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Shortest Paths

Learn about the concept of shortest paths in weighted graphs and how it can be applied to various scenarios like internet packet routing, flight reservations, and driving directions. Understand Dijkstra's algorithm and its step-by-step process.

patricklynch
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Shortest Paths

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  1. 0 A 4 8 2 8 2 3 7 1 B C D 3 9 5 8 2 5 E F Shortest Paths Shortest Paths

  2. Weighted Graphs • In a weighted graph, each edge has an associated numerical value, called the weight of the edge • Edge weights may represent, distances, costs, etc. • Example: • In a flight route graph, the weight of an edge represents the distance in miles between the endpoint airports 849 PVD 1843 ORD 142 SFO 802 LGA 1205 1743 337 1387 HNL 2555 1099 1233 LAX 1120 DFW MIA Shortest Paths

  3. Shortest Paths • Given a weighted graph and two vertices u and v, we want to find a path of minimum total weight between u and v. • Example: • Shortest path between Providence and Honolulu • Applications • Internet packet routing • Flight reservations • Driving directions Shortest path From PVD to HNL 849 PVD 1843 ORD 142 SFO 802 LGA 1205 1743 337 1387 HNL 2555 1099 1233 LAX 1120 DFW MIA Shortest Paths

  4. Shortest Path Properties Property 1: A subpath of a shortest path is itself a shortest path Property 2: There is a tree of shortest paths from a start vertex to all the other vertices Example: Tree of shortest paths from Providence DP! Connected & no cycles 849 PVD 1843 ORD 142 SFO 802 LGA 1205 1743 337 1387 HNL 2555 1099 1233 LAX 1120 DFW MIA Shortest Paths

  5. Dijkstra’s algorithm computes the distances of all the vertices from a given start vertex s  Single-source all-destination Assumption: the edge weights are nonnegative We grow a “cloud” of vertices, beginning with s and eventually covering all the vertices We store with each vertex v a labeld(v) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent vertices At each step We add to the cloud the vertex u outside the cloud with the smallest distance label, d(u) We update the labels of the vertices adjacent to u Dijkstra’s Algorithm It was designed in 20 min without pencil and paper! (source) Shortest Paths

  6. Edge Relaxation • Consider an edge e =(u,z) such that • uis the vertex most recently added to the cloud • z is not in the cloud • The relaxation of edge e updates distance d(z) as follows: d(z)min{d(z),d(u) + weight(e)} d(u) = 50 d(z) = 75 10 e u z s d(u) = 50 d(z) = 60 10 e u z s Shortest Paths

  7. 0 A 4 8 2 8 2 3 7 1 B C D 3 9 5 8 2 5 E F Step-by-step Example 0 A 4 8 2 8 2 4 7 1 B C D 3 9   2 5 E F 0 0 A A 4 4 8 8 2 2 8 2 3 7 2 3 7 1 7 1 B C D B C D 3 9 3 9 5 11 5 8 2 5 2 5 E F E F Shortest Paths

  8. Example (cont.) 0 A 4 8 2 7 2 3 7 1 B C D 3 9 5 8 2 5 E F 0 A 4 8 2 7 2 3 7 1 B C D 3 9 5 8 2 5 E F Shortest Paths

  9. Exampleby Table Filling (1/2) Quiz! Source node A 4 8 2 7 1 B C D 3 9 2 5 E F Shortest Paths

  10. Exampleby Table Filling (1/2) Quiz! Source node A 4 8 2 7 1 B C D 3 9 2 5 E F Selected nodes Shortest Paths

  11. Example by Table Filling (2/2) Quiz! Source node B C A F D E Animation Shortest Paths

  12. Example by Table Filling (2/2) Quiz! Source node B C A F D E Animation Selected nodes Shortest Paths

  13. Exercise 1 Quiz! Source node Shortest Paths

  14. Exercise 2 Source node Solution! Shortest Paths

  15. Exercise 3 Source node Solution! Shortest Paths

  16. Exercise 4 Source node Shortest Paths

  17. Dijkstra’s Algorithm AlgorithmDijkstraDistances(G, s) Q new heap-based priority queue for all v  G.vertices() ifv= s v.setDistance(0) else v.setDistance() l  Q.insert(v.getDistance(),v) v.setEntry(l) while Q.empty() l Q.removeMin() u l.getValue() for all e  u.incidentEdges() { relax e } z  e.opposite(u) r  u.getDistance() + e.weight() ifr< z.getDistance() z.setDistance(r)Q.replaceKey(z.getEntry(), r) • A heap-based adaptable priority queue with location-aware entries stores the vertices outside the cloud • Key: distance • Value: vertex • Recall that method replaceKey(l,k) changes the key of entry l • We store two labels with each vertex: • Distance • Entry in priority queue Shortest Paths

  18. Graph operations Method incidentEdges is called once for each vertex Label operations We set/get the distance and locator labels of vertex zO(deg(z)) times Setting/getting a label takes O(1) time Priority queue operations Each vertex is inserted once into and removed once from the priority queue, where each insertion or removal takes O(log n) time The key of a vertex in the priority queue is modified at most deg(w) times, where each key change takes O(log n) time Dijkstra’s algorithm runs in O((n + m) log n) time provided the graph is represented by the adjacency list structure Recall that Sv deg(v)= 2m The running time can also be expressed as O(m log n) since the graph is connected Analysis of Dijkstra’s Algorithm Shortest Paths

  19. Shortest Paths Tree AlgorithmDijkstraShortestPathsTree(G, s) … for all v  G.vertices() … v.setParent() … for all e  u.incidentEdges() { relax edge e } z  e.opposite(u) r  u.getDistance() + e.weight() ifr< z.getDistance() z.setDistance(r) z.setParent(e) Q.replaceKey(z.getEntry(),r) • Using the template method pattern, we can extend Dijkstra’s algorithm to return a tree of shortest paths from the start vertex to all other vertices • We store with each vertex a third label: • parent edge in the shortest path tree • In the edge relaxation step, we update the parent label Easy for backtracking Shortest Paths

  20. Why Dijkstra’s Algorithm Works • Dijkstra’s algorithm is based on DP. It adds vertices by increasing distance. • Suppose it didn’t find all shortest distances. Let F be the first wrong vertex the algorithm processed. • When the previous node, D, on the true shortest path was considered, its distance was correct • But the edge (D,F) was relaxed at that time! • Thus, so long as d(F)>d(D), F’s distance cannot be wrong. That is, there is no wrong vertex 0 A 4 8 2 7 2 3 7 1 B C D 3 9 5 5 2 E 8 11 F 4 H 9 G Shortest Paths

  21. Why It Doesn’t Work for Negative-Weight Edges • Dijkstra’s algorithm is based on the DP. It adds vertices by increasing distance, which does not hold if we have negative-weight edges. 0 A Quiz! 4 8 2 7 2 3 7 1 B C D 3 9 5 5 2 E 8 11 F 4 H 9 G Shortest Paths

  22. Bellman-Ford Algorithm (not in book) AlgorithmBellmanFord(G, s) for all v  G.vertices() ifv= s v.setDistance(0) else v.setDistance() for i 1 to n - 1do for each e  G.edges() { relax edge e } u  e.origin() z  e.opposite(u) r  u.getDistance() + e.weight() ifr< z.getDistance() z.setDistance(r) • Works even with negative-weight edges • Must assume directed edges (for otherwise we would have negative-weight cycles) • Iteration i finds all shortest paths that use i edges. • Running time: O(nm). • Can be extended to detect a negative-weight cycle if it exists • How? Shortest Paths

  23. Bellman-Ford Example Nodes are labeled with their d(v) values 0 0 4 4 8 8 -2 -2 -2 4 8 7 1 7 1       3 9 3 9 -2 5 -2 5     0 0 4 4 8 8 -2 -2 7 1 -1 7 1 5 8 -2 4 5 -2 -1 1 3 9 3 9 6 9 4 -2 5 -2 5   1 9 Shortest Paths

  24. DAG-based Algorithm (not in book) AlgorithmDagDistances(G, s) for all v  G.vertices() ifv= s v.setDistance(0) else v.setDistance() { Perform a topological sort of the vertices } for u 1 to n do {in topological order} for each e  u.outEdges() { relax edge e } z  e.opposite(u) r  u.getDistance() + e.weight() ifr< z.getDistance() z.setDistance(r) • Works even with negative-weight edges • Uses topological order • Doesn’t use any fancy data structures • Is much faster than Dijkstra’s algorithm • Running time: O(n+m). Shortest Paths

  25. DAG Example Nodes are labeled with their d(v) values 1 1 0 0 4 4 8 8 -2 -2 -2 4 8 4 4 3 7 2 1 3 7 2 1       3 9 3 9 -5 5 -5 5     6 5 6 5 1 1 0 0 4 4 8 8 -2 -2 5 4 4 3 7 2 1 -1 3 7 2 1 8 -2 4 5 -2 -1 9 3 3 9 0 4 1 7 -5 5 -5 5   1 7 6 5 6 5 Shortest Paths (two steps)

  26. Resources • Youtube tutorials • Dijkstra’s algorithm: Single source all destination • Bellman-Ford algorithm: Single source all destination • Floyd-Warshall algorithm: All pairs shortest path • Animation • Step-by-step animation Shortest Paths

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