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Shortest Paths and Dijkstra's Algorithm. CS 110: Data Structures and Algorithms First Semester, 2010-2011. Single-Source Shortest Paths. Given a weighted graph G and a source vertex v in G, determine the shortest paths from v to all other vertices in G Path length: sum of all edges in path
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Shortest PathsandDijkstra's Algorithm CS 110: Data Structures and Algorithms First Semester, 2010-2011
Single-Source Shortest Paths • Given a weighted graph G and a source vertex v in G, determine the shortest paths from v to all other vertices in G • Path length: sum of all edges in path • Useful in road map applications (e.g., Google maps or map quest) for example
Dijkstra’s Algorithm • Solves the single-source shortest path problem • Involves keeping a table of current shortest path lengths from a source vertex (initialize to infinity for all vertices except v, which has length 0)
Dijkstra’s Algorithm • Repeatedly select the vertex u with shortest path length, and update other lengths by considering the path that passes through that vertex • Stop when all vertices have been selected
Dijkstra’s Algorithm • Can make use of a priority queue to facilitate selection of shortest lengths • Need to refine the data structure to allow the updating of key values • Time complexity: O(n+m) or O(n2 log n) • O(n2 ) if computation of minimum is simplified
Dijkstra’s Algorithm 2704 BOS 867 849 1846 ORD PVD 187 740 621 802 144 JFK SFO 1258 1391 184 1464 BWI 1090 337 DFW 946 1235 LAX 1121 MIA 2342
Dijkstra’s Algorithm ∞ 2704 BOS 867 ∞ 849 1846 ORD PVD 187 740 ∞ 621 802 144 JFK ∞ ∞ SFO 1258 1391 0 184 1464 BWI 1090 337 DFW ∞ 946 1235 LAX 1121 ∞ MIA 2342 ∞
Dijkstra’s Algorithm ∞ 2704 BOS 867 621 849 1846 ORD PVD 187 740 ∞ 621 802 144 JFK ∞ 184 SFO 1258 1391 0 184 1464 BWI 1090 337 DFW ∞ 946 1235 LAX 1121 ∞ MIA 2342 946
Dijkstra’s Algorithm ∞ 2704 BOS 867 621 849 1846 ORD PVD 187 740 ∞ 621 802 144 JFK ∞ 184 SFO 1258 1391 0 184 1464 BWI 1090 337 DFW ∞ 946 1235 LAX 1121 ∞ MIA 2342 946
Dijkstra’s Algorithm 371 2704 BOS 867 621 849 1846 ORD PVD 187 740 328 621 802 144 JFK ∞ 184 SFO 1258 1391 0 184 1464 BWI 1090 337 DFW 1575 946 1235 LAX 1121 ∞ MIA 2342 946
Dijkstra’s Algorithm 371 2704 BOS 867 621 849 1846 ORD PVD 187 740 328 621 802 144 JFK ∞ 184 SFO 1258 1391 0 184 1464 BWI 1090 337 DFW 1575 946 1235 LAX 1121 ∞ MIA 2342 946
Dijkstra’s Algorithm 371 2704 BOS 867 621 849 1846 ORD PVD 187 740 328 621 802 144 JFK ∞ 184 SFO 1258 1391 0 184 1464 BWI 1090 337 DFW 1575 946 1235 LAX 1121 ∞ MIA 2342 946
Dijkstra’s Algorithm 371 2704 BOS 867 621 849 1846 ORD PVD 187 740 328 621 802 144 JFK ∞ 184 SFO 1258 1391 0 184 1464 BWI 1090 337 DFW 1575 946 1235 LAX 1121 ∞ MIA 2342 946
Dijkstra’s Algorithm 371 2704 BOS 867 621 849 1846 ORD PVD 187 740 328 621 802 144 JFK 3075 184 SFO 1258 1391 0 184 1464 BWI 1090 337 DFW 1575 946 1235 LAX 1121 ∞ MIA 2342 946
Dijkstra’s Algorithm 371 2704 BOS 867 621 849 1846 ORD PVD 187 740 328 621 802 144 JFK 3075 184 SFO 1258 1391 0 184 1464 BWI 1090 337 DFW 1575 946 1235 LAX 1121 ∞ MIA 2342 946
Dijkstra’s Algorithm 371 2704 BOS 867 621 849 1846 ORD PVD 187 740 328 621 802 144 JFK 2467 184 SFO 1258 1391 0 184 1464 BWI 1090 337 DFW 1423 946 1235 LAX 1121 ∞ MIA 2342 946
Dijkstra’s Algorithm 371 2704 BOS 867 621 849 1846 ORD PVD 187 740 328 621 802 144 JFK 2467 184 SFO 1258 1391 0 184 1464 BWI 1090 337 DFW 1423 946 1235 LAX 1121 ∞ MIA 2342 946
Dijkstra’s Algorithm 371 2704 BOS 867 621 849 1846 ORD PVD 187 740 328 621 802 144 JFK 2467 184 SFO 1258 1391 0 184 1464 BWI 1090 337 DFW 1423 946 1235 LAX 1121 3288 MIA 2342 946
Dijkstra’s Algorithm 371 2704 BOS 867 621 849 1846 ORD PVD 187 740 328 621 802 144 JFK 2467 184 SFO 1258 1391 0 184 1464 BWI 1090 337 DFW 1423 946 1235 LAX 1121 3288 MIA 2342 946
Dijkstra’s Algorithm 371 2704 BOS 867 621 849 1846 ORD PVD 187 740 328 621 802 144 JFK 2467 184 SFO 1258 1391 0 184 1464 BWI 1090 337 DFW 1423 946 1235 LAX 1121 3288 MIA 2342 946
Dijkstra’s Algorithm 371 2704 BOS 867 621 849 1846 ORD PVD 187 740 328 621 802 144 JFK 2467 184 SFO 1258 1391 0 184 1464 BWI 1090 337 DFW 1423 946 1235 LAX 1121 2658 MIA 2342 946
Pseudo-Code: Dijkstra function Dijkstra( Graph g, Vertex source ) for each vertex v in g dist[v] <-- infinity previous[v] <-- undefined dist[source] <-- 0 Q <-- the set of all nodes in Graph while Q is not empty u <-- vertex in Q with smallest dist[] if dist[u] == infinity break remove u from Q for each neighbor v of u alt = dist[u] + dist_between( u, v ) if alt < dist[v] dist[v] = alt previous[v] = u return dist[]
