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L19cs2110fa08-6up

L19cs2110fa08-6up - More Graphs Lecture 19 CS2110 Fall 2008...

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More Graphs Lecture 19 CS2110 – Fall 2008 2 3 2 4 3 1 2 3 4 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 1 2 3 4 1 2 3 4 Representations of Graphs Adjacency List Adjacency Matrix 1 2 3 4 Adjacency Matrix or Adjacency List? n = number of vertices m = number of edges d(u) = outdegree of u • Adjacency Matrix – Uses space O(n 2 ) – Can iterate over all edges in time O(n 2 ) – Can answer “Is there an edge from u to v?” in O(1) time – Better for dense graphs (lots of edges) • Adjacency List – Uses space O(m+n) – Can iterate over all edges in time O(m+n) – Can answer “Is there an edge from u to v?” in O(d(u)) time – Better for sparse graphs (fewer edges) Shortest Paths in Graphs • Finding the shortest (min-cost) path in a graph is a problem that occurs often –Find the shortest route between Ithaca and West Lafayette, IN –Result depends on our notion of cost Least mileage Least time • Cheapest Least boring –All of these “costs” can be represented as edge weights • How do we find a shortest path? Dijkstra’s Algorithm dijkstra(s) { D[s] = 0; D[t] = c(s,t), t ! s; mark s; while (some vertices are unmarked) { v = unmarked node with smallest D; mark v; for (each w adjacent to v) { D[w] = min(D[w], D[v] + c(v,w)); } } 2.4 1.5 1 2 3 4 2.4 0.9 1.5 3.1 0.1 X "
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2.4 1.5 1 2 3 4 2.4 0.9 1.5 3.1 0.1 X " 1.6 1.5 1 2 3 4 2.4 0.9 1.5 3.1 0.1 X 4.6 1.6 1.5 1 2 3 4 2.4 0.9 1.5 3.1 0.1 X 4.6 1.6 1.5 1 2 3 4 2.4 0.9 1.5 3.1 0.1 X 4.6 1.6 1.5 1 2 3 4 2.4 0.9 1.5 3.1 0.1 X 2.5 1.6 1.5 1 2 3 4 2.4 0.9 1.5 3.1 0.1 X 2.5
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1.6 1.5 1 2 3 4 2.4 0.9 1.5 3.1 0.1 X 2.5 The following are invariants of the loop: For u # X, D(u) = d(s,u) For u # X and v $ X, d(s,u) % d(s,v) For all u, D(u) is the length of the shortest path from s to u such that all nodes on the path (except possibly u) are in X Implementation: Use a priority queue for the nodes not yet taken – priority is D(u) Proof of Correctness Shortest Paths for Unweighted Graphs – A Special Case • Use breadth-first search • Time is O(n + m) in adj list representation, O(n 2 ) in adj matrix representation S B A C D E F Undirected Trees • An undirected graph is a tree if there is exactly one simple path between any pair of vertices Facts About Trees • |E| = |V| – 1 • connected • no cycles In fact, any two of these properties imply the third, and imply that the graph is a tree Spanning Trees A spanning tree of a connected undirected graph (V,E) is a subgraph (V,E') that is a tree
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Spanning Trees A spanning tree of a connected undirected graph (V,E) is a subgraph (V,E') that is a tree • Same set of vertices V • E' & E • (V,E') is a tree Finding a Spanning Tree A subtractive method If there is a cycle, pick an edge on the cycle, throw it out – the graph is still connected (why?) Repeat until no more cycles • Start with the whole graph – it is connected If there is a cycle, pick
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