Graph Algorithms: Understanding Tree, Back, Forward, and Cross Edges in Depth-First Search, Slides of Computer Science

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Algorithms

Graph Algorithms

Administrative ● Test postponed to Friday ● Homework: ■ Turned in last night by midnight: full credit ■ Turned in tonight by midnight: 1 day late, 10% off ■ Turned in tomorrow night: 2 days late, 30% off ■ Extra credit lateness measured separately

Review: Representing Graphs ● Assume V = {1, 2, …, n } ● An adjacency matrix represents the graph as a n x n matrix A: ■ A[ i , j ] = 1 if edge ( i , j ) ∈ E (or weight of edge) = 0 if edge ( i , j ) ∉ E ■ Storage requirements: O(V 2 ) ○ A dense representation ■ But, can be very efficient for small graphs ○ Especially if store just one bit/edge ○ Undirected graph: only need one diagonal of matrix

Review: Graph Searching ● Given: a graph G = (V, E), directed or undirected ● Goal: methodically explore every vertex and every edge ● Ultimately: build a tree on the graph ■ Pick a vertex as the root ■ Choose certain edges to produce a tree ■ Note: might also build a forest if graph is not connected

Review: Breadth-First Search ● Again will associate vertex “colors” to guide the algorithm ■ White vertices have not been discovered ○ All vertices start out white ■ Grey vertices are discovered but not fully explored ○ They may be adjacent to white vertices ■ Black vertices are discovered and fully explored ○ They are adjacent only to black and gray vertices ● Explore vertices by scanning adjacency list of grey vertices

Review: Breadth-First Search BFS(G, s) { initialize vertices; Q = {s}; // Q is a queue (duh); initialize to s while (Q not empty) { u = RemoveTop(Q); for each v ∈ u->adj { if (v->color == WHITE) v->color = GREY; v->d = u->d + 1; v->p = u; Enqueue(Q, v); } u->color = BLACK; } } What does v->p represent? What does v->d represent?

Breadth-First Search: Example ∞ ∞

r s t u v w x y Q:^ s

Breadth-First Search: Example 1 ∞

r s t u v w x y Q: w r

Breadth-First Search: Example 1 2

r s t u v w x y Q: t^ x^ v

Breadth-First Search: Example 1 2

r s t u v w x y Q: x^ v^ u

Breadth-First Search: Example 1 2

r s t u v w x y Q: u^ y

Breadth-First Search: Example 1 2

r s t u v w x y Q: y

BFS: The Code Again BFS(G, s) { initialize vertices; Q = {s}; while (Q not empty) { u = RemoveTop(Q); for each v ∈ u->adj { if (v->color == WHITE) v->color = GREY; v->d = u->d + 1; v->p = u; Enqueue(Q, v); } u->color = BLACK; } } What will be the running time? Touch every vertex: O(V) u = every vertex, but only once (Why?) So v = every vertex that appears in some other vert’s adjacency list Total running time: O(V+E)

BFS: The Code Again BFS(G, s) { initialize vertices; Q = {s}; while (Q not empty) { u = RemoveTop(Q); for each v ∈ u->adj { if (v->color == WHITE) v->color = GREY; v->d = u->d + 1; v->p = u; Enqueue(Q, v); } u->color = BLACK; } } What will be the storage cost in addition to storing the graph? Total space used: O(max(degree(v))) = O(E)