Lecture 16 Shortest Path Algorithms n The singlesource

Lecture 16. Shortest Path Algorithms n The single-source shortest path problem is the following: given a source vertex s, and a sink vertex v, we'd like to find the shortest path from s to v. Here shortest path means a sequence of directed edges from s to v with the smallest total weight. n There are some subtleties here. Should we allow negative edges? Of course, there are no negative distances; nevertheless, there actually some cases where negative edges make logical sense. But then there may not be a shortest path, because if there is a cycle with negative weight, we could simply go around that cycle as many times as we want and reduce the cost of the path as much as we like. To avoid this, we might want to detect negative cycles. This can be done in the algorithm itself. n Non-negative cycles aren't helpful, either. Suppose our shortest path contains a cycle of non-negative weight. Then by cutting it out we get a path with the same weight or less, so we might as well cut it out.

Weighted Graphs n In a weighted graph, each edge has an associated numerical value, called the weight of the edge n Edge weights may represent distances, costs, etc. n 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 PVD 42 1205 337 HNL 2555 1843 802 n LGA 1120 10 99 MIA

Shortest Path Problem n 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. n Example: n Shortest path between Providence and Honolulu n Applications n Internet packet routing n Flight reservations n Driving directions LAX 1233 DFW 7 8 3 1 2 PVD 14 1205 337 HNL 2555 3 4 7 1 849 ORD 802 SFO 1843 LGA 1120 10 99 MIA

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 7 8 3 1 2 PVD 14 1205 337 HNL 2555 3 4 17 849 ORD 802 SFO 1843 LGA 1120 10 99 MIA

Dijkstra’s Algorithm n The distance of a vertex v n We grow a “cloud” of vertices, from a vertex s is the length of a shortest path between s and v n Dijkstra’s algorithm computes the distances of all the vertices from a given start vertex s n Assumptions: n the graph is connected n the edges are undirected n the edge weights are nonnegative beginning with s and eventually covering all the vertices n 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 n At each step n We add to the cloud the vertex u outside the cloud with the smallest distance label, d(u) n We update the labels of the vertices adjacent to u

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

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

Dijkstra’s Algorithm n A priority queue stores the vertices outside the cloud n n Key: distance Element: vertex n Locator-based methods n insert(k, e) returns a locator n replace. Key(l, k) changes the key (distance) of an item n 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)

Analysis n Graph operations Method incident. Edges is called once for each vertex n Label operations n We set/get the distance and locator labels of vertex z O(deg(z)) times n Setting/getting a label takes O(1) time n Priority queue operations n Each vertex is inserted once into and removed once from the priority queue, where each insertion or removal takes O(log n) time n 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 n 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 n The running time can also be expressed as O(m log n) since the graph is connected n

Extension n 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 n We store with each vertex a third label: n parent edge in the shortest path tree n 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)

Why Dijkstra’s Algorithm Works n 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. A 8 B 2 2 7 7 C 3 5 E 0 2 4 1 9 D 8 F 5 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! 4

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

Correctness n Lemma. After i repetitions of the "for" loop in BELLMAN-FORD, if there is a path from s to u with at most i edges, then d[u] is at most the length of the shortest path from s to u with at most i edges. n Proof. By induction on i. The base case is i=0. Trivial. For the induction step, consider the shortest path from s to u with at most i edges. Let v be the last vertex before u on this path. Then the part of the path from s to v is the shortest path from s to v with at most i -1 edges. By the inductive hypothesis, d[v] after i-1 executions of the (*) "for" loop is at most the length of this path. Therefore, d[v] + w(u, v) is at most the length of the path from s to u, via v (as i-1 st node). In the i'th iteration, d[u] gets compared with d[v] + w(u, v), and is set equal to it if d[v] + w(u, v) is smaller. Therefore, after i iterations of the (*) "for" loop, d[u] is at most the length of the shortest path from s to u that uses at most i edges. So the lemma holds.

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 1 1 -2 9 4 9 -1 5 4

DAG-based Algorithm n Works even with negativen n 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)

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 5 5 -1 3 5 -5 2 7 0 1 3 6 4 1 -2 9 7 (two steps) 4 5 -1 5 4

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