Download presentation

Presentation is loading. Please wait.

Published byJerry Sculthorpe Modified over 3 years ago

1
CSE 2331/5331 CSE 780: Design and Analysis of Algorithms Lecture 14: Directed Graph BFS DFS Topological sort

2
CSE 2331/5331 ba f c de

3
Vertex Degree CSE 2331/5331 ba f c de

4
Representations of Graphs

5
CSE 2331/5331 Adjacency Lists For vertex v V, its adjacency list has size: outdeg(v) decide whether (v, u) E or not in time O(outdeg(v)) Size of data structure (space complexity): O(V+E)

6
CSE 2331/5331 Adjacency Matrix

7
CSE 2331/5331 Adjacency Matrix Size of data structure: O ( V V) Time to determine if (v, u) E : O(1) Though larger, it is simpler compared to adjacency list.

8
Sample Graph Algorithm Input: Directed graph G represented by adjacency list CSE 2331/5331 Running time: O(V + E)

9
Connectivity CSE 2331/5331

10
Reachability Test CSE 2331/5331

11
BFS and DFS The algorithms for BFS and DFS remain the same Each edge is now understood as a directed edge BFS(V,E, s) : visits all nodes reachable from s in non-decreasing order

12
CSE 2331/5331 BFS Starting from source node s, Spread a wavefront to visit other nodes First visit all nodes one edge away from s Then all nodes two edges away from s …

13
CSE 2331/5331 Pseudo-code Time complexity: O(V+E) Use adjacency list representation

14
Number of Reachable Nodes CSE 2331/5331 Time complexity: O(V+E)

15
DFS: Depth-First Search CSE 2331/5331

16
More Example CSE 2331/5331

17
DFS above can be replaced with BFS CSE 2331/5331 DFS(G, k);

18
Topological Sort CSE 2331/5331

19
Directed Acyclic Graph

20
CSE 2331/5331 Topological Sort A topological sort of a DAG G = (V, E) A linear ordering A of all vertices from V If edge (u,v) E => A[u] < A[v] undershorts pants belt shirt tie jacket shoes socks watch

21
Another Example CSE 2331/5331 Is the sorting order unique? Why requires DAG?

22
A topological sorted order of graph G exists if and only if G is a directed acyclic graph (DAG). CSE 2331/5331

23
Question How to topologically sort a given DAG? Intuition: Which node can be the first node in the topological sort order? A node with in-degree 0 ! After we remove this, the process can be repeated. CSE 2331/5331

24
Example CSE 2331/5331 undershorts pants belt shirt tie jacket shoes socks watch

25
CSE 2331/5331

26
Topological Sort – Simplified Implementation CSE 2331/5331

27
Time complexity O(V+E) Correctness: What if the algorithm terminates before we finish visiting all nodes? Procedure TopologicalSort(G) outputs a sorted list of all nodes if and only if the input graph G is a DAG If G is not DAG, the algorithm outputs only a partial list of vertices. CSE 2331/5331

28
Remarks Other topological sort algorithm by using properties of DFS CSE 2331/5331

Similar presentations

OK

Graphs. Types of Records One link – For stack and queue. Two links – For double ended queue. Many links:

Graphs. Types of Records One link – For stack and queue. Two links – For double ended queue. Many links:

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on descriptive vs inferential statistics Ppt on reaction intermediates Ppt on coalition government uk Ppt on pi in maths what is median Ppt on ms excel formulas Ppt on obesity management dog Ppt on online education system Ppt on conservation of minerals Free ppt on cost accounting Ppt on any scientific topic