Single Source Shortest Paths Definition Algorithms Bellman Ford DAG shortest path algorithm Dijkstra

Definition Input Task Weighted, connected directed graph G=(V,E) Weight (length) function w on each edge e in E Source node s in V Task Compute a shortest path from s to all nodes in V

Comments If edges are not weighted, then BFS works Optimal substructure A shortest path between s and t contains other shortest paths within it No known algorithm is better at finding a shortest path from s to a specific destination node t in G than finding the shortest path from s to all nodes in V

Negative weight edges Negative weight edges can be allowed as long as there are no negative weight cycles If there are negative weight cycles, then there cannot be a shortest path from s to any node t (why?) If we disallow negative weight cycles, then there always is a shortest path that contains no cycles

Relaxation technique For each vertex v, we maintain an upper bound d[v] on the weight of shortest path from s to v d[v] initialized to infinity Relaxing an edge (u,v) Can we improve the shortest path to v by going through u? If d[v] > d[u] + w(u,v), d[v] = d[u] + w(u,v) This can be done in O(1) time

Bellman-Ford Algorithm Bellman-Ford (G, w, s) Initialize-Single-Source(G,s) for (i=1 to V-1) for each edge (u,v) in E relax(u,v); if d[v] > d[u] + w(u,v) return NEGATIVE WEIGHT CYCLE

Running Time for (i=1 to V-1) The above takes (V-1)O(E) = O(VE) time for each edge (u,v) in E relax(u,v); The above takes (V-1)O(E) = O(VE) time if d[v] > d[u] + w(u,v) return NEGATIVE WEIGHT CYCLE The above takes O(E) time

Proof of Correctness If there is a shortest path from s to any node v, then d[v] will have this weight at end Let p = (e1, e2, …, ek) = (v1, v2, v3, …, v,+1) be a shortest path from s to v s = v1, v = vk+1, ei = (vi, vi+1) At beginning of ith iteration, during which we relax edge ei, d[vi] must be correct, so at end of ith iteration, d[vi+1] will e correct Iteration is the full sweep of relaxing all edges Proof by induction

Negative weight cycle for each edge (u,v) in E if d[v] > d[u] + w(u,v) return NEGATIVE WEIGHT CYCLE If no negative cycle, d[v] <= d[u] + w(u,v) for all edges (u,v) If there is a negative weight cycle, for some edge (u,v) on the cycle, it must be the case that d[v] > d[u] + (u,v).

DAG shortest path algorithm DAG-SP (G, w, s) Initialize-Single-Source(G,s) Topologically sort vertices in G for each vertex u, taken in sorted order for each edge (u,v) in E relax(u,v);

Running time Improvement for each vertex u, taken in sorted order for each edge (u,v) in E relax(u,v); We only do 1 relaxation for each edge: O(E) time In addition, O(V+E) for the sorting and we get O(V+E) running time overall

Proof of Correctness If there is a shortest path from s to any node v, then d[v] will have this weight at end Let p = (e1, e2, …, ek) = (v1, v2, v3, …, v,+1) be a shortest path from s to v s = v1, v = vk+1, ei = (vi, vi+1) Since we sort edges in topological order, we will process node vi (and edge ei) before processing later edges in the path.

Dijkstra’s Algorithm Dijkstra (G, w, s) /* Assumption: all edge weights non-negative */ Initialize-Single-Source(G,s) Completed = {}; ToBeCompleted = V; While ToBeCompleted is not empty u =EXTRACT-MIN(ToBeCompleted); Completed += {u}; for each edge (u,v) relax(u,v);

Running Time Analysis While ToBeCompleted is not empty u =EXTRACT-MIN(ToBeCompleted); Completed += {u}; for each edge (u,v) relax(u,v); Each edge relaxed once: O(E) Need to decrease-key potentially once per edge Need to extract-min once per node

Running Time Analysis cont’d O(E) edge relaxations Priority Queue operations O(E) decrease key operations O(V) extract-min operations Three implementations of priority queues Array: O(V2) time decrease-key is O(1) and extract-min is O(V) Binary heap: O(E log V) time assuming E >= V decrease-key and extract-min are O(log V) Fibonacci heap: O(V log V + E) time decrease-key is O(1) amortized time and extract-min is O(log V)

Proof of Correctness s u v Only nodes in S Assume that Dijkstra’s algorithm fails to compute length of all shortest paths from s Let v be the first node whose shortest path length is computed incorrectly Let S be the set of nodes whose shortest paths were computed correctly by Dijkstra prior to adding v to the processed set of nodes. Dijkstra’s algorithm has used the shortest path from s to v using only nodes in S when it added v to S. The shortest path to v must include one node not in S Let u be the first such node. s u v Only nodes in S

Proof of Correctness The length of the shortest path to u must be at least that of the length of the path computed to v. Why? The length of the path from u to v must be < 0. No path can have negative length since all edge weights are non-negative, and thus we have a contradiction. s u v Only nodes in S

Computing paths (not just distance) Maintain for each node v a predecessor node p(v) p(v) is initialized to be null Whenever an edge (u,v) is relaxed such that d(v) improves, then p(v) can be set to be u Paths can be generated from this data structure