(cid:12)x property if for 1(cid:20)r (cid:20)k, the total lengths of the lows C1;:::;Cr(cid:0)1 is at leasth(Cr)(cid:0)1. A similar pre(cid:12)x property an be formalized for partial y le overs. Formally, Definition 3.3. An l- low sequen e C = hC1;:::;Cki is said to have the pre(cid:12)x property if it satis(cid:12)es the following ondition: Xr(cid:0)1 8r2[k℄; jCtj(cid:21)h(Cr)(cid:0)1(cid:0)(n(cid:0)l) t=1 Theinterestingfa tisthattheinvolution onstru tedin theprevioussub-se tion for low sequen es workseven overthis restri ted set! Theorem 3.4 (Theorem 2 in [25℄). X l l =((cid:0)1) sgn(C)wt(C) C is an l- low sequen e with the pre(cid:12)x property A new proof of the above theorem. In[25℄,Valiantobservesthatpre(cid:12)x lowsequen esaretheterms omputedbySamuel- son'smethodforevaluating(cid:31)(cid:21)(A). Hen ethe orre tnessofthetheoremfollowsfrom the orre tness of Samuelson's method. And the orre tness of Samuelson's method is traditionally shown using linear algebra. Here is a simple alternative ombinatorial proof of this theorem. Observe that the involution de(cid:12)ned in the proof of Theorem 3.2 maps low sequen es with pre(cid:12)x property to low sequen es with pre(cid:12)x property. Why? Let C be an l- low sequen e with the pre(cid:12)x property, satisfying ase 1 in the proof. Sin e the length of low Ci only in reases in the pro ess, the pre(cid:12)x property ontinues to hold. Now let C be an l- low sequen e with the pre(cid:12)x property, satisfying ase 2. The involution 0 onstru ts a new l- low sequen e C by deta hing y le C from low Ci and inserting it later in the sequen e, say between Cj(cid:0)1 and Cj. This does not hange h(Ci). 0 Let C = D = hD1;:::;Dk+1i; here Dt = Ct for t 2 [i (cid:0)1℄ or for t = i +1 to j (cid:0)1, Di = CinC, Dj = C and Dt+1 = Ct for t = j to k. We must show that D has the pre(cid:12)x property. For r 2 [i℄, and for r = j +1 to k +1, the ondition Pr(cid:0)1 t=1 jDtj (cid:21) h(Dr)(cid:0)1(cid:0)(n(cid:0)l) holds be ause C has the pre(cid:12)x property. Now let i+1 (cid:20) r (cid:20) j. Sin e Ci was hosen from C for modi(cid:12) ation, and sin e i+1 (cid:20) r, we know that Dr;:::;Dk+1 form a partial y le over i.e. they form simple disjoint y les. Andthe headsofthese y lesarearrangedin in reasingorder. Sotheverti es overed in Dr;:::;Dk+1 must all be at least as large as h(Dr) and all distin t. But 8 M.MAHAJANANDVVINAY there are only n(cid:0)h(Dr)+1 su h verti es. Hen e Pk+1 t=r jDtj (cid:20) n(cid:0)h(Dr)+1 Pr(cid:0)1 l(cid:0) t=1 jDtj (cid:20) n(cid:0)h(Dr)+1 Pr(cid:0)1 t=1 jDtj (cid:21) h(Dr)(cid:0)1(cid:0)(n(cid:0)l) and D satis(cid:12)es the pre(cid:12)x property. Thus in summing over all l- low sequen es with the pre(cid:12)x property, the only l- lowsequen esthatdonot an eloutarethel- y le overs,givingthe laimedresult. Algorithm using pre(cid:12)x low sequen es. To ompute l usingthis hara terization, wemustsumupthe ontributionofalll- lowsequen eswiththepre(cid:12)xproperty. One wayistomodifythedynami programmingapproa husedintheprevioussub-se tion for low sequen es. This an be easily done. Let us instead do things di(cid:11)erently; the reason will be ome lear later. Adopt the onvention that there an be lows of length 0. Then ea h l- low sequen e C has exa tly one low Ci with head i, for i = 1 to n. So we write C = hC1;:::;Cni. De(cid:12)ne the signed weight of a low C as sw(C) = (cid:0)w(C) if C has non-zero length, and sw(C)=1otherwise. And de(cid:12)ne the signedweightof an l- lowsequen e Qn l as sw(C) = i=1sw(C). Then sgn(C)w(C) = ((cid:0)1) sw(C). So from the pre eding theorem, X l = sw(C) C is an l- low sequen e with thepre(cid:12)x property We say that a sequen e of non-negative integers l1;:::;ln satis(cid:12)es the property pre(cid:12)x(l) if Pn 1. t=1lt =l, and Pr(cid:0)1 Pn 2. For r 2[n℄, t=1 lt (cid:21)r(cid:0)1(cid:0)(n(cid:0)l). Alternatively t=rlt (cid:20)n(cid:0)r+1. Su h sequen es are \allowed" as lengths of lows in the low sequen es we onstru t; no other sequen es are allowed. We group the low sequen es with pre(cid:12)x property based on the lengths of the individual lows. In a low sequen e with pre(cid:12)x property C, if the length of low Ci (the possibly empty low with head i) is li, then any low with head i and length li anrepla eCi inC andstillgivea lowsequen esatisfyingthepre(cid:12)xproperty. Thus, if z(i;p) denotes the total signed weight of all lows that have vertex i as head and length p, then X Yn l = z(i;li) l1;:::;ln:pre(cid:12)x(l) i=1 To ompute l eÆ iently, wepla ethevaluesz(i;p)appropriatelyinaseriesofmatri- es B1;:::;Bn. The matrix Bk has entries z(k;p). Sin e we only onsider sequen es satisfying pre(cid:12)x(l), it suÆ es to onsider z(k;p) for p (cid:20) n(cid:0)k+1. Matrix Bk is of dimension(n(cid:0)k+2)(cid:2)(n(cid:0)k+1)andhasz(k;p)onthepthlowerdiagonalasshown below. DETERMINANT:OLDALGORITHMS,NEWINSIGHTS 9 2 3 z(k;0) 0 0 (cid:1)(cid:1)(cid:1) 0 0 6 z(k;1) z(k;0) 0 (cid:1)(cid:1)(cid:1) 0 0 7 6 7 6 z(k;2) z(k;1) z(k;0) (cid:1)(cid:1)(cid:1) 0 0 7 6 7 Bk =6 . . . . 7 6 .. .. .. .. 0 7 6 7 4 z(k;n(cid:0)k) z(k;n(cid:0)k(cid:0)1) z(k;n(cid:0)k(cid:0)2) (cid:1)(cid:1)(cid:1) z(k;1) z(k;0) 5 z(k;n(cid:0)k+1) z(k;n(cid:0)k) z(k;n(cid:0)k(cid:0)1) (cid:1)(cid:1)(cid:1) z(k;2) z(k;1) Now from the equation for l, it is lear that ! ! X Yn Yn l = Bi[ji(cid:0)1;ji℄= Bi [l+1;1℄ l+1=j0(cid:21)j1(cid:21)j2(cid:21):::(cid:21)jn=1: i=1 i=1 j0(cid:0)j1;j1(cid:0)j2;:::;jn(cid:0)1(cid:0)jn:pre(cid:12)x(l) or, more su intly, Yn T [ 0 1 2 3 (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) n ℄ = Bk k=1 It remains now to ompute z(i;p), the entries in the B matri es. We know that z(i;0) = 1 and z(i;1) = (cid:0)aii. For p (cid:21) 2, a low of length p with head i must (cid:12)rst visit a vertex u > i, then perform a walk of length p(cid:0)2 via verti es greater than i to some vertex v > i, and then return to i. To onstru t the path, we exploit the p fa tthat the (j;k)th entry in amatrix A givesthe sum ofthe weightsof all paths in GA of length exa tly p from j to k. So we must onsider the indu ed subgraph with verti es i+1;:::;n. This has an adja en y matrix Ai+1 obtained by removing the (cid:12)rst i rows and the (cid:12)rst i olumns of A. So A1 =A. Consider the submatri es of Ai as shown below. 0 (cid:0) (cid:1) 1 0 aii 1 Ri B 0 1 C B C B B C C B B C B C C Ai =BBBB BBBB Si CCCC BBB Ai+1 CCCA CCCC A A Then the lows ontributing to z(i;p) must use an edge in Ri, perform a walk p(cid:0)2 orresponding to Ai+1, and then return to i via an edge in Si. In other words, p(cid:0)2 z(i;p)=(cid:0)Ri Ai+1 Si 10 M.MAHAJANANDVVINAY So the matri es Bk look like this: 2 3 1 0 0 (cid:1)(cid:1)(cid:1) 0 0 6 (cid:0)akk 1 0 (cid:1)(cid:1)(cid:1) 0 0 7 6 7 6 (cid:0)RkSk (cid:0)akk 1 (cid:1)(cid:1)(cid:1) 0 0 7 6 7 6 . . . . 7 6 .. .. .. .. 0 7 6 7 Bk =66 ... ... ... ... 0 77 6 7 6 7 6 7 66 (cid:0)RkAnk+(cid:0)1k(cid:0)2Sk (cid:0)RkAnk+(cid:0)1k(cid:0)3Sk (cid:0)RkAnk+(cid:0)1k(cid:0)4Sk (cid:1)(cid:1)(cid:1) (cid:0)akk 1 77 4 5 n(cid:0)k(cid:0)1 n(cid:0)k(cid:0)2 n(cid:0)k(cid:0)3 (cid:0)RkAk+1 Sk (cid:0)RkAk+1 Sk (cid:0)RkAk+1 S1 (cid:1)(cid:1)(cid:1) (cid:0)RkSk (cid:0)akk This method of omputing (cid:31)A((cid:21)) is pre isely Samuelson's method [19, 8, 1, 25℄. SamuelsonarrivedatthisformulationusingLapla e'sTheoremonthematrix(cid:21)I(cid:0)A, whereas we have arrived at it via low sequen es with the pre(cid:12)x property. This interpretationoftheSamuelson-BerkowitzalgorithmisduetoValiant[25℄;the ombi- natorialproofof orre tness(proofofTheorem 3.4)isnew. (It ismentioned, without details, in [15℄.) 3.3. From Clows to Tour Sequen es Tables: Getting to Chistov's Al- gorithm. We now move in the other dire tion { generalize further beyond low sequen es. Firstly, we relax the ondition that the head of a low may be visited only on e. Thisgivesusmoregeneralized losedwalkswhi hwe alltours. To(cid:12)xa anon- i al representation, we do require the edges of the tour to be listed beginning from an o urren e of the head. Sin e there ould be multiple su h o urren es, we get di(cid:11)erent tours with the same multiset of edges. For instan e, the tour orresponding to the vertex sequen e 253246 is di(cid:11)erent from the tour orresponding to the vertex sequen e 246253. Se ondly, we deal with not just sequen es but ordered lists, or tables, of sequen es. Within a sequen e, the tours are ordered by their heads (and all heads are distin t). However, there is no restri tion on how the sequen es must be ordered in the table. In fa t, for the same multi-set of sequen es, di(cid:11)erent orderings ofthesequen eswillgivedi(cid:11)erenttablesthatwetreatasdistin t. Thirdly,theparity of a tour sequen e table depends on the number of sequen es in it, not the number of tours in it. A low sequen e is thus a tour sequen e table where (i) ea h sequen e ontains a single tour whi h is a low, and (ii) within the table the sequen es are ordered by their tour heads. Formally, Definition 3.5. A tour is an ordered sequen e of edges C =he1;e2;:::;epi su h that ei =hui;ui+1i for i2[p(cid:0)1℄ and ep =hup;u1i and ui =minfu1;:::;umg. The vertex u1 is alled the head of the tour and denQomted h(C). The length of the tour is jTj=p, and the weight of the tour is wt(T)= i=1w(ei). A j-tour sequen e T is an ordered sequen e of tours T = hT1;:::;Tki su h that h(T1) < ::: < h(Tk) and jT1j+:::+jTkj = j. The weight of the tour sequen e is Qk wt(T)= j=1wt(Tj), and the length is jTj=j. An l-tour sequen e table TST is an ordered sequen e of tour sequen es F = hQT1r;:::;Tri su h that jT1j + (cid:1)(cid:1)(cid:1) +l+jrTrj = l. The weight of the TST is wt(F) = j=1wt(Tj), and the sign is ((cid:0)1) . The following theorem shows that even TSTs an be used to ompute the har- a teristi polynomial.

Determinant: old algorithms, new insights

Similar Determinant: old algorithms, new insights