Algorithms. Algorithms 4.4 S HORTEST P ATHS. APIs shortest-paths properties. Dijkstra's algorithm edge-weighted DAGs negative weights

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1 Algorithm Shortet path in an edge-weighted digraph R OBERT S EDGEWICK K EVIN W AYNE Given an edge-weighted digraph, find the hortet path from to t. edge-weighted digraph ->. ->. ->7.7 -> >.28 ->.2 ->.8 ->2.2 7->. ->.2 2->7. ->2. ->.2 ->.8 ->.. S HORTEST P ATHS API hortet-path propertie Algorithm F O U R T H E D I T I O N R OBERT S EDGEWICK K EVIN W AYNE Dijktra' algorithm edge-weighted DAG negative weight hortet path from to ->2 2->7 7-> -> An edge-weighted digraph and a hortet path 2 Google map Shortet path application PERT/CPM. Map routing. Seam carving. Texture mapping. Robot navigation. Typeetting in TeX. Urban traffic planning. Optimal pipelining of VLSI chip. Telemarketer operator cheduling. Routing of telecommunication meage. Network routing protocol (OSPF, BGP, RIP). Exploiting arbitrage opportunitie in currency exchange. Optimal truck routing through given traffic congetion pattern. Reference: Network Flow: Theory, Algorithm, and Application, R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Prentice Hall,.

2 Shortet path variant Which vertice? Single ource: from one vertex to every other vertex. Single ink: from every vertex to one vertex t. Source-ink: from one vertex to another t. All pair: between all pair of vertice.. SHORTEST PATHS Retriction on edge weight? Nonnegative weight. Euclidean weight. Cycle? Arbitrary weight. No directed cycle. No "negative cycle." which variant? Algorithm ROBERT SEDGEWICK KEVIN WAYNE API hortet-path propertie Dijktra' algorithm edge-weighted DAG negative weight Simplifying aumption. Shortet path from to each vertex v exit. Weighted directed edge API Weighted directed edge: implementation in Java Similar to Edge for undirected graph, but a bit impler. public cla DirectedEdge DirectedEdge(int v, int w, double weight) weighted edge v w int from() vertex v int to() vertex w double weight() weight of thi edge String tostring() tring repreentation public cla DirectedEdge private final int v, w; private final double weight; public DirectedEdge(int v, int w, double weight) thi.v = v; thi.w = w; thi.weight = weight; v weight w public int from() return v; public int to() return w; from() and to() replace either() and other() Idiom for proceing an edge e: int v = e.from(), w = e.to(); public int weight() return weight; 7 8

3 Edge-weighted digraph API Edge-weighted digraph: adjacency-lit repreentation public cla EdgeWeightedDigraph EdgeWeightedDigraph(int V) edge-weighted digraph with V vertice EdgeWeightedDigraph(In in) edge-weighted digraph from input tream void addedge(directededge e) add weighted directed edge e Iterable<DirectedEdge> adj(int v) edge pointing from v int V() number of vertice int E() number of edge Iterable<DirectedEdge> edge() all edge String tostring() tring repreentation V tinyewd.txt 8 E adj Bag object reference to a DirectedEdge object Convention. Allow elf-loop and parallel edge. Edge-weighted digraph: adjacency-lit implementation in Java Single-ource hortet path API Same a EdgeWeightedGraph except replace Graph with Digraph. Goal. Find the hortet path from to every other vertex. public cla EdgeWeightedDigraph private final int V; private final Bag<DirectedEdge>[] adj; public EdgeWeightedDigraph(int V) thi.v = V; adj = (Bag<DirectedEdge>[]) new Bag[V]; for (int v = ; v < V; v++) adj[v] = new Bag<DirectedEdge>(); public void addedge(directededge e) int v = e.from(); adj[v].add(e); public Iterable<DirectedEdge> adj(int v) return adj[v]; add edge e = v w to only v' adjacency lit public cla SP SP p = new SP(G, ); for (int v = ; v < G.V(); v++) StdOut.printf("%d to %d (%.2f): ",, v, p.ditto(v)); for (DirectedEdge e : p.pathto(v)) StdOut.print(e + " "); StdOut.println(); SP(EdgeWeightedDigraph G, int ) hortet path from in graph G double ditto(int v) length of hortet path from to v Iterable <DirectedEdge> pathto(int v) hortet path from to v boolean hapathto(int v) i there a path from to v? 2

4 Single-ource hortet path API Goal. Find the hortet path from to every other vertex. public cla SP % java SP tinyewd.txt to (.): SP(EdgeWeightedDigraph G, int ) to (.): ->.8 ->. ->.2 to 2 (.2): ->2.2 to (.): ->2.2 2->7. 7->. to (.8): ->.8 to (.7): ->.8 ->. to (.): ->2.2 2->7. 7->. ->.2 to 7 (.): ->2.2 2->7. hortet path from in graph G double ditto(int v) length of hortet path from to v Iterable <DirectedEdge> pathto(int v) hortet path from to v boolean hapathto(int v) i there a path from to v? Algorithm ROBERT SEDGEWICK KEVIN WAYNE SHORTEST PATHS API hortet-path propertie Dijktra' algorithm edge-weighted DAG negative weight Data tructure for ingle-ource hortet path Data tructure for ingle-ource hortet path Goal. Find the hortet path from to every other vertex. Goal. Find the hortet path from to every other vertex. Obervation. A hortet-path tree (SPT) olution exit. Why? Obervation. A hortet-path tree (SPT) olution exit. Why? Conequence. Can repreent the SPT with two vertex-indexed array: ditto[v] i length of hortet path from to v. edgeto[v] i lat edge on hortet path from to v. Conequence. Can repreent the SPT with two vertex-indexed array: ditto[v] i length of hortet path from to v. edgeto[v] i lat edge on hortet path from to v. hortet-path tree from edgeto[] ditto[] null -> > >.7.7 ->.8.8 ->..7 -> >7.. parent-link repreentation public double ditto(int v) return ditto[v]; public Iterable<DirectedEdge> pathto(int v) Stack<DirectedEdge> path = new Stack<DirectedEdge>(); for (DirectedEdge e = edgeto[v]; e!= null; e = edgeto[e.from()]) path.puh(e); return path;

5 Edge relaxation Edge relaxation Relax edge e = v w. ditto[v] i length of hortet known path from to v. ditto[w] i length of hortet known path from to w. edgeto[w] i lat edge on hortet known path from to w. If e = v w give horter path to w through v, update both ditto[w] and edgeto[w]. Relax edge e = v w. ditto[v] i length of hortet known path from to v. ditto[w] i length of hortet known path from to w. edgeto[w] i lat edge on hortet known path from to w. If e = v w give horter path to w through v, update both ditto[w] and edgeto[w]. v w uccefully relaxe private void relax(directededge e) v. int v = e.from(), w = e.to(); if (ditto[w] > ditto[v] + e.weight()). ditto[w] = ditto[v] + e.weight(); w 7.2. edgeto[w] = e; black edge are in edgeto[] 7 8 Shortet-path optimality condition Shortet-path optimality condition Propoition. Let G be an edge-weighted digraph. Then ditto[] are the hortet path ditance from iff: ditto[] =. For each vertex v, ditto[v] i the length of ome path from to v. For each edge e = v w, ditto[w] ditto[v] + e.weight(). Pf. [ neceary ] Suppoe that ditto[w] > ditto[v] + e.weight() for ome edge e = v w. Then, e give a path from to w (through v) of length le than ditto[w]. v.. w ditto[v] 7.2 ditto[w] Propoition. Let G be an edge-weighted digraph. Then ditto[] are the hortet path ditance from iff: ditto[] =. For each vertex v, ditto[v] i the length of ome path from to v. For each edge e = v w, ditto[w] ditto[v] + e.weight(). Pf. [ ufficient ] Suppoe that = v v v2 vk = w i a hortet path from to w. Then, ditto[v] ditto[v] + e.weight() ditto[v2] ditto[v] + e2.weight()... ei = i th edge on hortet path from to w ditto[vk] ditto[vk-] + ek.weight() Add inequalitie; implify; and ubtitute ditto[v] = ditto[] = : ditto[w] = ditto[vk] e.weight() + e2.weight() + + ek.weight() weight of hortet path from to w Thu, ditto[w] i the weight of hortet path to w.! weight of ome path from to w 2

6 Generic hortet-path algorithm Generic hortet-path algorithm Generic algorithm (to compute SPT from ) Initialize ditto[] = and ditto[v] = for all other vertice. Repeat until optimality condition are atified: - Relax any edge. Generic algorithm (to compute SPT from ) Initialize ditto[] = and ditto[v] = for all other vertice. Repeat until optimality condition are atified: - Relax any edge. Propoition. Generic algorithm compute SPT (if it exit) from. Pf ketch. The entry ditto[v] i alway the length of a imple path from to v. Each ucceful relaxation decreae ditto[v] for ome v. The entry ditto[v] can decreae at mot a finite number of time.! Efficient implementation. How to chooe which edge to relax? Ex. Dijktra' algorithm (nonnegative weight). Ex 2. Topological ort algorithm (no directed cycle). Ex. Bellman-Ford algorithm (no negative cycle) Edger W. Dijktra: elect quote Do only what only you can do.. SHORTEST PATHS In their capacity a a tool, computer will be but a ripple on the urface of our culture. In their capacity a intellectual challenge, they are without precedent in the cultural hitory of mankind. Algorithm ROBERT SEDGEWICK KEVIN WAYNE API hortet-path propertie Dijktra' algorithm edge-weighted DAG negative weight The ue of COBOL cripple the mind; it teaching hould, therefore, be regarded a a criminal offence. It i practically impoible to teach good programming to tudent that have had a prior expoure to BASIC: a potential programmer they are mentally mutilated beyond hope of regeneration. Edger W. Dijktra Turing award 72 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 new generation of coding bum. 2

7 Edger W. Dijktra: elect quote Dijktra' algorithm demo Conider vertice in increaing order of ditance from (non-tree vertex with the lowet ditto[] value). Add vertex to tree and relax all edge pointing from that vertex an edge-weighted digraph Dijktra' algorithm demo Dijktra' algorithm viualization Conider vertice in increaing order of ditance from (non-tree vertex with the lowet ditto[] value). Add vertex to tree and relax all edge pointing from that vertex. v ditto[] edgeto[] hortet-path tree from vertex 27 28

8 Dijktra' algorithm viualization Dijktra' algorithm: correctne proof Propoition. Dijktra' algorithm compute a SPT in any edge-weighted digraph with nonnegative weight. Pf. Each edge e = v w i relaxed exactly once (when vertex v i relaxed), leaving ditto[w] ditto[v] + e.weight(). Inequality hold until algorithm terminate becaue: ditto[w] cannot increae ditto[] value are monotone decreaing ditto[v] will not change we chooe lowet ditto[] value at each tep (and edge weight are nonnegative) Thu, upon termination, hortet-path optimality condition hold.! 2 Dijktra' algorithm: Java implementation Dijktra' algorithm: Java implementation public cla DijktraSP private DirectedEdge[] edgeto; private double[] ditto; private IndexMinPQ<Double> pq; public DijktraSP(EdgeWeightedDigraph G, int ) edgeto = new DirectedEdge[G.V()]; ditto = new double[g.v()]; pq = new IndexMinPQ<Double>(G.V()); for (int v = ; v < G.V(); v++) ditto[v] = Double.POSITIVE_INFINITY; ditto[] =.; pq.inert(,.); while (!pq.iempty()) int v = pq.delmin(); for (DirectedEdge e : G.adj(v)) relax(e); relax vertice in order of ditance from private void relax(directededge e) int v = e.from(), w = e.to(); if (ditto[w] > ditto[v] + e.weight()) ditto[w] = ditto[v] + e.weight(); edgeto[w] = e; if (pq.contain(w)) pq.decreaekey(w, ditto[w]); ele pq.inert (w, ditto[w]); update PQ 2

9 Dijktra' algorithm: which priority queue? Computing a panning tree in a graph Depend on PQ implementation: V inert, V delete-min, E decreae-key. PQ implementation inert delete-min decreae-key total unordered array V V 2 binary heap log V log V log V E log V d-way heap log d V d log d V log d V E log E/V V Dijktra' algorithm eem familiar? Prim' algorithm i eentially the ame algorithm. Both are in a family of algorithm that compute a panning tree. Main ditinction: Rule ued to chooe next vertex for the tree. Prim: Cloet vertex to the tree (via an undirected edge). Dijktra: Cloet vertex to the ource (via a directed path). Fibonacci heap log V E + V log V amortized Bottom line. Array implementation optimal for dene graph. Binary heap much fater for pare graph. -way heap worth the trouble in performance-critical ituation. Fibonacci heap bet in theory, but not worth implementing. Note: DFS and BFS are alo in thi family of algorithm. Acyclic edge-weighted digraph Q. Suppoe that an edge-weighted digraph ha no directed cycle. I it eaier to find hortet path than in a general digraph? Algorithm ROBERT SEDGEWICK KEVIN WAYNE SHORTEST PATHS API hortet-path propertie Dijktra' algorithm edge-weighted DAG negative weight A. Ye!

10 Acyclic hortet path demo Conider vertice in topological order. Relax all edge pointing from that vertex. Acyclic hortet path demo Conider vertice in topological order. Relax all edge pointing from that vertex v ditto[] edgeto[] an edge-weighted DAG 7. hortet-path tree from vertex Shortet path in edge-weighted DAG Shortet path in edge-weighted DAG Propoition. Topological ort algorithm compute SPT in any edgeweighted DAG in time proportional to E + V. Pf. Each edge e = v w i relaxed exactly once (when vertex v i relaxed), leaving ditto[w] ditto[v] + e.weight(). Inequality hold until algorithm terminate becaue: ditto[w] cannot increae ditto[] value are monotone decreaing ditto[v] will not change edge weight can be negative! becaue of topological order, no edge pointing to v will be relaxed after v i relaxed Thu, upon termination, hortet-path optimality condition hold.! public cla AcyclicSP private DirectedEdge[] edgeto; private double[] ditto; public AcyclicSP(EdgeWeightedDigraph G, int ) edgeto = new DirectedEdge[G.V()]; ditto = new double[g.v()]; for (int v = ; v < G.V(); v++) ditto[v] = Double.POSITIVE_INFINITY; ditto[] =.; Topological topological = new Topological(G); for (int v : topological.order()) for (DirectedEdge e : G.adj(v)) relax(e); topological order

11 Content-aware reizing Content-aware reizing Seam carving. [Avidan and Shamir] Reize an image without ditortion for Seam carving. [Avidan and Shamir] Reize an image without ditortion for diplay on cell phone and web brower. diplay on cell phone and web brower. In the wild. Photohop CS, Imagemagick, GIMP, Content-aware reizing Content-aware reizing To find vertical eam: To find vertical eam: Grid DAG: vertex = pixel; edge = from pixel to downward neighbor. Weight of pixel = energy function of 8 neighboring pixel. Seam = hortet path (um of vertex weight) from top to bottom. Grid DAG: vertex = pixel; edge = from pixel to downward neighbor. Weight of pixel = energy function of 8 neighboring pixel. Seam = hortet path (um of vertex weight) from top to bottom. eam

12 Content-aware reizing Content-aware reizing To remove vertical eam: Delete pixel on eam (one in each row). To remove vertical eam: Delete pixel on eam (one in each row). eam Longet path in edge-weighted DAG Longet path in edge-weighted DAG: application Formulate a a hortet path problem in edge-weighted DAG. Negate all weight. Find hortet path. Negate weight in reult. equivalent: revere ene of equality in relax() Parallel job cheduling. Given a et of job with duration and precedence contraint, chedule the job (by finding a tart time for each) o a to achieve the minimum completion time, while repecting the contraint. longet path input ->. ->7.7 ->7.28 ->.2 ->.8 ->2.2 ->7. ->.2 7->2. ->2. ->.2 ->.8 ->. hortet path input -> -. -> > > -.2 -> -.8 -> >7 -. -> >2 -. ->2 -. -> -.2 -> -.8 -> -. job duration mut complete before Parallel job cheduling olution Key point. Topological ort algorithm work even with negative weight. 7 8

13 Critical path method Critical path method CPM. To olve a parallel job-cheduling problem, create edge-weighted DAG: Source and ink vertice. job duration mut complete before Two vertice (begin and end) for each job. Three edge for each job. begin to end (weighted by duration) ource to begin ( weight) end to ink ( weight) One edge for each precedence contraint ( weight) CPM. Ue longet path from the ource to chedule each job. 7 Parallel job cheduling olution job tart duration zero-weight edge to each job tart job finih precedence contraint (zero weight) 2 2 zero-weight edge from each job finih duration critical path Shortet path with negative weight: failed attempt Dijktra. Doen t work with negative edge weight.. SHORTEST PATHS 2-8 Dijktra elect vertex immediately after. But hortet path from to i 2. Algorithm ROBERT SEDGEWICK KEVIN WAYNE API hortet-path propertie Dijktra' algorithm edge-weighted DAG negative weight Re-weighting. Add a contant to every edge weight doen t work. 2 2 Adding 8 to each edge weight change the hortet path from 2 to. 2 Concluion. Need a different algorithm. 2

14 Negative cycle Bellman-Ford algorithm Def. A negative cycle i a directed cycle whoe um of edge weight i negative. Bellman-Ford algorithm digraph ->. -> -. ->7.7 -> >.28 ->.2 ->.8 ->2.2 7->. ->.2 2->7. ->2. ->.2 ->.8 ->. negative cycle ( ) ->->7-> hortet path from to ->->7->->->7->...->->-> Initialize ditto[] = and ditto[v] = for all other vertice. Repeat V time: - Relax each edge. for (int i = ; i < G.V(); i++) for (int v = ; v < G.V(); v++) for (DirectedEdge e : G.adj(v)) relax(e); pa i (relax each edge) Propoition. A SPT exit iff no negative cycle. auming all vertice reachable from Bellman-Ford algorithm demo Bellman-Ford algorithm demo Repeat V time: relax all E edge. Repeat V time: relax all E edge v ditto[] edgeto[] an edge-weighted digraph hortet-path tree from vertex

15 Bellman-Ford algorithm: viualization Bellman-Ford algorithm: analyi pae 7 Bellman-Ford algorithm Initialize ditto[] = and ditto[v] = for all other vertice. Repeat V time: - Relax each edge. SPT Propoition. Dynamic programming algorithm compute SPT in any edgeweighted digraph with no negative cycle in time proportional to E V. Pf idea. After pa i, found path that i at leat a hort a any hortet path containing i (or fewer) edge. 7 8 Bellman-Ford algorithm: practical improvement Single ource hortet-path implementation: cot ummary Obervation. If ditto[v] doe not change during pa i, no need to relax any edge pointing from v in pa i+. algorithm retriction typical cae wort cae extra pace FIFO implementation. Maintain queue of vertice whoe ditto[] changed. Overall effect. be careful to keep at mot one copy of each vertex on queue (why?) The running time i till proportional to E V in wort cae. But much fater than that in practice. topological ort Dijktra (binary heap) Bellman-Ford Bellman-Ford (queue-baed) no directed cycle no negative weight no negative cycle E + V E + V V E log V E log V V E V E V V E + V E V V Remark. Directed cycle make the problem harder. Remark 2. Negative weight make the problem harder. Remark. Negative cycle make the problem intractable.

16 Finding a negative cycle Finding a negative cycle Negative cycle. Add two method to the API for SP. boolean hanegativecycle() i there a negative cycle? Obervation. If there i a negative cycle, Bellman-Ford get tuck in loop, updating ditto[] and edgeto[] entrie of vertice in the cycle. Iterable <DirectedEdge> negativecycle() negative cycle reachable from 2 digraph ->. -> -. ->7.7 -> >.28 ->.2 ->.8 ->2.2 7->. ->.2 2->7. ->2. ->.2 ->.8 ->. negative cycle ( ) ->->7-> edgeto[v] v Propoition. If any vertex v i updated in pa V, there exit a negative cycle (and can trace back edgeto[v] entrie to find it). In practice. Check for negative cycle more frequently. 2 Negative cycle application: arbitrage detection Negative cycle application: arbitrage detection Problem. Given table of exchange rate, i there an arbitrage opportunity? USD EUR GBP CHF CAD USD Currency exchange graph. Vertex = currency. Edge = tranaction, with weight equal to exchange rate. Find a directed cycle whoe product of edge weight i >. EUR GBP *. *. = EUR..888 CHF USD GBP CAD Ex. $, 7 Euro,2.2 Canadian dollar $, CAD. CHF.7.. = 7.7 Challenge. Expre a a negative cycle detection problem.

17 Negative cycle application: arbitrage detection Shortet path ummary Model a a negative cycle detection problem by taking log. Let weight of edge v w be -ln (exchange rate from currency v to w). Multiplication turn to addition; > turn to <. Find a directed cycle whoe um of edge weight i < (negative cycle). -ln(.7) -ln(.) -ln(.) = -.7 USD Remark. Fatet algorithm i extraordinarily valuable!.2 CAD -. EUR -.8 replace each weight w with ln(w) CHF GBP Nonnegative weight. Arie in many application. Dijktra' algorithm i nearly linear-time. Acyclic edge-weighted digraph. Arie in ome application. Topological ort algorithm i linear time. Edge weight can be negative. Negative weight and negative cycle. Arie in ome application. If no negative cycle, can find hortet path via Bellman-Ford. If negative cycle, can find one via Bellman-Ford. Shortet-path i a broadly ueful problem-olving model.