Minimum Spanning Trees and Shortest Paths

Transcription

1 Minimum Spanning Trees and Shortest Paths Prim's algorithm ijkstra's algorithm November, 017 inda eeren / eoffrey Tien 1

2 Recall: S spanning tree Starting from vertex (,), (,6), (,16) (,), (,3), (,7), (,13) (,6), (,3), (,), (,17), (,13) (,16), (,), (,1), (,9) (,7), (,17), (,5), (,16) (,13), (,1), (,5), (,10), (,11) (,9), (,10), (,4) (,16), (,11), (,4) Queue: Identified: T T T T T T T T What if we use a priority queue (with neighbours' edge weights) instead of an ordinary queue? November, 017 inda eeren / eoffrey Tien

3 Prim's algorithm reedy algorithm uilds a spanning tree from initially one vertex. Repeatedly chooses the minimum-weight edge from a vertex in the tree, to a vertex outside the tree adds that vertex to the tree Primslgorithm(v) { mark v as visited, add v to spanning tree while (graph has unvisited vertices) { ind least cost edge (w, u) from a visited vertex w to unvisited vertex u Mark u as visited dd vertex u and edge (w, u) to the minimum spanning tree } } November, 017 inda eeren / eoffrey Tien 3

4 Prim's algorithm (,), (,6), (,16) (,), (,3), (,7), (,13) (,6), (,3), (,), (,17), (,13) (,16), (,), (,1), (,9) (,7), (,17), (,5), (,16) (,13), (,1), (,5), (,10), (,11) (,9), (,10), (,4) (,16), (,11), (,4) prq: (,,) (,,6) (,,16) (,,3) (,,7) (,,13) (,,) (,,17) (,,13) (,,5) (,,16) (,,1) (,,10) (,,11) (,,9) (,,4) Visited: T T T T T T T T ll vertices visited MST weight: 3 November, 017 inda eeren / eoffrey Tien 4

5 Prim's algorithm omplexity Unlike Kruskal's algorithm, we will intersperse insertion and removal operations to the priority queue Maximum number of insertions into the priority queue? ssuming heap implementation of prq in the worst case, then total cost of all insertions O log or dense graphs, O V, then log O log V O log V Thus the complexity of Prim's algorithm is O log V ctually we can describe Kruskal's algorithm the same way November, 017 inda eeren / eoffrey Tien 5

6 Single-source shortest path iven a graph = V, and a vertex s V, find the shortest path from s to every vertex in V Variations Weighted vs unweighted yclic vs acyclic Positive weights only vs negative weights allowed Multiple weight types to optimize November, 017 inda eeren / eoffrey Tien 6

7 Single-Source Shortest Path Un/directed, weighted graphs, no negative cycles What is the least cost path from one vertex to another? or weighted graphs, this is the path that has the smallest sum of its edge weights The shortest path between and is: ---- (7) and not - () November, 017 inda eeren / eoffrey Tien 7

8 ijkstra's lgorithm lassic algorithm for solving shortest path in weighted graphs without negative weights greedy algorithm est local choice is made at each step, without considering future consequences Intuition: Shortest path from source vertex to itself is 0 ost of going to adjacent nodes is at most edge weights heapest of these must be shortest path to that node Update paths for new node and continue picking shortest path November, 017 inda eeren / eoffrey Tien

9 ijkstra's lgorithm Initialize the cost of reaching each vertex to Initialize the cost of the source to 0 While there are unvisited vertices left in the graph Select the unvisited vertex with the lowest cost: u Mark u as visited, and note the vertex v which was used to reach u or each vertex w which is adjacent to u w's cost = min(w's old cost, u's cost + cost of (u, w)) nd note the "parent" vertex which can be used to reach w with the lowest cost November, 017 inda eeren / eoffrey Tien 9

10 ijkstra's lgorithm November, 017 inda eeren / eoffrey Tien 10 xample: directed graph Visited ost Parent Vertex ost Parent Source node:

11 xercise iven the following undirected, weighted graph, use Kruskal's algorithm and Prim's algorithm to find a minimum spanning tree and write its weight. Is the tree you found the unique MST for this graph? J K 7 I November, 017 inda eeren / eoffrey Tien 11

12 Readings for this lesson Koffman hapter 1.6 (Minimum spanning trees) hapter 1.6 (Shortest path) Next class: Koffman hapter 1.6 (Shortest path) November, 017 inda eeren / eoffrey Tien 1