Graph Searching Algorithms

S

Tree

Breadth-First Search (BFS)

u v w Breadth-First Search (BFS) x y z Not discovered Discovered, adjacent white nodes Discovered, no adjacent white nodes

u v w Breadth-First Search (BFS) x y z BFS(G, u): 1. Initialize the graph color[u]  gray π[u]  Nil d[u]  0 for each other vertex color[u]  white

u v w Breadth-First Search (BFS) x y Q BFS(G, u): 2. Initialize the queue Q  Ø Enqueue(Q, u) z

u v Breadth-First Search (BFS) x y t = u Q BFS(G, u): 3. While Q ≠ Ø 1) t  Dequeue(Q) w z

u v w Breadth-First Search (BFS) x y z Q t = u r = x, v BFS(G, u): 3. While Q ≠ Ø 2) for each r adj to t if color[r] = white color[r]  gray π[r]  t d[r]  d[t] + 1 Enqueue(Q, r)

u v Breadth-First Search (BFS) x y Q t = u r = x, v BFS(G, u): 3. While Q ≠ Ø 3) color[t]  black w z

u v w Breadth-First Search (BFS) x y z t = v Q BFS(G, u): 3. While Q ≠ Ø 1) t …  Dequeue(Q) 2) for each r adj to t 3) color[t]  black

u v w Breadth-First Search (BFS) x y z Q t = v r = y BFS(G, u): 3. While Q ≠ Ø 1) t …  Dequeue(Q) 2) for each r adj to t 3) color[t]  black

u v w Breadth-First Search (BFS) x y z Q t = v r = y BFS(G, u): 3. While Q ≠ Ø 1) t …  Dequeue(Q) 2) for each r adj to t 3) color[t]  black

u v w Breadth-First Search (BFS) x y z Q t = x r = BFS(G, u): 3. While Q ≠ Ø 1) t …  Dequeue(Q) 2) for each r adj to t 3) color[t]  black

u v w Breadth-First Search (BFS) x y z Q t = y r = w BFS(G, u): 3. While Q ≠ Ø 1) t …  Dequeue(Q) 2) for each r adj to t 3) color[t]  black

u v w Breadth-First Search (BFS) x y z Q t = w r = z BFS(G, u): 3. While Q ≠ Ø 1) t …  Dequeue(Q) 2) for each r adj to t 3) color[t]  black

u v w Breadth-First Search (BFS) x y z Q t = z r = BFS(G, u): 3. While Q ≠ Ø 1) t …  Dequeue(Q) 2) for each r adj to t 3) color[t]  black

u v w Breadth-First Search (BFS) x BFS(G, u): - the shortest-path distance from u y z

u v w Breadth-First Search (BFS) x y BFS(G, u): - the shortest-path distance from u - construct a tree z

u v w Breadth-First Search (BFS) x y BFS(G, u): - Initialization: |V| - Enqueuing/dequeuing: |V| - Scanning adj vertices: |E| z

u v w Breadth-First Search (BFS) x y z BFS(G, u): - Initialization: O(|V|) - Enqueuing/dequeuing: O(|V|) - Scanning adjacent vertices: O(|E|) => total running time: O(|V| + |E|)

Depth-First Search (DFS)

Depth-First Search (DFS) u v w d[u]: when u is discovered f[u]: when searching adj of u is finished

Depth-First Search (DFS) timestamp: t d[u] = t u v w d[u]: when u is discovered f[u]: when searching adj of u is finished

Depth-First Search (DFS) timestamp: t+1 d[u] = t u v d[v] = t+1 w d[u]: when u is discovered f[u]: when searching adj of u is finished

Depth-First Search (DFS) timestamp: t+2 d[u] = t u v d[v] = t+1 f[v] = t+2 w d[u]: when u is discovered f[u]: when searching adj of u is finished

Depth-First Search (DFS) timestamp: t+3 d[u] = t u v d[v] = t+1 f[v] = t+2 w d[w] = t+3 d[u]: when u is discovered f[u]: when searching adj of u is finished

Depth-First Search (DFS) timestamp: t+4 d[u] = t u v d[v] = t+1 f[v] = t+2 w d[w] = t+3 f[v] = t+4 d[u]: when u is discovered f[u]: when searching adj of u is finished

Depth-First Search (DFS) timestamp: t+5 d[u] = t f[u] = t+5 u v d[v] = t+1 f[v] = t+2 w d[w] = t+3 f[w] = t+4 d[u]: when u is discovered f[u]: when searching adj of u is finished

Depth-First Search (DFS) d[u] = t f[u] = t+5 u v d[v] = t+1 f[v] = t+2 w d[w] = t+3 f[w] = t+4 d[u]: when u is discovered f[u]: when searching adj of u is finished 1. d[u] < f[u] 2. [ d[u], f[u] ] entirely contains [ d[v], f[v] ] 3. [ d[v], f[v] ] and [ d[w], f[w] ] are entirely disjoint

u v w Depth-First Search (DFS) x y z Not discovered Discovered, adjacent white nodes Discovered, no adjacent white nodes

u v w Depth-First Search (DFS) x y z Not discovered Discovered, adjacent white nodes Discovered, no adjacent white nodes

u v w Depth-First Search (DFS) x y DFS(G): 1. Initialization for each u V[G], color[u]  white π[u]  Nil time  0 z

u v w Depth-First Search (DFS) x y z DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) DFS-Visit(u): 1. Initial Setting color[u]  gray d[u]  time  time + 1

u v w Depth-First Search (DFS) x y z DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) DFS-Visit(u): 1. Initial Setting 2. for each adj v of white π[v]  u DFS-Visit[v]

u v w Depth-First Search (DFS) x y z DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) DFS-Visit(u): 1. Initial Setting 2. for each adj v of white π[v]  u DFS-Visit[v]

u v w Depth-First Search (DFS) x y z DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) DFS-Visit(u): 1. Initial Setting 2. for each adj v of white π[v]  u DFS-Visit[v]

u v w Depth-First Search (DFS) x y z DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1

u v w Depth-First Search (DFS) x y z DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1

u v w Depth-First Search (DFS) x y z DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1

u v w Depth-First Search (DFS) x y z DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1

u v w Depth-First Search (DFS) x y z DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1

u v w Depth-First Search (DFS) x y z DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1

u v w Depth-First Search (DFS) x y z DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1

u v w Depth-First Search (DFS) x y z DFS(G): 1. Initialization 2. For each u V[G] if color[u] = white DFS-Visit(u) DFS-Visit(u): 1. Initial Setting 2. Handling adj vertices 3. color[u]  black f[u]  time  time + 1

u v w Depth-First Search (DFS) x DFS(G): - construct a forest y z

u v w Depth-First Search (DFS) x y DFS(G): - Initialization: O(|V|) - Traversing vertices: O(|V|) - Scanning adjacent vertices: O(|E|) => total running time: O(|V| + |E|) z

Topological Sorting

Topological Sorting m O(|V| 2 + |V||E|) Or O(|V| 2 ) Brute-Force way 1. Find a vertex without edges.

2. Put it onto the front of the list, and remove it from G. 3. Remove all edges to the removed edge.

4. Repeat 1~3.

m Topological Sorting Using DFS 1. Call DFS(G) 2. When a vertex is finished, put it onto the front of the list.

O(|V| + |E|)

Topological Sorting v enters the list before u?

At d[u]: Using DFS 1. Call DFS(G) 2. When a vertex is finished, put it onto the front of the list.

1) v is white: d[u] < d[v] < f[u] 2) v is black: f[v] < d[u] 3) v is gray: d[v] < d[u] < f[v]

Topological Sorting v enters the list before u?

At d[u]: 1) v is white: d[u] < d[v] t is gray: d[t] < d[u] < f[t] t is black: f[t] < d[u] Using DFS 1. Call DFS(G) 2. When a vertex is finished, put it onto the front of the list.

Topological Sorting u v w 1. If v is a descendant, d[u] < d[v] < f[v] < f[u] 2. If v is an anscestor, d[v] < d[u] < f[u] < f[v] 3. Otherwise, d[v] < f[v] < d[u] < f[u] or d[u] < f[u] < d[v] < f[v]

Depth-First Search (DFS) u v w 1. d[u] < f[u] 2. [ d[u], f[u] ] entirely contains [ d[v], f[v] ] 3. [ d[v], f[v] ] and [ d[w], f[w] ] are entirely disjoint In a depth-first forest, v is a descendant of u if and only if d[u] < d[v] < f[v] < f[u]

Topological Sorting 1. If v is a descendant, d[u] < d[v] < f[v] < f[u] 2. If v is an anscestor, d[v] < d[u] < f[u] < f[v] 3. Otherwise, Contradiction: G is not acyclic.

d[v] < f[v] < d[u] < f[u] or d[u] < f[u] < d[v] < f[v] At d[u]: 1) v is white: d[u] < d[v] t is gray: d[t] < d[u] < f[t] t is black: f[t] < d[u]

Topological Sorting 1. If v is a descendant, d[u] < d[v] < f[v] < f[u] 2. If v is an anscestor, d[v] < d[u] < f[u] < f[v] 3. Otherwise, Contradiction: v should be black.

d[v] < f[v] < d[u] < f[u] or d[u] < f[u] < d[v] < f[v] At d[u]: 1) v is white: d[u] < d[v] t is gray: d[t] < d[u] < f[t] t is black: f[t] < d[u]

Topological Sorting 1. If v is a descendant, d[u] < d[v] < f[v] < f[u] 2. If v is an anscestor, d[v] < d[u] < f[u] < f[v] 3. Otherwise, d[v] < f[v] < d[u] < f[u] or d[u] < f[u] < d[v] < f[v] At d[u]: 1) v is white: d[u] < d[v] t is white.

Topological Sorting 1. If v is a descendant, d[u] < d[v] < f[v] < f[u] 2. If v is an anscestor, d[v] < d[u] < f[u] < f[v] 3. Otherwise, d[v] < f[v] < d[u] < f[u] or d[u] < f[u] < d[v] < f[v] At d[u]: 1) v is white: d[u] < d[v] < f[u] t is white.

Topological Sorting 1. If v is a descendant, d[u] < d[v] < f[v] < f[u] 2. If v is an anscestor, d[v] < d[u] < f[u] < f[v] 3. Otherwise, v will enter the list before u.

d[v] < f[v] < d[u] < f[u] or d[u] < f[u] < d[v] < f[v] At d[u]: 1) v is white: d[u] < d[v] < f[u] 2) v is black: f[v] < d[u] 3) v is gray: d[v] < d[u] < f[v]

Topological Sorting 1. If v is a descendant, d[u] < d[v] < f[v] < f[u] 2. If v is an anscestor, d[v] < d[u] < f[u] < f[v] 3. Otherwise, v is already in the list.

d[v] < f[v] < d[u] < f[u] or d[u] < f[u] < d[v] < f[v] At d[u]: 1) v is white: d[u] < d[v] < f[u] 2) v is black: f[v] < d[u] 3) v is gray: d[v] < d[u] < f[v]

Topological Sorting 1. If v is a descendant, d[u] < d[v] < f[v] < f[u] 2. If v is an anscestor, d[v] < d[u] < f[u] < f[v] 3. Otherwise, Contradiction: G is not acyclic.

d[v] < f[v] < d[u] < f[u] or d[u] < f[u] < d[v] < f[v] At d[u]: 1) v is white: d[u] < d[v] < f[u] 2) v is black: f[v] < d[u] 3) v is gray: d[v] < d[u] < f[v]

Strongly Connected Component

Strongly Connected Component

Strongly Connected Component

Strongly Connected Component 1 3 1. Call DFS(G) 2. Arrange the vertices in order of decreasing f(u) 9 4 8 5 6 2 7

Strongly Connected Component 1 3 1. Call DFS(G) 2. Arrange the vertices in order of decreasing f(u) 3. Compute G T 9 4 8 5 6 2 7

Strongly Connected Component 1 3 1. Call DFS(G) 2. Arrange the vertices in order of decreasing f(u) 3. Compute G T 4. Run DFS(G T ) 9 4 8 5 6 2 7

Strongly Connected Component 1 3 1. Call DFS(G) 2. Arrange the vertices in order of decreasing f(u) 3. Compute G T 4. Run DFS(G T ) 9 4 8 5 6 2 7

Topological Sorting

Strongly Connected Component

Strongly Connected Component Last f[u] > Last f[u]

Strongly Connected Component 1 3 1. Call DFS(G) 2. Arrange the vertices in order of decreasing f(u) 9 4 8 5 6 2 7

Strongly Connected Component 1 4 2 3

Strongly Connected Component 1 4 2 3

Adjacency Matrix of Graphs Directed graph 5 3 4 1 2 Adjacency-Matrix 1 2 3 4 5

A = Adjacency Matrix of Graphs A T =