Data Structures in Java Lecture 19: Applications of DFS 11/30/2015 - PowerPoint PPT Presentation

Depth First Spanning Tree • The steps taken by DFS can be illustrated as a (directed) spanning tree. • Add a tree edge for every graph edge taken by DFS. • Add a back edge for every skipped edge. A B D E C 60

Depth First Spanning Tree • The steps taken by DFS can be illustrated as a (directed) spanning tree. • Add a tree edge for every graph edge taken by DFS. A A B D E C 61

Depth First Spanning Tree • The steps taken by DFS can be illustrated as a (directed) spanning tree. • Add a tree edge for every graph edge taken by DFS. A A B D E B C 62

Depth First Spanning Tree • The steps taken by DFS can be illustrated as a (directed) spanning tree. • Add a tree edge for every graph edge taken by DFS. A A B D E B C C 63

Depth First Spanning Tree • The steps taken by DFS can be illustrated as a (directed) spanning tree. • Add a tree edge for every graph edge taken by DFS. A A B E B D C C D 64

Depth First Spanning Tree • The steps taken by DFS can be illustrated as a (directed) spanning tree. • Add a tree edge for every graph edge taken by DFS. • Add a back edge for every skipped edge. A A B E B D C C D 65

Depth First Spanning Tree • The steps taken by DFS can be illustrated as a (directed) spanning tree. • Add a tree edge for every graph edge taken by DFS. • Add a back edge for every skipped edge. A A B E B D C C D 66

Depth First Spanning Tree • The steps taken by DFS can be illustrated as a (directed) spanning tree. • Add a tree edge for every graph edge taken by DFS. • Add a back edge for every skipped edge. A A B B D E C C E D 67

Depth First Spanning Tree • The steps taken by DFS can be illustrated as a (directed) spanning tree. • Add a tree edge for every graph edge taken by DFS. • Add a back edge for every skipped edge. A A B B D E C C E D 68

Identifying Articulation Points (1) • If the root of the DFS spanning tree has two outgoing 
 tree edges, the root is an articulation point. C A G B B A C D F C is an D articulation G E point. E F 69

Identifying Articulation Points (1) • If the root of the DFS spanning tree has two outgoing 
 tree edges, the root is an articulation point. A • Depends on which vertex we start DFS from. B A B C C D G D F A is not an E G E articulation point. F 70

Identifying Articulation Points (2) • Any non-root vertex v is an articulation point iff • v has a child w such that there is no 
 back-edge from the subtree below w 
 A to any ancestor of v . B A B C D C G D F C is an articulation E E G point because F of G. 71

Identifying Articulation Points (2) • Any non-root vertex v is an articulation point iff • v has a child w such that there is no 
 back-edge from the subtree below w 
 A to any ancestor of v. B A B C C D G D F D is an articulation E G E point because F of E. 72

Preorder Numbers • Assign numbers to each vertex in the order in which they 
 are visited by DFS. • For every tree edge (u,v): Num(u) < Num(v) A 1 • For every back edge (u,v): Num(u) > Num(v) B 2 v Num(v) C A 1 3 B 2 C 3 G D 7 D 4 4 E 5 E F 6 5 G 7 F 6 73

Low numbers • For each vertex, find the lowest numbered vertex that is 
 reachable by following a path that contains 
 A at most one back edge. 1 B 2 v Num(v) Low(v) C A 1 1 3 B 2 C 3 G D 7 4 D 4 E 5 E F 6 5 G 7 F 6 74

Low numbers • For each vertex, find the lowest numbered vertex that is 
 reachable by following a path that contains 
 A at most one back edge. 1 B 2 v Num(v) Low(v) C A 1 1 3 1 B 2 C 3 1 G D 7 4 D 4 1 E 5 E F 6 5 G 7 F 6 75

Low numbers • For each vertex, find the lowest numbered vertex that is 
 reachable by following a path that contains 
 A at most one back edge. 1 B 2 v Num(v) Low(v) C A 1 1 3 1 B 2 C 3 1 G D 7 4 D 4 1 E 5 4 E 4 F 6 5 7 G 7 F 6 76

Identifying Articulation Points • Any non-root vertex v is an articulation point iff • v has a child w such that there is no 
 A 1 back-edge from the subtree below w 
 to any ancestor of v. B 2 v Num(v) Low(v) C A 1 1 3 1 B 2 C 3 1 G D 7 4 D 4 1 E 5 4 E 4 F 6 5 7 G 7 F 6 77

Identifying Articulation Points • Any non-root vertex v is an articulation point iff • v has a child w such that Low(w) ≥ Num(v) A 1 B 2 v Num(v) Low(v) C A 1 1 3 1 B 2 C 3 1 G 7 D 4 D 4 1 E 5 4 E 4 F 6 5 7 G 7 F 6 78

Computing Low Numbers Recursively compute_low(v) { A Low(v) = Num(v) 1 for all back edges (v,u) { B if ( Num(u) < Low(v) ) 2 Low(v) = Num(u); C } 3 for all tree edges (v,u) { G compute_low(u); D 7 4 if ( Low(u) < Low(v) ) 
 Low(v) = Low(u) ; E } 5 } F 6 79

Computing Low Numbers Recursively compute_low(v) { A Low(v) = Num(v) 1 1 for all back edges (v,u) { B if ( Num(u) < Low(v) ) 2 Low(v) = Num(u); C } 3 for all tree edges (v,u) { G compute_low(u); D 7 4 if ( Low(u) < Low(v) ) 
 Low(v) = Low(u) ; E } 5 } F 6 80

Computing Low Numbers Recursively compute_low(v) { A Low(v) = Num(v) 1 1 for all back edges (v,u) { B if ( Num(u) < Low(v) ) 2 2 Low(v) = Num(u); C } 3 for all tree edges (v,u) { G compute_low(u); D 7 4 if ( Low(u) < Low(v) ) 
 Low(v) = Low(u) ; E } 5 } F 6 81

Computing Low Numbers Recursively compute_low(v) { A Low(v) = Num(v) 1 1 for all back edges (v,u) { B if ( Num(u) < Low(v) ) 2 2 Low(v) = Num(u); C } 3 3 for all tree edges (v,u) { G compute_low(u); D 7 4 if ( Low(u) < Low(v) ) 
 Low(v) = Low(u) ; E } 5 } F 6 82

Computing Low Numbers Recursively compute_low(v) { A Low(v) = Num(v) 1 1 for all back edges (v,u) { B if ( Num(u) < Low(v) ) 2 2 Low(v) = Num(u); C } 3 3 for all tree edges (v,u) { 7 G compute_low(u); D 7 4 1 if ( Low(u) < Low(v) ) 
 Low(v) = Low(u) ; E } 5 } F 6 83

Computing Low Numbers Recursively compute_low(v) { A Low(v) = Num(v) 1 1 for all back edges (v,u) { B if ( Num(u) < Low(v) ) 2 2 Low(v) = Num(u); C } 3 3 for all tree edges (v,u) { 7 G compute_low(u); D 7 4 1 if ( Low(u) < Low(v) ) 
 Low(v) = Low(u) ; E 5 } 5 } F 6 84

Computing Low Numbers Recursively compute_low(v) { A Low(v) = Num(v) 1 1 for all back edges (v,u) { B if ( Num(u) < Low(v) ) 2 2 Low(v) = Num(u); C } 3 3 for all tree edges (v,u) { 7 G compute_low(u); D 7 4 1 if ( Low(u) < Low(v) ) 
 Low(v) = Low(u) ; E 5 } 5 } F 6 4 85

Computing Low Numbers Recursively compute_low(v) { A Low(v) = Num(v) 1 1 for all back edges (v,u) { B if ( Num(u) < Low(v) ) 2 2 Low(v) = Num(u); C } 3 3 for all tree edges (v,u) { 7 G compute_low(u); D 7 4 1 if ( Low(u) < Low(v) ) 
 Low(v) = Low(u) ; E 4 } 5 } F 6 4 86

Computing Low Numbers Recursively compute_low(v) { A Low(v) = Num(v) 1 1 for all back edges (v,u) { B if ( Num(u) < Low(v) ) 2 2 Low(v) = Num(u); C } 3 3 for all tree edges (v,u) { 7 G compute_low(u); D 7 4 1 if ( Low(u) < Low(v) ) 
 Low(v) = Low(u) ; E 4 } 5 } F 6 4 87

Computing Low Numbers Recursively compute_low(v) { A Low(v) = Num(v) 1 1 for all back edges (v,u) { B if ( Num(u) < Low(v) ) 2 2 Low(v) = Num(u); C } 1 3 for all tree edges (v,u) { 7 G compute_low(u); D 7 4 1 if ( Low(u) < Low(v) ) 
 Low(v) = Low(u) ; E 4 } 5 } F 6 4 88

Computing Low Numbers Recursively compute_low(v) { A Low(v) = Num(v) 1 1 for all back edges (v,u) { B if ( Num(u) < Low(v) ) 2 1 Low(v) = Num(u); C } 1 3 for all tree edges (v,u) { 7 G compute_low(u); D 7 4 1 if ( Low(u) < Low(v) ) 
 Low(v) = Low(u) ; E 4 } 5 } F 6 4 89

Computing Low Numbers Time to compute preorder numbers: O(|V| +|E|) compute_low(v) { A Low(v) = Num(v) 1 1 Time to compute low numbers: O(|V|+|E|) for all back edges (v,u) { Time to check for articulation points: O(|V|+|E|) B if ( Num(u) < Low(v) ) 2 1 Low(v) = Num(u); Total: O(|V|+|E|) C } 1 3 for all tree edges (v,u) { 7 G compute_low(u); D 7 4 1 if ( Low(u) < Low(v) ) 
 Low(v) = Low(u) ; E 4 } 5 } F 6 4 90

Contents • Applications of DFS • Euler Circuits • Biconnectivity in Undirected Graphs. • Finding Strongly Connected Components for Directed Graphs. 91

Connectivity in Directed Graphs • A directed graph is weakly connected if there is an undirected path from every vertex to every other vertex. weakly connected graph 92

Strongly Connected Graphs • A directed graph is strongly connected if there is a path from every vertex to every other vertex. v Weakly connected, but not strongly connected (no other vertex can be reached from v). 93

Testing if a Graph is Strongly Connected • Run DFS to see if all vertices are reachable from some start node s. s • Reverse direction of edges and run DFS again. 94

Testing if a Graph is Strongly Connected • Run DFS to see if all vertices are reachable from some start node s. s • Reverse direction of edges and run DFS again. 95

Testing if a Graph is Strongly Connected • Run DFS to see if all vertices are reachable from some start node s. s • Reverse direction of edges and run DFS again. 96

Testing if a Graph is Strongly Connected • Run DFS to see if all vertices are reachable from some start node s. s • Reverse direction of edges and run DFS again. 97

Testing if a Graph is Strongly Connected • Run DFS to see if all vertices are reachable from some start node s. s • Reverse direction of edges and run DFS again. 98

Testing if a Graph is Strongly Connected • Run DFS to see if all vertices are reachable from some start node s. s • Reverse direction of edges and run DFS again. 99

Testing if a Graph is Strongly Connected • Run DFS to see if all vertices are reachable from some start node s. s • Reverse direction of edges and run DFS again. 100