Shortest Paths. Dijkstra's algorithm implementation negative weights. Google maps. Shortest paths in a weighted digraph. Shortest path versions

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1 Google map Shortet Path Dijktra' algorithm implementation negatie eight Reference: Algorithm in Jaa, Chapter Algorithm in Jaa, th Edition Robert Sedgeick and Kein Wayne Copyright 8 Noember, 8 :: PM Shortet path in a eighted digraph Shortet path erion Gien a eighted digraph G, find the hortet directed path from to t Which ertice? From one ertex to another. From one ertex to eery other. Beteen all pair of ertice. Edge eight. Nonnegatie eight. Arbitrary eight. Euclidean eight. t hortet path: t cot: =

2 Early hitory of hortet path algorithm Shortet path application Shimbel (9). Information netork. Ford (9). RAND, economic of tranportation. Leyzorek, Gray, Johnon, Lade, Meaker, Petry, Seitz (97). Combat Deelopment Dept. of the Army Electronic Proing Ground. Dantzig (98). Simplex method for linear programming. Bellman (98). Dynamic programming. Moore (99). Routing long-ditance telephone call for Bell Lab. Dijktra (99). Simpler and fater erion of Ford' algorithm. Map. Robot naigation. Texture mapping. Typeetting in TeX. Urban traffic planning. Optimal pipelining of VLSI chip. Telemarketer operator cheduling. Subroutine in adanced algorithm. Routing of telecommunication meage. Approximating pieceie linear function. Netork routing protocol (OSPF, BGP, RIP). Exploiting arbitrage opportunitie in currency exchange. Optimal truck routing through gien traffic congetion pattern. Reference: Netork Flo: Theory, Algorithm, and Application, R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Prentice Hall, 99. Edger W. Dijktra: elect quote The quetion of hether computer can think i like the quetion of hether ubmarine can im. Do only hat only you can do. Dijktra' algorithm implementation negatie eight In their capacity a a tool, computer ill be but a ripple on the urface of our culture. In their capacity a intellectual challenge, they are ithout precedent in the cultural hitory of mankind. The ue of COBOL cripple the mind; it teaching hould, therefore, be regarded a a criminal offence. Edger Dijktra Turing aard 97 APL i a mitake, carried through to perfection. It i the language of the future for the programming technique of the pat: it create a ne generation of coding bum. 7 8

3 Single-ource hortet-path Edge relaxation Gien. Weighted digraph G, ource ertex. Goal. Find hortet path from to eery other ertex. Oberation. Ue parent-link repreentation to tore hortet path tree dit[] 9 pred[] - Relaxation along edge e from to. dit[] i length of ome path from to. dit[] i length of ome path from to. If gie a horter path to through, update dit[] and pred[]. dit[] = dit[] = dit[] = 7 int = e.from(), = e.to(); if (dit[] > dit[] + e.eight()) dit[] = dit[] + e.eight()); pred[] = e; 9 Dijktra' algorithm Dijktra' algorithm Initialize S to, dit[] to, dit[] to for all other. Repeat until S contain all ertice connected to : - find edge e ith in S and not in S that minimize dit[] + e.eight(). - relax along edge e - add to S Initialize S to, dit[] to, dit[] to for all other. Repeat until S contain all ertice connected to : - find edge e ith in S and not in S that minimize dit[] + e.eight(). - relax along edge e - add to S dit[] = dit[] + e.eight(); pred[] = e; dit[] e dit[] e S S

4 Dijktra algorithm example Dijktra' algorithm: correctne proof edge ith in S and not in S edge in hortet path tree Inariant. For in S, dit[] i the length of the hortet path from to. (.9) (.) (.) (. =.9 +.) (.8 =.9 +.9) (.) (.7 =. +.) (.9 =. +.) Pf. (by induction on S ) Let be next ertex added to S. Let P* be the path through. Conider any other path P, and let x be firt node on path outide S. P i already longer than P* a oon a it reache x by greedy choice. Thu, dit[] i the length of the hortet path from to. P x S P* (.8 =. +.) (.8 =. +.) (.9) (.8) (. =.8 +.) Shortet path tree Remark. Dijktra examine ertice in increaing ditance from ource. % % Dijktra' algorithm implementation negatie eight 7% %

5 Weighted directed graph API Weighted digraph: adjacency-et implementation in Jaa public cla DirectedEdge implement Comparable<Edge> DirectedEdge(int, int, double eight) create a eighted edge int from() ertex public cla WeightedDigraph priate final int V; priate final SET<Edge>[] adj; int to() ertex double eight() the eight String tostring() tring repreentation public WeightedDigraph(int V) thi.v = V; adj = (SET<DirectedEdge>[]) ne SET[V]; for (int = ; < V; ++) adj[] = ne SET<DirectedEdge>(); public cla WeightedDigraph eighted digraph data type WeightedDigraph(int V) create an empty digraph ith V ertice WeightedDigraph(In in) create a digraph from input tream oid inert(directededge e) add an edge from to public oid addedge(directededge e) int = e.from(); adj[].add(e); ame a eighted undirected graph, but only add edge to ' adjacency et Iterable<DirectedEdge> adj(int ) return an iterator oer edge leaing public Iterable<DirectedEdge> adj(int ) return adj[]; int V() return number of ertice String tostring() return a tring repreentation 7 8 Weighted directed edge: implementation in Jaa Shortet path data type public cla DirectedEdge implement Comparable<DirectedEdge> priate final int, ; priate final double eight; public DirectedEdge(int, int, double eight) thi. = ; thi. = ; thi.eight = eight; public int from() return ; public int to() return ; public int eight() return eight; ame a Edge, except from() and to() replace either() and other() Deign pattern. Dijktra cla i a WeightedDigraph client. Client query method return ditance and path iterator. public cla Dijktra Dijktra(WeightedDigraph G, int ) double ditance(int ) Iterable <DirectedEdge> path(int ) hortet path from in graph G length of hortet path from to hortet path from to public int compareto(edge that) if (thi. < that.) return -; if (thi. > that.) return +; if (thi. < that.) return -; if (thi. > that.) return +; return ; for ue in a ymbol table In = ne In("netork.txt"); WeightedDigraph G = ne WeightedDigraph(in); int =, t = G.V() - ; Dijktra dijktra = ne Dijktra(G, ); StdOut.println("ditance = " + dijktra.ditance(t)); for (int : dijktra.path(t)) StdOut.println(); 9

6 Dijktra implementation challenge Dijktra' algorithm implementation Find edge e ith in S and not in S that minimize dit[] + e.eight(). Ho difficult? Intractable. O(E) time. O(V) time. O(log V) time. O(log* V) time. Contant time. try all edge Dijktra ith an array priority queue Dijktra ith a binary heap Q. What goe onto the priority queue? A. Fringe ertice connected by a ingle edge to a ertex in S (priority = hortet path to lat ertex in S + eight of ingle edge) S dit[] e S Starting to look familiar? Dijktra' algorithm: PQ implementation approach Lazy Dijktra algorithm example Maintain thee inariant. For in S, dit[] i the length of the hortet path from to. For not in S, dit[] minimize dit[] + e.eight() among all edge e ith in S. PQ contain ertice not in S, ith dit[] a priority. Implication. To find next ertex to add to S, delmin from PQ. To maintain inariant, update dit[] by relaxing all edge leaing (and update PQ if ertex not in S get cloer to a ertex in S) (.9) (.) priority queue (.) (. =.9 +.) (.8 =.9 +.9) (.) (.7 =. +.) (.9 =. +.) Total running time. Depend on PQ implementation. Exactly V delmin() operation. Exactly E edge relaxation. At mot inert() operation. (.7) (.8 =. +.) (.8 =. +.) (.9) (.8) (. =.8 +.) (.9) (.9) (. =.8 +.8) (.9 =.8 +.) (.)

7 Lazy implementation of Dijktra' SPT algorithm Dijktra' algorithm: hich priority queue? Initialize dit[] = and marked[] = fale for all ertice. Running time of Dijktra depend on PQ implementation. MinPQplu<Double, Integer> pq = ne MinPQplu<Double, Integer>(); dit[] =.; pq.put(dit[], ); hile (!pq.iempty()) int = pq.delmin(); if (marked[]) continue; marked() = true; for (Edge e : G.adj()) int = e.to(); if (dit[] > dit[] + e.eight()) dit[] = dit[] + e.eight(); pred[] = e; pq.inert(dit[], ); Remark. Same a LazyPrim except dit[] i ditance from to. relax edge e PQ implementation inert delmin total array V V binary heap log V log V E log V d-ay heap (Johnon) log d V log d V E log d V Fibonacci heap (Sleator-Tarjan) log V E + V log V Bottom line. Array implementation optimal for dene graph. Binary heap far better for pare graph. d-ay heap orth the trouble in performance-critical ituation. Fibonacci heap bet in theory, but not orth implementing. Priority-firt earch Inight. All of our graph-earch method are the ame algorithm! Maintain a et of explored ertice S. Gro S by exploring edge ith exactly one endpoint leaing S. DFS. Take edge from ertex hich a dicoered mot recently. BFS. Take from ertex hich a dicoered leat recently. Prim. Take edge of minimum eight. Dijktra. Take edge to ertex that i cloet to. e dit[] S Dijktra' algorithm implementation negatie eight Challenge. Expre thi inight in reuable Jaa code. 7 8

8 Currency conerion Currency conerion Problem. Gien currencie and exchange rate, hat i bet ay to conert one ounce of gold to US dollar? oz. gold $7.. oz. gold 8. $7.. oz. gold. Franc.9 Euro $7.8. [ 8..7 ] [ ] Graph formulation. Vertex = currency. Edge = tranaction, ith eight equal to exchange rate. Find path that maximize product of eight. 7. currency Euro Franc $ Gold G. $ UK pound Euro Japanee Yen Si Franc US dollar F.77 E 9. Gold (oz.) Challenge. Expre a a hortet path problem. 9 Currency conerion Shortet path ith negatie eight: failed attempt Reduce to hortet path problem by taking log. Let eight of edge be - lg (exchange rate from currency to ). Multiplication turn to addition. Shortet path ith gien eight correpond to bet exchange equence. 7. Dijktra. Doen t ork ith negatie edge eight. Dijktra elect ertex immediately after. But hortet path from to i. G. $ lg(.) = Re-eighting. Add a contant to eery edge eight alo doen t ork. 9. Adding 9 to each edge change the hortet path becaue it add 9 to each edge;.9 F.77 E.87 rong thing to do for path ith many edge. Challenge. Sole hortet path problem ith negatie eight. Bad ne. Need a different algorithm.

9 Negatie cycle Shortet path ith negatie eight Def. A negatie cycle i a directed cycle hoe um of edge eight i negatie. Problem. Doe a gien digraph contain a negatie cycle? Problem. Find the hortet imple path from to t C cot(c) < t Oberation. If negatie cycle C i on a path from to t, then hortet path can be made arbitrarily negatie by pinning around cycle. Bad ne. Problem i intractable. C cot(c) < t Good ne. Can ole problem in O(VE) tep; if no negatie cycle, can ole problem ith ame algorithm! Wore ne. Need a different problem. Edge relaxation Shortet path ith negatie eight: dynamic programming algorithm Relaxation along edge e from to. dit[] i length of ome path from to. dit[] i length of ome path from to. If gie a horter path to through, update dit[] and pred[]. A imple olution that ork! Initialize dit[] =, dit[]=. Repeat V time: relax each edge e. dit[] = dit[] = dit[] = 7 int = e.from(), = e.to(); if (dit[] > dit[] + e.eight()) dit[] = dit[] + e.eight()); pred[] = e; for (int i = ; i <= G.V(); i++) for (int = ; < G.V(); ++) for (Edge e : G.adj()) int = e.to(); if (dit[] > dit[] + e.eight()) dit[] = dit[] + e.eight()) pred[] = e; phae i relax edge -

10 Dynamic programming algorithm trace Dynamic programming algorithm dit[].9. (. = +.) (. = +.) relaxed edge that update dit[] (.9 =. +.) (.7 =. +.) (. =.9 +.) Running time. Proportional to E V. Inariant. At end of phae i, dit[] length of any path from to uing at mot i edge. Propoition. If there are no negatie cycle, upon termination dit[] i the length of the hortet path from from to. and pred[] gie the hortet path can top early ince no entrie in dit[] updated (. =.8 +.) (.8 =. +.) (.8 =. +.) 7 8 Bellman-Ford-Moore algorithm Single ource hortet path implementation: cot ummary Oberation. If dit[] doen't change during phae i, no need to relax any edge leaing in phae i+. FIFO implementation. Maintain queue of ertice hoe ditance changed. Running time. Proportional to EV in ort cae. Much fater than that in practice. be careful to keep at mot one copy of each ertex on queue nonnegatie cot no negatie cycle algorithm ort cae typical cae Dijktra (array heap) V V Dijktra (binary heap) E lg V E dynamic programming E V E V Bellman-Ford E V E Remark. Negatie eight make the problem harder. Remark. Negatie cycle make the problem intractable. 9

11 Shortet path application: arbitrage Negatie cycle detection I there an arbitrage opportunity in currency graph? Ex: $.9 Franc.98 Euro $.8. I there a negatie cot cycle? If there i a negatie cycle reachable from. Bellman-Ford-Moore get tuck in loop, updating ertice in cycle. 7. G. $ pred[] F.77 E < Finding a negatie cycle. If any ertex i updated in phae V, there exit a negatie cycle, and e can trace back pred[] to find it. Remark. Fatet algorithm i aluable! Negatie cycle detection Shortet path ummary Goal. Identify a negatie cycle (reachable from any ertex). -8. Dijktra algorithm. Eay and optimal for dene digraph. PQ yield near optimal for pare graph G $ -..9 Priority-firt earch. Generalization of Dijktra algorithm. Encompae DFS, BFS, and Prim. Enable eay olution to many graph-proceing problem. -. F.87 E -7.7 Negatie eight. Arie in application. If negatie cycle, problem i intractable (!) If no negatie cycle, olable ia claic algorithm. Solution. Initialize Bellman-Ford by etting dit[] = for all ertice. Shortet-path i a broadly ueful problem-oling model.