ECE250: Algorithms and Data Structures Elementary Graph Algorithms Part A

Transcription

1 ECE250: Algorithms and Data Structures Elementary Graph Algorithms Part A Ladan Tahvildari, PEng, SMIEEE Associate Professor Software Technologies Applied Research (STAR) Group Dept. of Elect. & Comp. Eng. University of Waterloo Materials from CLRS: Chapter 22.1 and B.4

2 Acknowledgements v The following resources have been used to prepare materials for this course: Ø MIT OpenCourseWare Ø Introduction To Algorithms (CLRS Book) Ø Data Structures and Algorithm Analysis in C++ (M. Wiess) Ø Data Structures and Algorithms in C++ (M. Goodrich) v Thanks to many people for pointing out mistakes, providing suggestions, or helping to improve the quality of this course over the last ten years: Ø ECE250 2

3 Graphs Definition v A graph G = (V,E) is composed of: Ø V: set of vertices Ø E V V: set of edges connecting the vertices v An edge e = (u,v) is a pair of vertices v We assume directed graphs. Ø If a graph is undirected, we represent an edge between u and v by having (u,v) E and (v,u) E A B A B V = {A, B, C, D} C D C D E = {(A,B), (B,A), (A,C), (C,A), (C,D), (D,C), (B,C), (C,B)} ECE250 3

4 Applications v Electronic circuits, pipeline networks v Transportation and communication networks v Modeling any sort of relationtionships (between components, people, processes, concepts) ECE250 4

5 Graph Terminology v Vertex v is adjacent to vertex u iff (u,v) E v degree (of a vertex): # of adjacent vertices v Path a sequence of vertices v 1,v 2,...v k such that v i+1 is adjacent to v i for i = 1.. k 1 ECE250 5

6 Graph Terminology (cont ) v Simple path a path with no repeated vertices v Cycle a simple path, except that the last vertex is the same as the first vertex v Connected graph: any two vertices are connected by some path ECE250 6

7 Graph Terminology (cont ) v Subgraph a subset of vertices and edges forming a graph v Connected component maximal connected subgraph Ø For example, the graph below has 3 connected components Every vertex is reachable from every other vertex ECE250 7

8 Graph Terminology (cont ) v (free) tree connected graph without cycles v forest collection of trees ECE250 8

9 Data Structures for Graphs v The Adjacency list of a vertex v: a sequence of vertices adjacent to v v Represent the graph by the adjacency lists of all its vertices Space =Θ ( V + deg( v)) =Θ ( V + E) ECE250 9

10 Adjacency Matrix v Matrix M with entries for all pairs of vertices v M[i,j] = true there is an edge (i,j) in the graph v M[i,j] = false there is no edge (i,j) in the graph v Space = O( V 2 ) Values of 1 or 0 too. 1 implies connected. ECE250 10

11 Pseudocode Assumptions v Graph ADT with an operation Ø V():VertexSet v A looping construct for each v V, where V is of a type VertexSet, and v is of a type Vertex v Vertex ADT with operations: Ø adjacent():vertexset Ø d():int and setd(d:int) Ø f():int and setf(f:int) Ø parent():vertex and setparent(p:vertex) Ø color():{white, gray, black} and setcolor(c:{white, gray, black}) ECE250 11

12 Graph Searching Algorithms v Systematic search of every edge and vertex of the graph v Graph G = (V,E) is either directed or undirected v Applications Ø Compilers Ø Graphics Ø Maze-solving Ø Mapping Ø Networks: routing, searching, clustering, etc. ECE250 12

13 Breadth First Search v A Breadth-First Search (BFS) traverses a connected component of a graph, and in doing so defines a spanning tree with several useful properties v The starting vertex s, it is assigned a distance 0. v In the first round, the string is unrolled the length of one edge, and all of the edges that are only one edge away from the anchor are visited (discovered), and assigned distances of 1 ECE250 13

14 Breadth-First Search (cont ) v In the second round, all the new edges that can be reached by unrolling the string 2 edges are visited and assigned a distance of 2 v This continues until every vertex has been assigned a level v The label of any vertex v corresponds to the length of the shortest path (in terms of edges) from s to v ECE250 14

15 BFS Algorithm BFS(G,s) 01 for each vertex u G.V() 02 u.setcolor(white) 03 u.setd( ) 04 u.setparent(nil) 05 s.setcolor(gray) 06 s.setd(0) 07 Q.init() 08 Q.enqueue(s) 09 while not Q.isEmpty() 10 u Q.head() 11 for each v u.adjacent() do 12 if v.color() = white then 13 v.setcolor(gray) 14 v.setd(u.d() + 1) 15 v.setparent(u) 16 Q.enqueue(v) 17 Q.dequeue() 18 u.setcolor(black) Init all vertices Init BFS with s Handle all u s children before handling any children of children ECE250 15

16 Coloring of vertices v A vertex is white if it is undiscovered v A vertex is gray if it has been discovered but not all of its edges have been explored v A vertex is black after all of its adjacent vertices have been discovered (the adj. list was examined completely) v Let s do an example of BFS: A S B D G C E F ECE250 16

17 BFS Running Time v Given a graph G = (V,E) Ø Vertices are enqueued if their color is white Ø Assuming that en- and dequeuing takes O(1) time the total cost of this operation is O(V) Ø Adjacency list of a vertex is scanned when the vertex is dequeued (and only then ) Ø The sum of the lengths of all lists is Θ(E). Consequently, O(E) time is spent on scanning them Ø Initializing the algorithm takes O(V) v Total running time O(V+E) (linear in the size of the adjacency list representation of G) ECE250 17

18 BFS Properties v Given a graph G = (V,E), BFS discovers all vertices reachable from a source vertex s v It computes the shortest distance to all reachable vertices v It computes a breadth-first tree that contains all such reachable vertices v For any vertex v reachable from s, the path in the breadth first tree from s to v, corresponds to a shortest path in G ECE250 18