Download presentation

Presentation is loading. Please wait.

Published byGrant Byrd Modified over 3 years ago

1
Graph Traversals Visit vertices of a graph G to determine some property: Is G connected? Is there a path from vertex a to vertex b? Does G have a cycle? Will removing a single edge disconnect G? If G is directed, what are the strongly connected components? If G is the WWW, follow links to locate information. Most graph algorithms need to examine or process each vertex and each edge.

2
Breadth-First Search Find the shortest path from a source node s to all other nodes. Outputs the distance of each vertex from s. a tree of shortest paths called the breadth-first tree. Idea: Find nodes at distance 0, then at distance 1, then at distance 2, …

3
A BFS Example s d b g f e c a L = s 0

4
s d b g f e a Visualized as many simultaneous explorations starting from s and spreading out independently. L = s 0 L = a, c 1 frontier of exploration c

5
s d b g f e c a L = s 0 L = a, c 1 L = d, e, f 2 Dashed edges were explored but had resulted in the discovery of no new vertices.

6
s d b g f e c a L = s 0 L = a, c 1 L = d, e, f 2 L = b, g 3

7
The Finish s d b g f e c a L = s 0 L = a, c 1 L = d, e, f 2 L = b, g 3

8
Breadth-First Tree s d b g f e c a 0 1 1 2 2 2 3 3 d(b): shortest distance from s to b # visited vertices = 1 + # tree edges 1 + # explored edges.

9
The BFS Algorithm BFS(G, s) for each v V(G) – s do d(v) // shortest distance from s d(s) 0 // initialization L s T i 0 while L ≠ do // all nodes at distance i from s L for each u L do for each v Adj(u) do if d(v) = then // not yet visited d(v) d(u) + 1 insert v into L T T {(u, v)} i i + 1 0 i i+1 i

10
Correctness Lemma For each i, the set L includes all nodes at distance i from s. i Proof By induction on the distance k from s. s … uv k Basis: L = s includes the only node at distance 0 from s. 0 Inductive step: Suppose L consists of all nodes at distance k ≥ 0 from s. k Corollary At the finish of BFS, d(v) is the length of the shortest path from s to v. k k+1 By hypothesis u L. Thus v L. Every node v at distance k+1 from s must be adjacent to a node u at distance k.

11
Running Time O(|E|) if G is represented using adjacency lists. # edge scans ≤ |E| for a directed graph ≤ 2 |E| for an undirected graph u v e u v e O(|V| ) if represented as an adjacency matrix. 2 Every vertex is inserted at most once into some list L. i # inserted vertices 1 + # scanned edges Courtesy of Dr. Fernandez-Baca |E| + 1.

Similar presentations

Presentation is loading. Please wait....

OK

Introduction to Algorithms

Introduction to Algorithms

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Download ppt on biogas power plant Ppt on balanced diet and nutrition Ppt on classical economics definition Ppt on sales closing techniques Ppt on negotiation skills Ppt on different types of dance forms software Ppt on intelligent manufacturing solutions Well made play ppt on apple Ppt on exploring number system Download ppt on first law of motion