Shortest Path With Negative Weights

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1 Shor Pah Wih Ngaiv Wigh Conn Conn. Dird hor pah wih ngaiv wigh. Ngaiv yl dion. appliaion: urrny xhang arbirag Tramp amr problm. appliaion: opimal piplining of VLSI hip Shor Pah wih Ngaiv Wigh Shor Pah wih Ngaiv Wigh Ngaiv o yl. OPT(i, v) = lngh of hor -v pah uing a mo i ar. L P b uh a pah. - - Ca 1: P u a mo i-1 ar. Ca : P u xaly i ar. if (u, v) i la ar, hn OPT l b -u pah uing a mo i-1 ar, and hn u (u, v) If om pah from o v onain a ngaiv o yl, hr do no xi a hor -v pah; ohrwi, hr xi on ha i impl. W v OPT ( i, v) = min OPT ( i 1, v), min ( u, v ) E { OPT ( i 1, u) + ( u, v) } if i = ohrwi (W) < Goal: ompu OPT(n-1, ) and find a orrponding - pah.

2 Shor Pah wih Ngaiv Wigh: Algorihm Dynami Programming Shor Pah INPUT: G = (V, E),, n = V ARRAY: OPT[..n, V] FOREACH v V OPT[, v] = OPT[,] = FOR i = 1 o n FOREACH v V min OPT ( i 1, u) + ( u, m = OPT[i-1, v] u, v ) E m = FOREACH (u, v) E m = min (m, OPT[i-1, u] + [u,v]) OPT[i, v] = min(m, m ) { ) } ( v Shor Pah: Running Tim Dynami programming algorihm rquir Θ(mn) im and pa. Our loop rpa n im. Innr loop for vrx v onidr indgr(v) ar. v V Finding h hor pah. indgr( v) = m Could mainain prdor variabl. Alrnaiv: ompu opimal dian, onidr only zro rdud o ar. RETURN OPT[n-1, ] Shor Pah: Ding Ngaiv Cyl L1: if OPT(n,v) < OPT(n-1,v) for om nod v, hn (any) hor pah from o v uing a mo n ar onain a yl; morovr any uh yl ha ngaiv o. Proof (by onradiion). Sin OPT(n,v) < OPT(n-1,v), P ha n ar. L C b any dird yl in P. Dling C giv u a pah from o v of fwr han n ar C ha ngaiv o. Shor Pah: Ding Ngaiv Cyl L1: if OPT(n,v) < OPT(n-1,v) for om nod v, hn (any) hor pah from o v uing a mo n ar onain a yl; morovr any uh yl ha ngaiv o. Proof (by onradiion). Sin OPT(n,v) < OPT(n-1,v), P ha n ar. L C b any dird yl in P. Dling C giv u a pah from o v of fwr han n ar C ha ngaiv o. C (C) < v Corollary: an d ngaiv o yl in O(mn) im. Nd o ra bak hrough ub-problm

3 Ding Ngaiv Cyl: Appliaion Shor Pah: Praial Improvmn Currny onvrion. Givn n urrni (finanial inrumn) and xhang ra bwn pair of urrni, i hr an arbirag opporuniy? Fa algorihm vry valuabl! Praial improvmn. If OPT(i, v) = OPT(i-1, v) for all nod v, hn OPT(i, v) ar h hor pah dian.! Conqun: an op algorihm a oon a hi happn. $ 8 1/ F Mainain only on array OPT(v).! U O(m+n) pa; ohrwi Θ(mn) b a. / / /1 / 8 IBM No nd o hk ar of h form (u, v) unl OPT(u) hangd in prviou iraion.! Avoid unnary work. 1 DM 1/1 Ovrall ff. Sill O(mn) wor a, bu O(m) bhavior in prai. 9 1 Shor Pah: Praial Improvmn Bllman-Ford FIFO Shor Pah INPUT: G = (V, E),, n = V ARRAY: OPT[V], prd[v] FOREACH v V OPT[v] =, prd[v] = φ Shor Pah: Sa of h Ar All im blow ar for ingl our hor pah in dird graph wih no ngaiv yl. O(mn) im, O(m + n) pa. Shor pah: raighforward. Ngaiv yl: Bllman-Ford prdor variabl onain hor pah or ngaiv yl (no provd hr). Ngaiv yl wak: op if any nod nquud n im. OPT[] =, Q = QUEUEini() WHILE (Q φ) u = QUEUEg() FOREACH (u, v) E IF (OPT[u] + [u,v] < OPT[v]) OPT[v] = OPT[u] + [u,v] prd[v] = u IF (v Q) QUEUEpu(v) RETURN OPT[n-1] O(mn 1/ log C) im if all ar o ar ingr bwn C and C. Rdu o wighd bipari mahing (aignmn problm). "Co-aling." Gabow-Tarjan (1989), Orlin-Ahuja (199). O(mn + n log n) undird hor pah, no ngaiv yl. Rdu o wighd non-bipari mahing. Byond h op of hi our. 11 1

4 Tramp-Samr Problm Tramp-amr (min o o im raio) problm. A ramp amr ravl from por o por arrying argo. A voyag from por v o por w arn p(v,w) dollar, and rquir (v,w) day. Capain wan a our ha ahiv larg man daily profi. Wward Ho (189 19) 1 p = = p = 1 = p = - = Tramp-Samr Problm Tramp-amr (min o o im raio) problm. Inpu: digraph G = (V, E), ar o, and ar ravral im >. Goal: find a dird yl W ha minimiz raio Novl appliaion. Minimiz yl im (maximiz frquny) of logi hip on IBM proor hip by adjuing loking hdul. Spial a. Find a ngaiv o yl. µ ( W ) = W W. man daily profi = + 1 = Tramp-Samr Problm Linariz objiv funion. L µ* b valu of minimum raio yl. L µ b a onan. Dfin l = µ. Tramp-Samr Problm Linariz objiv funion. L µ* b valu of minimum raio yl. L µ b a onan. Dfin l = µ. Ca 1: hr xi ngaiv o yl W uing lngh l. W ( µ ) < µ > µ *. W W Ca : vry dird yl ha poiiv o uing lngh l. ( W µ < µ W W ) > for vry ylw for vry ylw µ < µ *. Ca : vry dird yl ha nonngaiv o uing lngh l, and hr xi a zro o yl W*. ( W µ µ W W W * W * ) for vry yl W for vry ylw = µ µ = µ *. µ µ *. 1 1

5 Tramp-Samr Problm Tramp-Samr: Squnial Sarh Produr Linariz objiv funion. L µ* b valu of minimum raio yl. L µ b a onan. Dfin l = µ. Ca 1: hr xi ngaiv o yl W uing lngh l. µ* < µ Ca : vry dird yl ha poiiv o uing lngh l. µ* > µ Ca : vry dird yl ha nonngaiv o uing lngh l, and hr xi a zro o yl W*. µ* = µ Squnial Tramp Samr L µ b a known uppr bound on µ*. REPEAT (forvr) l µ Solv hor pah problm wih lngh l IF (ngaiv o yl W w.r.. l ) µ µ(w) ELSE Find a zro o yl W* w.r.. l. RETURN W*. Thorm: qunial algorihm rmina. Ca 1 µ rily dra from on iraion o h nx. µ i h raio of om yl, and only finily many yl Tramp-Samr: Binary Sarh Produr Tramp-Samr: Binary Sarh Produr W yl lf -C, righ C Binary Sarh Tramp Samr lf µ* righ REPEAT (forvr) IF (µ(w) = µ*) RETURN W µ (lf + righ) / l µ Solv hor pah problm wih lngh l IF (ngaiv o yl w.r.. l ) righ µ W ngaiv o yl w.r.. l ELSE IF (zro o yl W*) RETURN W*. ELSE lf µ Invarian: inrval [lf, righ] and yl W aify: lf µ* µ(w) < righ. Proof by induion follow from a 1-. Lmma. Upon rminaion, h algorihm rurn a min raio yl. Immdia from a. Aumpion. All ar o ar ingr bwn C and C. All ar ravral im ar ingr bwn T and T. Lmma. Th algorihm rmina afr O(log(nCT)) iraion. Proof on nx lid. Thorm. Th algorihm find min raio yl in O(mn log (nct)) im. 19

6 Tramp-Samr: Binary Sarh Produr Lmma. Th algorihm rmina afr O(log(nCT)) iraion. Iniially, lf = -C, righ = C. Eah iraion halv h iz of h inrval. L (W) and (W) dno o and ravral im of yl W. W how any inrval of iz l han 1 / (n T ) onain a mo on valu from h { (W) / (W) : W i a yl }. l W 1 and W yl wih µ(w 1 ) > µ(w ) ( W1 ) ( W) ( W ) ( ) ( ) ( ) 1 W W W > 1 > ( W1) ( W) ( W1) ( W) numraor of RHS i a la 1, dnominaor i a mo n T Afr 1 + log ((C) (n T )) = O(log (nct)) iraion, a mo on raio in h inrval. Algorihm mainain yl W and inrval [lf, righ].. lf µ* µ(w) < righ.. Tramp Samr: Sa of h Ar Min raio yl. O(mn log (nct)). O(n log n) dn. (Mgiddo, 199) O(n log n) par. (Mgiddo, 198) Minimum man yl. Spial a whn all ravral im = 1. Θ(mn). (Karp, 198) O(mn 1/ log C). (Orlin-Ahuja, 199) O(mn log n). (Karp-Orlin, 1981) paramri implx - b in prai 1 Opimal Piplining of VLSI Chip Novl appliaion. Minimiz yl im (maximiz frquny) of logi hip on IBM proor hip by adjuing loking hdul. If lok ignal arriv a lah imulanouly, min yl im = 1. Allow individual lok arrival im a lah. Clok ignal a lah: A:, 1,,,... A 9 B B: -1, 9, 19, 9,... C:, 1,,, D: -,, 1,,... Opimal yl im = 1. Max man wigh yl = 1. C Lah Graph D