Representations of Graphs

Transcription

1 ELEMENTARY GRAPH ALGORITHMS -- CS-5321 Presentation -- I am Nishit Kapadia Representations of Graphs There are two standard ways: A collection of adjacency lists - they provide a compact way to represent sparse graphs, when E << V 2 An adjacency matrix - may be preferred when the graph is dense or to quickly find out whether there is an edge connecting two given vertices -- If G is a directed (undirected) graph, the sum of the lengths of all the adjacency lists is E (2 E ) -- The adjacency-list representation requires Θ(V+E) memory, which is desirable -- The adjacency matrix representation consists of a V x V matrix : a ij = 1, if (i,j) є E a ij = 0, otherwise It requires a Θ (V 2 ) memory independent of the number of edges in the graph

2 Breath First Search BFS is one of the simplest algorithms for searching a graph (directed or undirected) 1. It systematically explores the edges of G to discover every vertex that is reachable from s (the source vertex) 2. It computes the distance (smallest number of edges) from s to each reachable vertex 3. It also produces a breath-first tree with root s that contains all reachable vertices 4. It is named so because it discovers all vertices at distance k from s before discovering any vertices at distance k+1 -- All vertices start out white and may later become gray and then black. Gray & black vertices, therefore, have been discovered. -- If (u,v) є E and vertex u is black, then vertex v is either gray or black. Thus, all vertices adjacent to black vertices have been discovered. -- Whenever a white vertex v is discovered while scanning the adjacency list of an already discovered vertex u, the vertex v and the edge (u,v) are added to the tree. So, u is called the predecessor or parent of v in the BFS -- Since a vertex is discovered at most once, it has at most one parent BFS(G,s) for each vertex u є V[G] {s} do color[u] WHITE d[u] π[u] NIL color[s] GRAY d[s] 0 π[s] NIL Q Ø ENQUEUE (Q,s) //fields are cleared //the root is visited while Q Ø do u DEQUEUE (Q) for each v є Adj[u] do if color[v] = WHITE then color[v] GRAY d[v] d[u] + 1 π[v] u ENQUEUE (Q,v) color[u] BLACK //queue will contain only gray nodes //black vertex is dequeued

3 Analysis of BFS After initialization, no vertex is ever whitened, which ensures that each vertex is enqueued at most once, and dequeued at most once. Enqueuing and dequeuing take O(1) time, so the total time devoted to queue operations is O(V). Because the adjacency list of each vertex is scanned only when the vertex is dequeued, each adjacency list is scanned at most once; which requires O(E) time. The overhead of initialization is O(V), and thus the total running time of BFS is O(V+E). The BFS computes the shortest path distances, denoted by (s,v) minimum distance from s to v ( (s,v) = when no path). Depth-first search DFS searches deeper in the graph whenever possible. Edges are explored out of the most recently discovered vertex v that still has unexplored edges leaving it. When all of the v s edges have been explored, the search backtracks to explore edges leaving the vertex from which v was discovered. The predecessor subgraph may be composed of several trees, because the search may be repeated from multiple sources. It forms a Depth-first forest. DFS also timestamps each vertex with d[v] and f[v]. These timestamps are integers between 1 and 2 V for the V vertices.

4 DFS(G) for each vertex u є V[G] do color[u] WHITE π [u] NIL time 0 for each vertex u є V[G] do if color[u] = WHITE then DFS-VISIT(u) //clear fields //vertex u becomes the root of a new tree // DFS returns with d[u] & f[u] for every vertex u DFS-VISIT(u) color[u] GRAY time time + 1 d[u] time for each v є Adj[u] do if color[v] = WHITE then π [v] u DFS-VISIT(v) color[u] BLACK f[u] time time +1 //coloring the current root // recursive call Running Time of DFS? The upper and the lower (exclusive of the calls to DFS-VISIT) for loop takes Θ(V). The procedure DFS-VISIT is called exactly once for each vertex. The for loop in DFS-VISIT is executed Adj[v] times, since Adj[v] = Θ(E) Therefore, the total time cost is Θ(V + E) Properties of DFS The predecessor subgraph forms a forest of trees. Vertex v is a descendant of vertex u in the depth-first forest iff v is discovered during the time in which u is gray. The discovery and finishing times have a parenthesis structure. Vertex v is a descendant of vertex u in the depth-first forest for a (directed or undirected) graph G iff d[u] < d[v] < f[v] < f[u].

5 Classification of edges in DFS It is useful to glean important information about the graph Tree edges are the edges in the depth-first forest Gπ. Non-Tree edges are: Back edges : from v to its ancestor Forward edges : from v to its descendant Cross edges : they go between unrelated (parentally) vertices in the same depth-first tree or between different depth-first trees. Topological Sort The DFS can be used to perform a Topological Sort on a directed acyclic graph. It can be viewed as an ordering of its vertices along a horizontal line, so that all directed edges go from left to right. Application: To indicate precedences among events. Algorithm: TOPOLOGICAL SORT 1. Call DFS to compute finishing times for each vertex 2. As each vertex is finished, insert it onto the front of a linked list 3. Return the link list of vertices Complexity: Θ(V+E) for DFS + O(1) per vertex to insert to L.L.

6 Strongly Connected Components An application of Depth-first Search: Decomposing a directed graph into its strongly connected components. SCC(G) 1. Call DFS(G) to compute finishing times f[u] for each vertex u 2. Compute G T 3. Call DFS(G T ), but in the main loop of DFS, consider the vertices in order of decreasing f[u] 4. Output the vertices of each tree in the depth-first forest formed in line 3 as a separate strongly connected component

7 C and C are distinct SSCs edge (u,v) є E; u є C & v є C. Then, f(c) > f(c ) If the edge (u,v) є E T. Then, f(c) < f(c ) There are no edges in G T from C to any other strongly connected component, and so the search from x will not visit vertices from any other component. Thus, the tree rooted at x contains exactly the vertices of C. In the end, each depth-first tree will be one strongly connected component.