Outline. Elimination tree. Jean-Yves L Excellent and Bora Uçar. GRAAL, LIP, ENS Lyon, France

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1 Outln Elmntn tr Jn-Yvs L Exllnt n Br Uçr GRAAL, LIP, ENS Lyn, Frn CR-07: Sprs Mtrx Cmputtns, Otr 2010 ttp://rl.ns-lyn.r/~ur/cr07/ 1/32 CR09

2 Outln Outln 1 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn 2/32 CR09

3 T lmntn tr T rl lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Tr r mny uss lmntn trs r sprs trztn. W wll sussn ts us n Clsky trztn LL T vn symmtr pstv mtrx A. Nnzrs n L n rtrz n trms pts n t lmntn tr. In prtulr, t rw n lumn struturs (r t numr nnzrs n rw n lumn L) n xprss n trms t lmntn tr. Prtn t strutur L. Rll tt n n ul t rp L usn t lmntn prss. Ts vs runnn tm prprtnl t t sz t ll-n rp G + (A) r t numr nnzrs n L. Nt, ut w wnt mr. 3/32 CR09

4 St-up T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Rmnr A spnnn tr nnt rp G = (V, E) s tr T = (V, F ), su tt F E. A tpll rrn rt tr s n rrn tt numrs lrn vrts r tr prnt. A pstrr s tpll rrn w numrs t vrts n ny sutr nsutvly. Lt A n n n symmtr pstv-nt n rrul mtrx, A = LL T ts Clsky trztn, n G + (A) ts ll rp (rp F = L + L T ). W ssum n rrn (t lst prtl) n A. 4/32 CR09

5 A rst ntn T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn 138 JOSEPH W. H. LIU Sn A s rrul, t rst n 1 lumns L s t lst n -nl nnzr (prv?). Fr lumn < n L, rmv ll t nnzrs n t lumn xpt t JOSEPH rst W. H. n LIU lw t nl. Lt L t nt t rmnn strutur n nsr t mtrx F t = L t + L T t. T rp G(F t ) s tr ll t lmntn tr. I 0 FIG An xmpl mtrx struturs. 5/32 CR09

6 A rst ntn T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn T lmntn tr A s spnnn tr G + (A) stsyn t rltn PARENT [] = mn{ > : l 0}. A = F = I ll n ntrs ntrs n T(A) FIG An xmpl 3 mtrx 7 struturs. 9 G(A) G(A G (A) = G(F) G(F) G(Ft)= T(A 2 FIG Grp 6/32strutursCR09 t xmpl n F T(A) 4 6

7 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn A sn ntn: Rprsnts lumn pnns Dpnny twn lumns L: Clumn > pns n lumn l 0 Us rt rp t xprss ts pnny ( rm t, lumn pns n lumn ) Smply runnt pnns (trnstv rutn) T trnstv rutn t rt ll rp vs t lmntn tr strutur. Rmv rt (, ) tr s pt lnt rtr tn n rm t. 7/32 CR09

8 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Drt ll rp n ts trnstv rutn Drt ll rp Trnstv rutn 1 T(A) 8/32 CR09

9 T lmntn tr A tr ntn: DFS tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Trm T lmntn tr T (A) nnt rp G(A) s pt-rst sr tr t ll rp G + (A). Pr. Lt x 1, x 2,..., x n t n rrn G + (A). Cnsr t pt-rst sr sut t t llwn t-rkn rul: wn tr s mr tn n n t xplr nxt, lwys pk t n wt lrst susrpt. Wt ts tnl rul t pt-rst sr wll nstrut T (A). 9/32 CR09

10 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn A pt-rst sr tr t ll rp Any DFS n n unrt rp prus nly Tr n Bk s. 6/7 5/8 4/11 3/12 13/14 9/10 1/20 2/19 15/18 16/17 10/32 CR09

11 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Pt rtrztn ll s Nnzrs L I l 0, tn n x s n nstr x n T (A). Sm zrs L Lt T [x ] n T [x ] tw snt sutrs T (A). Tn l st = 0 r ny x s T [x ] n x t T [x ]. Smply us tr s n rss n t DFS tr n unrt rp. T rllry ls sws tt t lmntn tr xprsss ptntl prlllsm (rsultn rm t sprsty). 11/32 CR09

12 Fll-n ntrs T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Fll-pt trm [Rs, Trn, Lukr 76] Lt G = (V, E, α) n rr rp. Tn (v, w) s n G α = (V, E F (G α )) tr xsts pt µ = [v = v 1, v 2,..., v k+1 = w] n G su tt α 1 (v ) < mn{α 1 (v), α 1 (w)}, 2 k Rsttn usn t lmntn tr Lt >. Tn l 0 tr xsts pt x, x p1,..., x pk, x n t rp A su tt {x p1,..., x pk } T [x ]. Hr T [x ] s t st ns n t sutr rt t x. 12/32 CR09

13 n xk n t lmntn tr, wr k O. [J Dntn n prprts Dtrmnn rw n lumn unts T T lmntn tr sm rsult wsdtrmntn ls vn y Trn n Ynnkks [59, rt t rsult t Wttn [60], w prsnt t t t 1978 SIA Rw strutur Sprs Mtrx t Cmputtns. Clsky Ptrlly, tr t rsult Trm 3.5 s llws: Rw strutur trm Lt > > k. W v l 0 n x s n nstr sm n x k n t lmntn tr wt k 0. T rsult Trm 3.5 n us t rtrz t rw Dn T r [x ] tclsky t strutur tr. t t rw L,.., T r [x ] = {x : l 0, }. W v T r [x ] T [x ]. Lt us n Tr [x] t t strutur t t rw A prvus trm (l 0 x s n nstr x ) stts T [x ]. In tr wrs, T r [x ] s tn rm T [x ] y prunn: Fn t lvs T r [x ] n prun T [x ] t ts lvs. 13/32 CR09

14 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Rw strutur t Clsky tr Crllry T n x s l n n t rw sutr T r [x ] 0 n r vry prpr ll n ntrs snnt ntrs n T(A) x k x, w v k = THE ROLE OF ELIMINATION TREES 143 FIG An xmpl mt F) 1 T(A) 14/32 CR09 G(A G(F)

15 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Rw strutur t Clsky tr Rmrk E l x rw sutr T r [x ] rrspns t n {x, x } n t rnl rp A. THE ROLE OF ELIMINATION TREES T rw sutr T r [x ] FIG An xmpl mt mrkr[x ] r k < n k 0 3 k wl mrkr[x ] mrkr[x ] 8 prnt() FIG Rw sutrs r t mtrx xmpl F /32 CR09 G(A

16 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Clumn strutur t Clsky tr ) A = Clumn strutur trm T strutur lumn L,.., {x : l 0, } s vn y A G(A) (T [x ]) {x } Hr, A G(A) (S) = {x / S : x A G(A) (v) r sm v S} Rmrk: Sn L L, t lumn s nlu n rw sutrs T r [] tt ntn. W n trvrs ll rw sutrs T r [] n trmn t ll n ntrs ntrs T(A) pttrn L y ntn su tt T r [] ntns ) F = A = G (A) = G(F) F = T(A) 16/32 5 CR JOSEPH W. H. LIU ll n ntrs ntrs n T(A) A G(A) (T []) {} = {,, } I

17 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Clumn strutur t Clsky tr Clumn strutur trm {x : l 0, } = A G(A) (T [x ]) {x } Wy? Frst, sn l 0, w v t v x n t strutur L. Rmnr: Rw strutur trm Lt > > k. W v l 0 n x s n nstr sm n x k n t lmntn tr wt k 0. Sn, l 0 wt >, y t trm n t lt, t n x s n nstr sm n x k wt k 0. In tr wrs, x A G(A) (x k ), wr x k T [x ]. Equvlntly, x A G(A) (T [x ]). 17/32 CR09

18 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn T rw n lumn unts: Slw vrsn Trvrs rw sutr t unt t nnzrs n rws n lumns. () 1 r = 1,..., n r = 1,..., n r() 1 mrkr[x ] r k < n k 0 k trvrs n mrk t ns n rw sutr T r [x ] wl mrkr[x ] r() r() + 1 () () + 1 mrkr[x ] prnt() Runs n tm prprtnl t t numr nnzrs n L 18/32 CR09

19 T lmntn tr T rw n lumn unts Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Glrt, N, n Pytn, 94 mput rw n lumn unts n tm prprtnl t O(mα(m, n)), wr m = nnz(a) n α(m, n) s t nvrs t Akrmnn s untn (w s 4 r snsl m n n). T ssntl s r s llws (t lt t mnry us): Rw unts: pstrr t lmntn tr, n unt t numr s n rw sutr wtut uln t. W knw t lvs, srt tm rn t t pstrr. Fn t lvl t lst mmn nstr tw nsutvly numr ns n sutrt t rm t rst ts tw ns; rn n lvls v t lnt t pt. A t rsultn pt lnts. Clumn unts: A uus ntn t lumn unt r n usn t wts snnts t n, wr t wt n r mput s ntrutns rm rw sutr. An, t pstrr n t lst mmn nstrs tw nsutv ns r us. 19/32 CR09

20 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Dtrmntn lmntn trs strt wt n mpty rst (n vrts) r = 1,..., n vrtx t t rst r k < n k 0 mk t prnt t rt t tr ntnn k Fr ny us snt st prtns t n t rt t tr ntnn. Ts wy mplxty O(mα(m, n)) n v. mkst(x): rt nw snltn st wt lmnt x. n(x): rturn t rprsnttv t st ntnn x. lnk(x, y): rm t unn t tw sts ntnn x n y, n rturn t rprsnttv t nw st. Smpl yt tv tnqus (pt mprssn n lnn) sul us r O(mα(m, n)) tm nsstts kpn tw struturs, n prt, lttl stlr vrsn s us. 20/32 CR09

21 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Dtrmntn lmntn trs 138 JOSEPH W. H. LIU r = 1,..., n mkst() prnt[] 0 r k < n k 0 u n(k) prnt[u] = 0 n u tn prnt[u] lnk(, u) I 0 FIG An xmpl mtrx struturs. 21/32 CR09

22 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Dtrmntn lmntn trs 138 JOSEPH W. H. LIU r = 1,..., n mkst() prnt[] 0 r k < n k 0 u n(k) prnt[u] = 0 n u tn prnt[u] lnk(, u) I 0 FIG An xmpl mtrx struturs. 22/32 CR09

23 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Dtrmntn lmntn trs 138 JOSEPH W. H. LIU r = 1,..., n mkst() prnt[] 0 r k < n k 0 u n(k) prnt[u] = 0 n u tn prnt[u] lnk(, u) I 0 FIG An xmpl mtrx struturs. 23/32 CR09

24 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Dtrmntn lmntn trs 138 JOSEPH W. H. LIU r = 1,..., n mkst() prnt[] 0 r k < n k 0 u n(k) prnt[u] = 0 n u tn prnt[u] lnk(, u) I 0 FIG An xmpl mtrx struturs. 24/32 CR09

25 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Dtrmntn lmntn trs 138 JOSEPH W. H. LIU r = 1,..., n mkst() prnt[] 0 r k < n k 0 u n(k) prnt[u] = 0 n u tn prnt[u] lnk(, u) I 0 FIG An xmpl mtrx struturs. 25/32 CR09

26 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Dtrmntn lmntn trs 138 JOSEPH W. H. LIU r = 1,..., n mkst() prnt[] 0 r k < n k 0 u n(k) prnt[u] = 0 n u tn prnt[u] lnk(, u) I 0 FIG An xmpl mtrx struturs. 26/32 CR09

27 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Dtrmntn lmntn trs 138 JOSEPH W. H. LIU r = 1,..., n mkst() prnt[] 0 r k < n k 0 u n(k) prnt[u] = 0 n u tn prnt[u] lnk(, u) I 0 FIG An xmpl mtrx struturs. 27/32 CR09

28 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Dtrmntn lmntn trs 138 JOSEPH W. H. LIU r = 1,..., n mkst() prnt[] 0 r k < n k 0 u n(k) prnt[u] = 0 n u tn prnt[u] lnk(, u) I 0 FIG An xmpl mtrx struturs. 28/32 CR09

29 T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Dtrmntn lmntn trs 138 JOSEPH W. H. LIU r = 1,..., n mkst() prnt[] 0 r k < n k 0 u n(k) prnt[u] = 0 n u tn prnt[u] lnk(, u) I 0 FIG An xmpl mtrx struturs. 29/32 CR09

30 T lmntn tr Equvlnt rrrns Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Dntn Tw rrns P n Q r quvlnt t struturs t ll rps PAP T n QAQ T r t sm (tt s ty r smrp). Equvlnt rrns rsult n t sm munt ll-n n mputtn urn trztn. T s t nttn, w suss nly n rrn wrt A,.., P s n quvlnt rrn A t ll rp A n tt PAP T r smrp. 30/32 CR09

31 T lmntn tr Equvlnt rrrns Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Any tpll rrn n T (A) r quvlnt Lt P t prmuttn mtrx rrspnn t tpll rrn T (A). Tn, G + (PAP T ) n G + (A) r smrp. Any tpll rrn n T (A) r quvlnt Lt P t prmuttn mtrx rrspnn t tpll rrn T (A). T lmntn tr T (PAP T ) n T (A) r smrp. A qu n? W n prss t lrn n ny rr (n prlll mplmnttn ts wul lp). Tr r sm tr quvlnt rrns tn t pstrrn vn lmntn tr. Fr sm tr spts trztn, n my prr n rrn t sm tr. S pprs y Lu n t w. 31/32 CR09

32 Qustns T lmntn tr Dntn n prprts Dtrmnn rw n lumn unts Dtrmntn Lt L n L tw lumns L n supps >. Wn ts lumns v t sm pttrn t ns k >? Wt s t tr? Gvn n rr mtrx n ts lmntn tr. Hw n w sy t vn rrn s prt? 32/32 CR09

Victims Compensation Claim Status of All Pending Claims and Claims Decided Within the Last Three Years

Victims Compensation Claim Status of All Pending Claims and Claims Decided Within the Last Three Years Claim#:021914-174 Initials: J.T. Last4SSN: 6996 DOB: 5/3/1970 Crime Date: 4/30/2013 Status: Claim is currently under review. Decision expected within 7 days Claim#:041715-334 Initials: M.S. Last4SSN: 2957