23 DECOMPOSITION OF GRAPHS

Transcription

1 DATA STRUCTURES AND ALGORITHMS 23 DECOMPOSITION OF GRAPHS REPRESENTING & EXPLORING GRAPHS IMRAN IHSAN ASSISTANT PROFESSOR, AIR UNIVERSITY, ISLAMABAD LECTURES ADAPTED FROM: DANIEL KANE, NEIL RHODES DEPARTMENT OF CS & ENGINEERING UNIVERSITY OF CALIFORNIA, SAN DIEGO

2 GRAPHS Represents connections between objects. Describe many important phenomena. 2

3 INTERNET Webpages connected by links. This is important for Google's page rank. 3

4 MAPS Intersections connected by roads. 4

5 SOCIAL NETWORKS People connected by friendships. 5

6 CONFIGURATION SPACES Possible configurations connected by motions. 6

7 FORMAL DEFINITION Definition An (undirected) Graph is a collection V of vertices, and a collection E of edges each of which connects a pair of vertices. 7

8 DRAWING GRAPHS Vertices: Points. Edges: Lines. Vertices: A,B,C,D Edges: (A, B), (A, C), (A,D), (C,D) 8

9 PROBLEM How many edges are in the graph given below? 9

10 PROBLEM 12 10

11 LOOPS AND MULTIPLE EDGES Loops connect a vertex to itself. Multiple edges between same vertices. If a graph has neither, it is simple. 11

12 GRAPH REPRESENTATIONS 12

13 REPRESENTING GRAPHS To compute things about graphs we first need to represent them. There are many ways to do this. 13

14 EDGE LIST List of all edges: Edges: (A, B), (A, C), (A,D), (C,D) 14

15 ADJACENCY MATRIX Matrix. Entries 1 if there is an edge, 0 if there is not. A B C D A B C D

16 ADJACENCY LIST For each vertex, a list of adjacent vertices. A adjacent to B,C,D B adjacent to A C adjacent to A,D D adjacent to A,C 16

17 PROBLEM What are the neighbors of C? 17

18 PROBLEM A,B,D,F,H,I 18

19 EXPLORING GRAPHS 19

20 MOTIVATION You're playing a video game and want to make sure that you've found everything in a level before moving on. How do you ensure that you accomplish this? 20

21 EXAMPLES This notion of exploring a graph has many applications: Finding routes Ensuring connectivity Solving puzzles and mazes 21

22 PATHS We want to know what is reachable f rom a given vertex. Definition A path in a graph G is a sequence of vertices v 0, v 1,..., v n so that for all i, (v i, v i+1 ) is an edge of G. 22

23 FORMAL DESCRIPTION Reachability Input: Output: Graph G and vertex s The collection of vertices v of G so that there is a path from s to v. 23

24 PROBLEM Which vertices are reachable from A? 24

25 SOLUTION A, C, D, F, H, I 25

26 BASIC IDEA We want to make sure that we have explored every edge leaving every vertex we have found. 26

27 PSEUDOCODE Components (s) DiscoveredNodes {s} while there is an edge e leaving DiscoveredNodes that has not been explored: add vertex at other end of e to DiscoveredNodes return DiscoveredNodes 27

28 FORMAL SPECIFICATION We need to do some work to handle the bookkeeping for this algorithm. How do we keep track of which edges/vertices we have dealt with? What order do we explore new edges in? 28

29 VISIT MARKERS To keep track of vertices found: Give each vertex boolean visited(v). 29

30 UNPROCESSED VERTICES Keep a list of vertices with edges left to check. This will end up getting hidden in the program stack. 30

31 DEPTH FIRST ORDERING We will explore new edges in Depth First order. We will follow a long path forward, only backtracking when we hit a dead end. 31

32 ALGORITHM Explore(v) visited(v) true for (v, w) E: if not visited(w): Explore(w) //Need adjacency list representation! 32

33 EXAMPLE 33

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49 RESULT Theorem If all vertices start unvisited, Explore(v) marks as visited exactly the vertices reachable from v. Proof Only explores things reachable from v. w not marked as visited unless explored. If w explored, all neighbors explored. 49

50 ALGORITHM DFS(G) for all v V : mark v unvisited for v V : if not visited(v): Explore(v) //Need adjacency list representation! 50

51 EXAMPLE 51

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67 RUNTIME Number of calls to explore: Each explored vertex is marked visited. No vertex is explored after visited once. Each vertex is explored exactly once. Checking for neighbors: Each vertex checks each neighbor. Total number of neighbors over all vertices is O( E ). Total runtime: O(1) work per vertex. O(1) work per edge. Total O( V + E ). 67