Graph Traversals. Slides by Carl Kingsford. Feb. 1, Based on/reading: Chapter 3 of Kleinberg & Tardos
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1 Graph Traversals Slides by Carl Kingsford Feb. 1, 2013 Based on/reading: Chapter 3 of Kleinberg & Tardos
2 Breadth-First Search Breadth-first search explores the nodes of a graph in increasing distance away from some starting vertex s. It decomposes the component into layers L i such that the shortest path from s to each of nodes in L i is of length i. Breadth-First Search: 1. L 0 is the set {s}. 2. Given layers L 0, L 1,..., L j, then L j+1 is the set of nodes that are not in a previous layer and that have an edge to some node in layer L j.
3 BFS Tree s
4 BFS Tree Example A BFS traversal of a graph results in a breadth-first search tree: s
5 BFS Tree Example A BFS traversal of a graph results in a breadth-first search tree: s Can we say anything about the non-tree edges?
6 Example BFS 8:-1 3:-1 12:-1 6:-1 14:-1 9:-1 2:-1 7:-1 1:-1 4:-1 0:0 5:-1 10:-1 13:-1 11:-1
7 Example BFS 8:-1 3:-1 12:-1 6:-1 14:-1 9:-1 2:-1 7:1 1:1 4:-1 0:0 5:1 10:1 13:1 11:1
8 Example BFS 8:-1 3:2 12:-1 6:-1 14:-1 9:-1 2:2 7:1 1:1 4:-1 0:0 5:1 10:1 13:1 11:1
9 Example BFS 8:-1 3:2 12:-1 6:-1 14:-1 9:-1 2:2 7:1 1:1 4:-1 0:0 5:1 10:1 13:1 11:1
10 Example BFS 8:-1 3:2 12:-1 6:2 14:-1 9:2 2:2 7:1 1:1 4:-1 0:0 5:1 10:1 13:1 11:1
11 Example BFS 8:-1 3:2 12:-1 6:2 14:2 9:2 2:2 7:1 1:1 4:2 0:0 5:1 10:1 13:1 11:1
12 Example BFS 8:-1 3:2 12:-1 6:2 14:2 9:2 2:2 7:1 1:1 4:2 0:0 5:1 10:1 13:1 11:1
13 Example BFS 8:-1 3:2 12:-1 6:2 14:2 9:2 2:2 7:1 1:1 4:2 0:0 5:1 10:1 13:1 11:1
14 Example BFS 8:-1 3:2 12:3 6:2 14:2 9:2 2:2 7:1 1:1 4:2 0:0 5:1 10:1 13:1 11:1
15 Example BFS 8:-1 3:2 12:3 6:2 14:2 9:2 2:2 7:1 1:1 4:2 0:0 5:1 10:1 13:1 11:1
16 Example BFS 8:3 3:2 12:3 6:2 14:2 9:2 2:2 7:1 1:1 4:2 0:0 5:1 10:1 13:1 11:1
17 Example BFS 8:3 3:2 12:3 6:2 14:2 9:2 2:2 7:1 1:1 4:2 0:0 5:1 10:1 13:1 11:1
18 Example BFS 8:3 3:2 12:3 6:2 14:2 9:2 2:2 7:1 1:1 4:2 0:0 5:1 10:1 13:1 11:1
19 Example BFS 8:3 3:2 12:3 6:2 14:2 9:2 2:2 7:1 1:1 4:2 0:0 5:1 10:1 13:1 11:1
20 Example BFS 8:3 3:2 12:3 6:2 14:2 9:2 2:2 7:1 1:1 4:2 0:0 5:1 10:1 13:1 11:1
21 Depth-First Search DFS keeps walking down a path until it is forced to backtrack. s It backtracks until it finds a new path to go down. 1 Think: Solving a maze. 2 It results in a search tree, called the depth-first search tree. In general, the DFS tree will be very different than the BFS tree
22 Depth-First Search s
23 Example DFS 0:0:-1 10:-1:-1 3:-1:-1 4:-1:-1 11:-1:-1 14:-1:-1 9:-1:-1 6:-1:-1 13:-1:-1 5:-1:-1 8:-1:-1 7:-1:-1 12:-1:-1 1:-1:-1 2:-1:-1
24 Example DFS 0:0:-1 10:-1:-1 3:-1:-1 4:-1:-1 11:-1:-1 14:1:-1 9:-1:-1 6:-1:-1 13:-1:-1 5:-1:-1 8:-1:-1 7:-1:-1 12:-1:-1 1:-1:-1 2:-1:-1
25 Example DFS 0:0:-1 10:-1:-1 3:-1:-1 4:-1:-1 11:-1:-1 14:1:-1 9:-1:-1 6:-1:-1 13:-1:-1 5:2:-1 8:-1:-1 7:-1:-1 12:-1:-1 1:-1:-1 2:-1:-1
26 Example DFS 0:0:-1 10:-1:-1 3:-1:-1 4:-1:-1 11:-1:-1 14:1:-1 9:-1:-1 6:-1:-1 13:-1:-1 5:2:-1 8:-1:-1 7:-1:-1 12:3:-1 1:-1:-1 2:-1:-1
27 Example DFS 0:0:-1 10:-1:-1 3:-1:-1 4:-1:-1 11:-1:-1 14:1:-1 9:-1:-1 6:-1:-1 13:-1:-1 5:2:-1 8:-1:-1 7:-1:-1 12:3:-1 1:-1:-1 2:4:-1
28 Example DFS 0:0:-1 10:-1:-1 3:-1:-1 4:-1:-1 11:-1:-1 14:1:-1 9:-1:-1 6:-1:-1 13:-1:-1 5:2:-1 8:-1:-1 7:-1:-1 12:3:-1 1:5:-1 2:4:-1
29 Example DFS 0:0:-1 10:-1:-1 3:-1:-1 4:-1:-1 11:-1:-1 14:1:-1 9:-1:-1 6:-1:-1 13:-1:-1 5:2:-1 8:-1:-1 7:6:-1 12:3:-1 1:5:-1 2:4:-1
30 Example DFS 0:0:-1 10:-1:-1 3:-1:-1 4:-1:-1 11:-1:-1 14:1:-1 9:-1:-1 6:7:-1 13:-1:-1 5:2:-1 8:-1:-1 7:6:-1 12:3:-1 1:5:-1 2:4:-1
31 Example DFS 0:0:-1 10:-1:-1 3:-1:-1 4:-1:-1 11:-1:-1 14:1:-1 9:-1:-1 6:7:-1 13:8:-1 5:2:-1 8:-1:-1 7:6:-1 12:3:-1 1:5:-1 2:4:-1
32 Example DFS 0:0:-1 10:-1:-1 3:-1:-1 4:-1:-1 11:9:-1 14:1:-1 9:-1:-1 6:7:-1 13:8:-1 5:2:-1 8:-1:-1 7:6:-1 12:3:-1 1:5:-1 2:4:-1
33 Example DFS 0:0:-1 10:10:-1 3:-1:-1 4:-1:-1 11:9:-1 14:1:-1 9:-1:-1 6:7:-1 13:8:-1 5:2:-1 8:-1:-1 7:6:-1 12:3:-1 1:5:-1 2:4:-1
34 Example DFS 0:0:-1 10:10:11 3:-1:-1 4:-1:-1 11:9:12 14:1:-1 9:-1:-1 6:7:14 13:8:13 5:2:-1 8:15:-1 7:6:-1 12:3:-1 1:5:-1 2:4:-1
35 Example DFS 0:0:-1 10:10:11 3:-1:-1 4:23:-1 11:9:12 14:1:22 9:-1:-1 6:7:14 13:8:13 5:2:21 8:15:16 7:6:17 12:3:20 1:5:18 2:4:19
36 Example DFS 0:0:-1 10:10:11 3:24:-1 4:23:-1 11:9:12 14:1:22 9:-1:-1 6:7:14 13:8:13 5:2:21 8:15:16 7:6:17 12:3:20 1:5:18 2:4:19
37 Example DFS 0:0:-1 10:10:11 3:24:-1 4:23:-1 11:9:12 14:1:22 9:25:-1 6:7:14 13:8:13 5:2:21 8:15:16 7:6:17 12:3:20 1:5:18 2:4:19
38 Example DFS 0:0:29 10:10:11 3:24:27 4:23:28 11:9:12 14:1:22 9:25:26 6:7:14 13:8:13 5:2:21 8:15:16 7:6:17 12:3:20 1:5:18 2:4:19
39 General Tree Growing (following Gross & Yellen) We can think of BFS and DFS (and several other algorithms) as special cases of tree growing: Let T be the current tree T, and Maintain a list of frontier edges: the set of edges of G that have one endpoint in T and one endpoint not in T : v Repeatedly choose a frontier edge (somehow) and add it to T.
40 Tree Growing TreeGrowing(graph G, vertex v, func nextedge): T = (v, ) S = set of edges incident to v While S is not empty: e = nextedge(g, S) T = T + e // add edge e to T S = updatefrontier(g, S, e) return T The function nextedge(g, S) returns a frontier edge from S. updatefrontier(g, S, e) returns the new frontier after we add edge e to T.
41 Tree Growing These algorithms are all special cases / variants of Tree Growing, with different versions of nextedge: 1. Depth-first search 2. Breadth-first search 3. Prim s minimum spanning tree algorithm 4. Dijkstra s shortest path 5. A*
42 BFS & DFS as Tree Growing What s nextedge for DFS? What s nextedge for BFS?
43 BFS & DFS as Tree Growing What s nextedge for DFS? Select a frontier edge whose tree endpoint was discovered most recently. Why? We can use a stack to implement DFS. Runtime: O( Edges ) What s nextedge for BFS?
44 BFS & DFS as Tree Growing What s nextedge for DFS? Select a frontier edge whose tree endpoint was discovered most recently. Why? We can use a stack to implement DFS. Runtime: O( Edges ) What s nextedge for BFS? Select a frontier edge whose tree endpoint was discovered earliest. Why? We can use a queue to implement BFS. Runtime: O( Edges )
45 Prim s Algorithm Prim s Algorithm: Run TreeGrowing starting with any root node, adding the frontier edge with the smallest weight. Theorem. Prim s algorithm produces a minimum spanning tree. r u e v S = set of nodes already in the tree when e is added
46 Implementations of BFS and DFS
47 BFS implementation procedure bfs(g, s): Q := queue containing only s while Q not empty v := Q.front(); Q.remove front() for w G.neighbors(v): if w not seen: mark w seen Q.enqueue(w)
48 Recursive implementation of DFS procedure dfs(g, u): while u has an unvisited neighbor in G v := an unvisited neighbor of u mark v visited dfs(g, v)
49 Stack-based implementation of DFS procedure dfs(g, s): S := stack containing only s while S not empty v := S.pop() if v not visited: mark v visited for w G.neighbors(v): S.push(w)
50 Properties of BFS and DFS
51 Property of Non-BFS-Tree Edges Theorem. Choose x L i and y L j such that {x, y} is an edge in undirected graph G. Then i and j differ by at most 1. In other words, edges of G that do not appear in the tree connect nodes either in the same layer or adjacent layer. Proof. Suppose not, and that i < j 1. All the neighbors of x will be found by layer i + 1. Therefore, the layer of y is less than i + 1, so j i + 1, which contradicts i < j 1.
52 A property of Non-DFS-Tree Edges Theorem. Let x and y be nodes in the DFS tree T G such that {x, y} is an edge in undirected graph G. Then one of x or y is an ancestor of the other in T G. Proof. Suppose, wlog, x is reached first in the DFS. All the nodes that are marked explored between first encountering x and leaving x for the last time are descendants of x in T G. When we reach x, node y must not yet have been explored. It must become explored before leaving x for the last time (otherwise, we should add {x, y} to T G ). Hence, y is a descendent of x in T G.
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