Shortest Paths A 8 B 8 2 Shortest

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

Outline and Reading Weighted graphs (§ 12. 1) n n Shortest path problem Shortest path properties Dijkstra’s algorithm (§ 12. 6. 1) n n Algorithm Edge relaxation The Bellman-Ford algorithm Shortest paths in DAGs All-pairs 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 SFO LAX 3 4 7 1 1233 849 ORD 1 7 138 DFW Shortest Paths PVD 42 1205 337 HNL 2555 1843 802 n LGA 1120 10 99 MIA 3

Shortest Path Problem Given a weighted graph and two vertices u and v, we want to find a path of minimum total weight between u and v. n Length of a path is the sum of the weights of its edges. Example: n Shortest path between Providence and Honolulu Applications n n Internet packet routing Flight reservations Driving directions SFO LAX 3 4 7 1 1233 849 ORD DFW Shortest Paths 7 8 3 1 2 PVD 14 1205 337 HNL 2555 1843 802 n LGA 1120 10 99 MIA 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 LAX 1233 DFW Shortest Paths 7 8 3 1 2 PVD 14 1205 337 HNL 2555 3 4 17 849 ORD 802 SFO 1843 LGA 1120 10 99 MIA 5

Dijkstra’s Algorithm The distance of a vertex v from a vertex s is the length of a shortest path between s and v Dijkstra’s algorithm computes the distances of all the vertices from a given start vertex s Assumptions: n n n the graph is connected the edges are undirected 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 label d(v) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent vertices At each step n n Shortest Paths 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 6

Edge Relaxation Consider an edge e = (u, z) such that n n d(u) = 50 u is the vertex most recently added to the cloud z is not in the cloud s The relaxation of edge e updates distance d(z) as follows: u e d(u) = 50 d(z) min{d(z), d(u) + weight(e)} s Shortest Paths u e 10 d(z) = 75 z 10 d(z) = 60 z 7

Example A 8 B 2 8 7 E 2 8 1 2 7 C 3 0 2 4 5 B 9 F 2 7 5 E C 3 3 5 B 2 Shortest Paths 7 4 2 1 D 8 2 7 C 3 0 2 3 5 F A 5 E 0 9 8 D 11 8 2 4 1 A 8 D F A 5 E 4 9 8 B 2 3 2 C 0 4 1 9 D 8 F 5 8 3

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

Dijkstra’s Algorithm A priority queue stores the vertices outside the cloud n n Key: distance Element: vertex Locator-based methods n n insert(k, e) returns a locator replace. Key(l, k) changes the key of an item We store two labels with each vertex: n n distance (d(v) label) locator in priority queue Algorithm Dijkstra. Distances(G, s) Q new heap-based priority queue for all v G. vertices() if v = s set. Distance(v, 0) else set. Distance(v, ) l Q. insert(get. Distance(v), v) set. Locator(v, l) while Q. is. Empty() u Q. remove. Min() for all e G. incident. Edges(u) { relax edge e } z G. opposite(u, e) r get. Distance(u) + weight(e) if r < get. Distance(z) set. Distance(z, r) Q. replace. Key(get. Locator(z), r) Shortest Paths 10

Analysis Graph operations n Method incident. Edges is called once for each vertex Label operations n n We set/get the distance and locator labels of vertex z O(deg(z)) times Setting/getting a label takes O(1) time Priority queue operations n n 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 n Recall that Sv deg(v) = 2 m The running time can also be expressed as O(m log n) since the graph is connected Shortest Paths 11

Extension 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: n parent edge in the shortest path tree In the edge relaxation step, we update the parent label Algorithm Dijkstra. Shortest. Paths. Tree(G, s) … for all v G. vertices() … set. Parent(v, ) … for all e G. incident. Edges(u) { relax edge e } z G. opposite(u, e) r get. Distance(u) + weight(e) if r < get. Distance(z) set. Distance(z, r) set. Parent(z, e) Q. replace. Key(get. Locator(z), r) Shortest Paths 12

Why Dijkstra’s Algorithm Works Dijkstra’s algorithm is based on the greedy method. It adds vertices by increasing distance. n n 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. Shortest Paths A 8 B 2 2 7 7 C 3 5 E 0 2 4 1 9 D 8 F 5 13 3

Why It Doesn’t Work for Negative-Weight Edges Dijkstra’s algorithm is based on the greedy method. It adds vertices by increasing distance. n If a node with a negative incident edge were to be added late to the cloud, it could mess up distances for vertices already in the cloud. A 8 B 2 6 7 7 C 0 5 E 0 4 5 1 -8 D 9 F 5 C’s true distance is 1, but it is already in the cloud with d(C)=5! Shortest Paths 14 4

Bellman-Ford Algorithm Works even with negative- Algorithm Bellman. Ford(G, s) weight edges for all v G. vertices() if v = s Must assume directed set. Distance(v, 0) edges (for otherwise we else would have negativeset. Distance(v, ) weight cycles) for i 1 to n-1 do Iteration i finds all shortest for each e G. edges() paths that use i edges. { relax edge e } Running time: O(nm). u G. origin(e) z G. opposite(u, e) Can be extended to detect r get. Distance(u) + weight(e) a negative-weight cycle if it if r < get. Distance(z) exists n set. Distance(z, r) How? Shortest Paths 15

Bellman-Ford Example Nodes are labeled with their d(v) values 0 8 4 -2 7 3 -2 1 0 8 8 -2 7 9 3 5 0 8 -2 4 7 1 -2 6 1 5 9 0 8 4 -2 1 -2 3 -2 -2 5 8 4 9 9 4 -1 5 7 3 5 -2 Shortest Paths 1 1 -2 9 4 9 -1 5 16 4

DAG-based Algorithm 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). Algorithm Dag. Distances(G, s) for all v G. vertices() if v = s set. Distance(v, 0) else set. Distance(v, ) Perform a topological sort of the vertices for u 1 to n do {in topological order} for each e G. out. Edges(u) { relax edge e } z G. opposite(u, e) r get. Distance(u) + weight(e) if r < get. Distance(z) set. Distance(z, r) Shortest Paths 17

1 DAG Example Nodes are labeled with their d(v) values 1 1 0 8 4 -2 3 2 7 3 -5 4 1 0 8 3 8 -2 2 7 9 3 6 5 -5 5 0 8 -5 2 1 3 6 9 6 4 5 5 0 8 -2 7 4 1 1 8 5 -2 1 3 4 4 -2 4 1 -2 9 7 4 -1 5 3 5 -5 5 Shortest Paths 2 7 0 1 3 6 4 1 -2 9 7 (two steps) 4 -1 5 5 18 4

All-Pairs Shortest Paths Algorithm All. Pair(G) {assumes vertices 1, …, n} Find the distance for all vertex pairs (i, j) between every pair of if i = j vertices in a weighted D 0[i, i] 0 directed graph G. else if (i, j) is an edge in G We can make n calls to D 0[i, j] weight of edge (i, j) Dijkstra’s algorithm (if no else negative edges), which D 0[i, j] + takes O(nmlog n) time. for k 1 to n do Likewise, n calls to for i 1 to n do for j 1 to n do Bellman-Ford would take Dk[i, j] min{Dk-1[i, j], Dk-1[i, k]+Dk-1[k, j]} O(n 2 m) time. return Dn We can achieve O(n 3) time using dynamic Uses only vertices numbered 1, …, k i (compute weight of this edge) programming (similar to j the Floyd-Warshall Uses only vertices algorithm). numbered 1, …, k-1 k Shortest Paths numbered 1, …, k-1 19