Weighted Graphs. Shortest Paths. Shortest Path Properties. Shortest Path Problem. Dijkstra s Algorithm. Edge Relaxation PVD ORD SFO LGA HNL LAX DFW

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1 4/14/1 :7 PM Shore Pah A Weighed Graph In a weighed graph, each edge ha an aociaed numerical value, called he weigh of he edge dge weigh may repreen, diance, co, ec. xample: In a fligh roue graph, he weigh of an edge repreen he diance in mile beween he endpoin airpor HNL SO LAX OR W LGA PV MIA Shore Pah 1 Shore Pah Shore Pah Problem Given a weighed graph and wo verice u and v, we wan o find a pah of minimum oal weigh beween u and v. Lengh of a pah i he um of he weigh of i edge. xample: Shore pah beween Providence and Honolulu Applicaion Inerne packe rouing ligh reervaion riving direcion PV OR SO LGA HNL LAX W MIA Shore Pah Shore Pah Properie Propery 1: A ubpah of a hore pah i ielf a hore pah Propery : There i a ree of hore pah from a ar verex o all he oher verice xample: Tree of hore pah from Providence HNL SO LAX OR W LGA PV MIA Shore Pah 4 ijkra Algorihm dge Relaxaion The diance of a verex v from a verex i he lengh of a hore pah beween and v ijkra algorihm compue he diance of all he verice from a given ar verex Aumpion: he graph i conneced he edge are undireced he edge weigh are nonnegaive We grow a cloud of verice, beginning wih and evenually covering all he verice We ore wih each verex v a label d(v) repreening he diance of v from in he ubgraph coniing of he cloud and i adjacen verice A each ep We add o he cloud he verex u ouide he cloud wih he malle diance label, d(u) We updae he label of he verice adjacen o u onider an edge e (u,z) uch ha u i he verex mo recenly added o he cloud z i no in he cloud The relaxaion of edge e updae diance d(z) a follow: d(z) min{d(z),d(u) weigh(e)} d(u) u d(u) u e e d(z) 7 z d(z) z Shore Pah Shore Pah 1

2 4/14/1 :7 PM xample A 4 A A 4 A Shore Pah 7 xample (con.) A A Shore Pah ijkra Algorihm A prioriy queue ore he verice ouide he cloud Key: diance lemen: verex Locaor-baed mehod iner(k,e) reurn he index of e replacekey(l,k) change he key of an iem We ore wo label wih each verex: iance (d(v) label) index in prioriy queue Algorihm ijkraiance(g, ) Q new heap-baed prioriy queue for all v G.verice() if v eiance(v, ) ele eiance(v, ) l Q.iner(geiance(v),v) elocaor(v,l) while Q.impy() u Q.removeMin() for all e G.incidendge(u) // relax edge e z G.oppoie(u,e) r geiance(u) weigh(e) if r geiance(z) eiance(z,r) Q.replaceKey(geLocaor(z),r) Shore Pah 9 Analyi Graph operaion Mehod incidendge i called once for each verex Label operaion We e/ge he diance and locaor label of verex zo(deg(z)) ime Seing/geing a label ake O(1) ime Prioriy queue operaion ach verex i inered once ino and removed once from he prioriy queue, where each inerion or removal ake O(log n) ime The key of a verex in he prioriy queue i modified a mo deg(w) ime, where each key change ake O(log n) ime ijkra algorihm run in O((n m) log n) ime provided he graph i repreened by he adjacency li rucure Recall ha v deg(v) m The running ime can alo be expreed a O(m log n) ince he graph i conneced Shore Pah 1 xenion Uing he emplae mehod paern, we can exend ijkra algorihm o reurn a ree of hore pah from he ar verex o all oher verice We ore wih each verex a hird label: paren edge in he hore pah ree In he edge relaxaion ep, we updae he paren label Algorihm ijkrashorepahtree(g, ) for all v G.verice() eparen(v, ) for all e G.incidendge(u) // relax edge e z G.oppoie(u,e) r geiance(u) weigh(e) if r geiance(z) eiance(z,r) eparen(z,e) Q.replaceKey(geLocaor(z),r) Why ijkra Algorihm Work ijkra algorihm i baed on he greedy mehod. I add verice by increaing diance. Suppoe i didn find all hore diance. Le be he fir wrong verex he algorihm proceed. When he previou node,, on he rue hore pah wa conidered, i diance wa correc. u he edge (,) wa relaxed a ha ime! Thu, o long a d()>d(), diance canno be wrong. Tha i, here i no wrong verex. A Shore Pah 11 Shore Pah 1

3 4/14/1 :7 PM Shore Pah Shore Pah: ailed Aemp Shore pah problem. Given a direced graph G = (V, ), wih edge weigh c vw, find hore pah from node o node. allow negaive weigh x. Node repreen agen in a financial eing and c vw i co of ranacion in which we buy from agen v and ell immediaely o w. ijkra. an fail if negaive edge co. u 1 - v Re-weighing. Adding a conan o every edge weigh can fail Shore Pah: Negaive o ycle Negaive co cycle. ynamic Programming Subproblem Propery: The problem can be recurively defined by he ubproblem of he ame kind. Obervaion. If ome pah from o conain a negaive co cycle, here doe no exi a hore - pah; oherwie, here exi one ha i imple. Trade pace for ime: A able i ued o ore he oluion of he ubproblem (he meaning of programming before he age of compuer i able ). W c(w) < 1 eigning a P oluion How are he ubproblem defined? Where are he oluion ored? How are he bae value compued? How do we compue each enry from oher enrie in he able? Wha i he order in which we fill in he able? Two P algorihm for All-pair hore pah oh are correc. oh produce correc value for all-pair hore pah. The difference i he ubproblem formulaion, and hence in he running ime. The reaon boh algorihm are given i o remind you how o do P algorihm! u, be prepared o provide one or boh of hee algorihm, and o be able o apply i o an inpu (on ome exam, for example).

4 4/14/1 :7 PM ynamic Programming ir aemp: le {1,,,n} denoe he e of verice. Subproblem formulaion: M[i,j,k] = min lengh of any pah from i o j ha ue a mo k edge. All pah have a mo n-1 edge, o 1 k n-1. When k=1, M[i,j,1] = w[i,j], he edge weigh from i o j. Minimum pah from i o j are found in M[i,j,n-1] How o e M[i,j,k] from oher enrie, for k>1? onider a minimum weigh pah from i o j ha ha a mo k edge. ae 1: The minimum weigh pah ha a mo k-1 edge. M[i,j,k] = M[i,j,k-1] ae : The minimum weigh pah ha exacly k edge. M[i,j,k] = min{ M[i,x,k-1] + w(x,j) : x in V} ombining he wo cae: M[i,j,k] = min{min{m[i,x,k-1] + w(x,j) : x in V}, M[i,j,k-1]} Queion: How o e M[i,j,k] from oher enrie? inihing he deign Where i he anwer ored? How are he bae value compued? How do we compue each enry from oher enrie? Wha i he order in which we fill in he marix? Running ime? Peudo-ode and Running ime analyi M[i,j,1] = W[i,j] for k = o n-1 M[i,j,k] = min{min{m[i,x,k-1] + w(x,j): x in V}, M[i,j,k-1]} How many enrie do we need o compue? O(n ) 1 i n; 1 j n; 1 k n-1 How much ime doe i ake o compue each enry? O(n) Toal ime: O(n 4 ) Nex P approach Try a new ubproblem formulaion! Q[i,j,k] = minimum weigh of any pah from i o j ha ue inernal verice drawn from {1,,,k}. eigning a P oluion How are he ubproblem defined? Where i he anwer ored? How are he bae value compued? How do we compue each enry from oher enrie? Wha i he order in which we fill in he marix? 4

5 4/14/1 :7 PM Solving ubproblem Q[i,j,k] = minimum weigh of any pah from i o j ha ue inernal verice (oher han i and j) drawn from {1,,,k}. ae cae: Q[i,j,] = w[i,j] for all i,j Minimum pah from i o j are found in Q[i,j,n] Once again, O(n ) enrie in he marix Q[i,j,k] = minimum weigh of any pah from i o j ha ue inernal verice drawn from {1,,,k}. Such minimum co pah eiher include verex k or doe no include verex k. If he minimum co pah P include verex k, hen you can divide P ino he pah P 1 from i o k, and P from k o j. Wha i he weigh of P 1? Wha i he weigh of P? Solving ubproblem Q[i,j,k] = minimum weigh of any pah from i o j ha ue inernal verice drawn from {1,,,k}. P i a minimum co pah from i o j ha ue verex k, and ha all inernal verice from {1,, k}. Pah P 1 from i o k, and P from k o j. The weigh of P 1 i Q[i,k,k-1] (why??). The weigh of P i Q[k,j,k-1] (why??). Thu he weigh of P i Q[i,k,k-1] + Q[k,j,k-1]. New P algorihm Q[i,j,] = w[i,j] for k= 1 o n Q[i,j,k] = min{q[i,j,k-1], Q[i,k,k-1] + Q[k,j,k-1]} ach enry only ake O(1) ime o compue There are O(n ) enrie Hence, O(n ) ime. Reuing he pace // Ue R[i,j] for Q[i,j,], Q[i,j,1],, Q[i,j,n]. R[i,j] = W[i,j]; for k= 1 o n R[i,j] = min{r[i,j], R[i,k] + R[k,j]} How o check negaive cycle // Ue R[i,j] for Q[i,j,], Q[i,j,1],, Q[i,j,n]. R[i,j] = W[i,j]; for k= 1 o n R[i,j] = min{r[i,j], R[i,k] + R[k,j]}; if (R[i,i] < ) prin( There i a negaive cycle );

6 4/14/1 :7 PM eecing Negaive ycle: Applicaion urrency converion. Given n currencie and exchange rae beween pair of currencie, i here an arbirage opporuniy? Remark. ae algorihm very valuable! $ 1/7 4/ / /1 / IM 1/1 17 M 1

Chapter 13. Network Flow III Applications. 13.1 Edge disjoint paths. 13.1.1 Edge-disjoint paths in a directed graphs

Chapter 13. Network Flow III Applications. 13.1 Edge disjoint paths. 13.1.1 Edge-disjoint paths in a directed graphs Chaper 13 Nework Flow III Applicaion CS 573: Algorihm, Fall 014 Ocober 9, 014 13.1 Edge dijoin pah 13.1.1 Edge-dijoin pah in a direced graph 13.1.1.1 Edge dijoin pah queiong: graph (dir/undir)., : verice.