6.8 Shortest Paths. Chapter 6. Dynamic Programming. Shortest Paths: Failed Attempts. Shortest Paths

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1 1 Chaper.8 Shore Pah Dynamic Programming Slide by Kein Wayne. Copyrigh 5 Pearon-Addion Weley. All righ reered. Shore Pah Shore Pah: Failed Aemp Shore pah problem. Gien a direced graph G = (V, E), wih edge weigh c w, find hore pah from node o node. allow negaie weigh Ex. Node repreen agen in a financial eing and c w i co of ranacion in which we buy from agen and ell immediaely o w. Dijkra. Can fail if negaie edge co. u 1 - Re-weighing. Adding a conan o eery edge weigh can fail

2 5 Shore Pah: Negaie Co Cycle Shore Pah: Dynamic Programming Negaie co cycle. Def. OPT(i, ) = lengh of hore - pah P uing a mo i edge Cae 1: P ue a mo i-1 edge. OPT(i, ) = OPT(i-1, ) Oberaion. If ome pah from o conain a negaie co cycle, here doe no exi a hore - pah; oherwie, here exi one ha i imple. W c(w) < Cae : P ue exacly i edge. if (, w) i fir edge, hen OPT ue (, w), and hen elec be w- pah uing a mo i-1 edge if i = OPT(i, ) = min OPT(i 1, ), min { OPT(i 1, w)+ c w oherwie (, w) E Remark. By preiou oberaion, if no negaie cycle, hen OPT(n-1, ) = lengh of hore - pah. Shore Pah: Implemenaion Shore Pah: Pracical Improemen Shore-Pah(G, ) { foreach node V M[, ] M[, ] for i = 1 o n-1 foreach node V M[i, ] M[i-1, ] foreach edge (, w) E M[i, ] min { M[i, ], M[i-1, w] + c w Analyi. Θ(mn) ime, Θ(n ) pace. Finding he hore pah. Mainain a "ucceor" for each able enry. Pracical improemen. Mainain only one array M[] = hore - pah ha we hae found o far. No need o check edge of he form (, w) unle M[w] changed in preiou ieraion. Theorem. Throughou he algorihm, M[] i lengh of ome - pah, and afer i round of updae, he alue M[] i no larger han he lengh of hore - pah uing i edge. Oerall impac. Memory: O(m + n). Running ime: O(mn) wor cae, bu ubanially faer in pracice. 8

3 9 Bellman-Ford: Efficien Implemenaion Puh-Baed-Shore-Pah(G,, ) { foreach node V { M[] ucceor[] φ.9 Diance Vecor Proocol M[] = for i = 1 o n-1 { foreach node w V { if (M[w] ha been updaed in preiou ieraion) { foreach node uch ha (, w) E { if (M[] > M[w] + c w ) { M[] M[w] + c w ucceor[] w If no M[w] alue changed in ieraion i, op. Diance Vecor Proocol Diance Vecor Proocol Communicaion nework. Node rouer. Edge direc communicaion link. Co of edge delay on link. naurally nonnegaie, bu Bellman-Ford ued anyway! Dijkra' algorihm. Require global informaion of nework. Diance ecor proocol. Each rouer mainain a ecor of hore pah lengh o eery oher node (diance) and he fir hop on each pah (direcion). Algorihm: each rouer perform n eparae compuaion, one for each poenial deinaion node. "Rouing by rumor." Bellman-Ford. Ue only local knowledge of neighboring node. Synchronizaion. We don' expec rouer o run in lockep. The order in which each foreach loop execue in no imporan. Moreoer, algorihm ill conerge een if updae are aynchronou. Ex. RIP, Xerox XNS RIP, Noell' IPX RIP, Cico' IGRP, DEC' DNA Phae IV, AppleTalk' RTMP. Caea. Edge co may change during algorihm (or fail compleely) "couning o infiniy" 1 deleed 11 1

4 1 Pah Vecor Proocol Link ae rouing. Each rouer alo ore he enire pah. Baed on Dijkra' algorihm. no ju he diance and fir hop Aoid "couning-o-infiniy" problem and relaed difficulie. Require ignificanly more orage..1 Negaie Cycle in a Graph Ex. Border Gaeway Proocol (BGP), Open Shore Pah Fir (OSPF). Deecing Negaie Cycle Deecing Negaie Cycle Lemma. If OPT(n,) = OPT(n-1,) for all, hen no negaie cycle. Pf. Bellman-Ford algorihm. Lemma. If OPT(n,) < OPT(n-1,) for ome node, hen (any) hore pah from o conain a cycle W. Moreoer W ha negaie co. Theorem. Can deec negaie co cycle in O(mn) ime. Add new node and connec all node o wih -co edge. Check if OPT(n, ) = OPT(n-1, ) for all node. if ye, hen no negaie cycle if no, hen exrac cycle from hore pah from o Pf. (by conradicion) Since OPT(n,) < OPT(n-1,), we know P ha exacly n edge. By pigeonhole principle, P mu conain a direced cycle W. Deleing W yield a - pah wih < n edge W ha negaie co. W c(w) < 15 1

5 1 Deecing Negaie Cycle: Applicaion Deecing Negaie Cycle: Summary Currency conerion. Gien n currencie and exchange rae beween pair of currencie, i here an arbirage opporuniy? Remark. Fae algorihm ery aluable! Bellman-Ford. O(mn) ime, O(m + n) pace. Run Bellman-Ford for n ieraion (inead of n-1). Upon erminaion, Bellman-Ford ucceor ariable race a negaie cycle if one exi. See p. 88 for improed erion and early erminaion rule. $ 8 1/ F 8 4/ / /1 /5 IBM 1/1 1 DM 5