Depth-First Search: Understanding Tree and Cross Edges in Graphs, Slides of Data Structures and Algorithms

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Depth-First Search

Problem of the Day

Prove that in a breadth-first search on a undirected graph G, every edge in G is either a tree edge or a cross edge, where a cross edge (x, y) is an edge where x is neither is an ancestor or descendent of y.

if (discovered[y] == FALSE) { parent[y] = v; process edge(v,y); dfs(g,y); } else if ((!processed[y]) || (g− >directed)) process edge(v,y); if (finished) return; p = p− >next; } process vertex late(v); time = time + 1; exit time[v] = time; processed[v] = TRUE; }

The Key Idea with DFS

A depth-first search of a graph organizes the edges of the graph in a precise way. In a DFS of an undirected graph, we assign a direction to each edge, from the vertex which discover it:

1 2 6 3 4 5

1

(^23)

4 5 6

Edge Classificatio Implementation

int edge classification(int x, int y) { if (parent[y] == x) return(TREE); if (discovered[y] && !processed[y]) return(BACK); if (processed[y] && (entry time[y]¿entry time[x])) return(FORWARD); if (processed[y] && (entry time[y]¡entry time[x])) return(CROSS); printf(”Warning: unclassified edge (%d,%d)”,x,y); }

DFS: Tree Edges and Back Edges Only

The reason DFS is so important is that it defines a very nice ordering to the edges of the graph. In a DFS of an undirected graph, every edge is either a tree edge or a back edge. Why? Suppose we have a forward edge. We would have encountered (4, 1) when expanding 4, so this is a back edge.

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2

3 4

DFS Application: Finding Cycles

Back edges are the key to finding a cycle in an undirected graph. Any back edge going from x to an ancestor y creates a cycle with the path in the tree from y to x.

process edge(int x, int y) { if (parent[x]! = y) { (* found back edge! *) printf(”Cycle from %d to %d:”,y,x); find path(y,x,parent); finished = TRUE; } }

Articulation Vertices

Suppose you are a terrorist, seeking to disrupt the telephone network. Which station do you blow up?

An articulation vertex is a vertex of a connected graph whose deletion disconnects the graph. Clearly connectivity is an important concern in the design of any network. Articulation vertices can be found in O(n(m+n)) – just delete each vertex to do a DFS on the remaining graph to see if it is connected.

Topological Sorting

A directed, acyclic graph has no directed cycles.

D A

G F

C E

B

A topological sort of a graph is an ordering on the vertices so that all edges go from left to right. DAGs (and only DAGs) has at least one topological sort (here G, A, B, C, F, E, D).

Applications of Topological Sorting

Topological sorting is often useful in scheduling jobs in their proper sequence. In general, we can use it to order things given precidence constraints. Example: Dressing schedule from CLR.

Topological Sorting via DFS

A directed graph is a DAG if and only if no back edges are encountered during a depth-first search. Labeling each of the vertices in the reverse order that they are marked processed finds a topological sort of a DAG. Why? Consider what happens to each directed edge {x, y} as we encounter it during the exploration of vertex x:

Case Analysis

• If y is currently undiscovered , then we then start a DFS of y before we can continue with x. Thus y is marked completed before x is, and x appears before y in the topological order, as it must.
• If y is discovered but not completed , then {x, y} is a back edge, which is forbidden in a DAG.
• If y is completed , then it will have been so labeled before x. Therefore, x appears before y in the topological order, as it must.

Strongly Connected Components

A directed graph is strongly connected iff there is a directed path between any two vertices. The strongly connected components of a graph is a partition of the vertices into subsets (maximal) such that each subset is strongly connected. a b

c d g h

e f

Observe that no vertex can be in two maximal components, so it is a partition.

There is an elegant, linear time algorithm to find the strongly connected components of a directed graph using DFS which is similar to the algorithm for biconnected components.