Minimum Spanning Trees and Shortest Paths

Transcription

1 Minimum Spanning Trees and Shortest Paths Kruskal's lgorithm Prim's lgorithm Shortest Paths pril 04, 018 inda eeren / eoffrey Tien 1

2 Kruskal's algorithm ata types for implementation Kruskalslgorithm() { set = ø while ( V 1) { ind minimum weight edge e such that dd e to } } e does not contain cycles We need Ts that support our required operations efficiently ow do we find the minimum weight edge? Priority queue! ow can we check for cycles and perform union? isjoint sets! pril 04, 018 inda eeren / eoffrey Tien

3 Kruskal's algorithm xample MST weight: prq,, 3, 4, 5, 10, 11, 1, 13, 6, 13, 7, 16 prq cost heap ordered array build removemin Note that no insertions performed after build, 8, 16 Overall cost?, 9, 17 Note: only one edge direction listed in prq for compactness in this slide pril 04, 018 inda eeren / eoffrey Tien 3

4 slight tangent maze construction What makes a good maze? bunch of adjacent rooms ach room is a vertex Open wall between rooms form edge in Unpredictable, not easily solved ighly branching, many dead ends Just enough walls to get from any room to any other room specially start and finish out pril 04, 018 inda eeren / eoffrey Tien 4

5 Maze under construction So far, a number of walls have been knocked down, while others remain Now we consider the wall between rooms and Should we knock it down? If and are otherwise connected If and are not otherwise connected lgorithm: While edges remain in Remove a random edge e = u, v from If u and v have not been connected dd e to Mark u and v as connected This is a lot like Kruskal's algorithm! Solve it using disjoint sets and random edge selection pril 04, 018 inda eeren / eoffrey Tien 5

6 Recall: S spanning tree Starting from vertex (,), (,6), (,16) (,), (,3), (,7), (,13) (,6), (,3), (,8), (,17), (,13) (,16), (,8), (,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? pril 04, 018 inda eeren / eoffrey Tien 6

7 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 } } pril 04, 018 inda eeren / eoffrey Tien 7

8 Prim's algorithm (,), (,6), (,16) (,), (,3), (,7), (,13) (,6), (,3), (,8), (,17), (,13) (,16), (,8), (,1), (,9) (,7), (,17), (,5), (,16) (,13), (,1), (,5), (,10), (,11) (,9), (,10), (,4) (,16), (,11), (,4) prq: (,,) (,,6) (,,16) (,,3) (,,7) (,,13) (,,8) (,,17) (,,13) (,,5) (,,16) (,,1) (,,10) (,,11) (,,9) (,,4) Visited: T T T T T T T T ll vertices visited MST weight: 38 pril 04, 018 inda eeren / eoffrey Tien 8

9 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 pril 04, 018 inda eeren / eoffrey Tien 9

10 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 pril 04, 018 inda eeren / eoffrey Tien 10

11 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 - (8) pril 04, 018 inda eeren / eoffrey Tien 11

12 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 pril 04, 018 inda eeren / eoffrey Tien 1

13 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 pril 04, 018 inda eeren / eoffrey Tien 13

14 ijkstra's lgorithm xample: directed graph Vertex ost Parent Priority queue: Source node: Visited ost Parent Note that we need access to arbitrary elements in this prq in order to update pril 04, 018 inda eeren / eoffrey Tien 14

15 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 8 K 7 I pril 04, 018 inda eeren / eoffrey Tien 15

16 Readings for this lesson arrano & enry hapter (Minimum spanning trees) hapter (Shortest paths) pril 04, 018 inda eeren / eoffrey Tien 16