Download presentation

Presentation is loading. Please wait.

Published byJimmy Paskell Modified over 3 years ago

1
Directed Graphs 3/6/121

2
Normal Person’s Graph x y y = f(x) 3/6/122

3
Computer Scientist’s Graph a f e c d b 3/6/123

4
Digraphs a set, V, of vertices aka “nodes” a set, E ⊆ V×V of directed edges (v,w) ∈ E v w notation: v → w 3/6/124

5
Relations and Graphs a c b d V= {a,b,c,d} E = {(a,b), (a,c), (c,b)} 3/6/125

6
Formally, a digraph with vertices V is the same as a binary relation on V. Digraphs 3/6/126

7
Walks & Paths Walk: follow successive edges length: 5 edges (not the 6 vertices) 3/6/127

8
Walks & Paths Path: walk thru vertices without repeat vertex length: 4 edges 3/6/128

9
Lemma: The shortest walk between two vertices is a path! Proof: (by contradiction ) suppose path from u to v crossed itself: u v c Walks & Paths 3/6/129

10
Proof: (by contradiction ) suppose path from u to v crossed itself: u v c then path without c---c is shorter! Lemma: The shortest walk between two vertices is a path! Walks & Paths 3/6/1210

11
Walks & Paths Digraph G defines walk relation G + u G + v iff ∃ walk u to v (the positive walk relation) “+” means 1 or more 3/6/1211

12
Walks & Paths Digraph G defines walk relation G * u G * v iff u to v (the walk relation) “*” means “0 or more” 3/6/1212

13
Cycles A cycle is a walk whose only repeat vertex is its start & end. (a single vertex is a length 0 cycle) 3/6/1213

14
… v0v0 v1v1 v2v2 v n-1 v0v0 v0v0 vivi v i+1 Cycles 3/6/1214

15
Closed walk starts & ends at the same vertex. Lemma: The shortest positive length closed walk containing a vertex is a positive length cycle! Proof: similar Closed Walks & Cycles 3/6/1215

16
has no positive length cycle Directed Acyclic Graph DAG lec 7M.16 3/6/1216

17
examples: < relation on integers ⊊ relation on sets prerequisite on classes Directed Acyclic Graph DAG 3/6/1217

18
Example: Tournament Graph Every team plays every other 3/6/1218 H Y P D H Y P D DAG => Unique ranking

Similar presentations

OK

Lecture 16: DFS, DAG, and Strongly Connected Components Shang-Hua Teng.

Lecture 16: DFS, DAG, and Strongly Connected Components Shang-Hua Teng.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on high voltage engineering netherlands Ppt on indian railway services Ppt on principles of object-oriented programming concepts Download ppt on new invention of computer technology Ppt on the road not taken analysis Ppt on bluetooth based smart sensor networks definition Ppt on limits and continuity ppt Ppt on fdi in retail sector Ppt on buddhist architecture in india Ppt on success of democracy in today's india