1 / 23

Breadth First Search

Breadth First Search. Representations of Graphs. Two standard ways to represent a graph Adjacency lists, Adjacency Matrix Applicable to directed and undirected graphs. Adjacency lists Graph G(V, E) is represented by array Adj of | V | lists

heath
Download Presentation

Breadth First Search

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Breadth FirstSearch

  2. Representations of Graphs • Two standard ways to represent a graph • Adjacency lists, • Adjacency Matrix • Applicable to directed and undirected graphs. Adjacency lists • Graph G(V, E) is represented by array Adj of |V| lists • For each u  V, the adjacency list Adj[u] consists of all the vertices adjacent to u in G • The amount of memory required is: (V + E)

  3. Adjacency Matrix • A graph G(V, E) assuming the vertices are numbered 1, 2, 3, … , |V| in some arbitrary manner, then representation of G consists of: |V| × |V| matrix A = (aij) such that 1 if (i, j)  E 0 otherwise • Preferred when graph is dense • |E| is close to |V|2 aij={

  4. 1 2 3 5 4 1 2 2 5 4 4 Adjacency matrix of undirected graph

  5. Adjacency Matrix • The amount of memory required is • (V2) • For undirected graph to cut down needed memory only entries on and above diagonal are saved • In an undirected graph, (u, v) and (v, u) represents the same edge, adjacency matrix A of an undirected graph is its own transpose A = AT • It can be adapted to represent weighted graphs.

  6. Breadth First Search

  7. Breadth First Search • One of simplest algorithm searching graphs • A vertex is discovered first time, encountered • Let G (V, E) be a graph with source vertex s, BFS • discovers every vertex reachable from s. • gives distance from s to each reachable vertex • produces BF tree root with s to reachable vertices • To keep track of progress, it colors each vertex • vertices start white, may later gray, then black • Adjacent to black vertices have been discovered • Gray vertices may have some adjacent white vertices

  8. Breadth First Search • It is assumed that input graph G (V, E) is represented using adjacency list. • Additional structures maintained with each vertex v  V are • color[u] – stores color of each vertex • π[u] – stores predecessor of u • d[u] – stores distance from source s to vertex u

  9. Breadth First Search BFS(G, s) 1 for each vertex u  V [G] – {s} 2 do color [u] ← WHITE 3 d [u] ← ∞ 4 π[u] ← NIL 5 color[s] ← GRAY 6 d [s] ← 0 7 π[s] ← NIL 8 Q ← Ø /* Q always contains the set of GRAY vertices */ 9 ENQUEUE (Q, s) 10 while Q ≠ Ø 11 do u ← DEQUEUE (Q) 12 for each v  Adj [u] 13 do if color [v] = WHITE /* For undiscovered vertex. */ 14 then color [v] ← GRAY 15 d [v] ← d [u] + 1 16 π[v] ← u 17 ENQUEUE(Q, v) 18 color [u] ← BLACK

  10. Q  Breadth First Search Except root node, s For each vertex u  V(G) color [u]  WHITE d[u] ∞ π [s]  NIL r s t u        v w x y

  11. Q s Breadth First Search Considering s as root node color[s]  GRAY d[s]  0 π [s]  NIL ENQUEUE (Q, s) r s t u  0       v w x y

  12. Q w r Breadth First Search • DEQUEUE s from Q • Adj[s] = w, r • color [w] = WHITE • color [w] ← GRAY • d [w] ← d [s] + 1 = 0 + 1 = 1 • π[w] ← s • ENQUEUE (Q, w) • color [r] = WHITE • color [r] ← GRAY • d [r] ← d [s] + 1 = 0 + 1 = 1 • π[r] ← s • ENQUEUE (Q, r) • color [s] ← BLACK r s t u 1 0    1   v w x y

  13. Q r t x Breadth First Search • DEQUEUE w from Q • Adj[w] = s, t, x • color [s] ≠ WHITE • color [t] = WHITE • color [t] ← GRAY • d [t] ← d [w] + 1 = 1 + 1 = 2 • π[t] ← w • ENQUEUE (Q, t) • color [x] = WHITE • color [x] ← GRAY • d [x] ← d [w] + 1 = 1 + 1 = 2 • π[x] ← w • ENQUEUE (Q, x) • color [w] ← BLACK r s t u 1 0 2   1 2  v w x y

  14. Q t x v Breadth First Search • DEQUEUE r from Q • Adj[r] = s, v • color [s] ≠ WHITE • color [v] = WHITE • color [v] ← GRAY • d [v] ← d [r] + 1 = 1 + 1 = 2 • π[v] ← r • ENQUEUE (Q, v) • color [r] ← BLACK r s t u 1 0 2  2 1 2  v w x y

  15. r s t u 1 0 2 3 2 1 2  v w x y Q x v u Breadth First Search DEQUEUE t from Q Adj[t] = u, w, x color [u] = WHITE color [u] ← GRAY d [u] ← d [t] + 1 = 2 + 1 = 3 π[u] ← t ENQUEUE (Q, u) color [w] ≠ WHITE color [x] ≠ WHITE color [t] ← BLACK

  16. r s t u 1 0 2 3 2 1 2 3 v w x y Q v u y Breadth First Search DEQUEUE x from Q Adj[x] = t, u, w, y color [t] ≠ WHITE color [u] ≠ WHITE color [w] ≠ WHITE color [y] = WHITE color [y] ← GRAY d [y] ← d [x] + 1 = 2 + 1 = 3 π[y] ← x ENQUEUE (Q, y) color [x] ← BLACK

  17. r s t u 1 0 2 3 2 1 2 3 v w x y Q u y Breadth First Search DEQUEUE v from Q Adj[v] = r color [r] ≠ WHITE color [v] ← BLACK

  18. r s t u 1 0 2 3 2 1 2 3 v w x y Q y Breadth First Search DEQUEUE u from Q Adj[u] = t, x, y color [t] ≠ WHITE color [x] ≠ WHITE color [y] ≠ WHITE color [u] ← BLACK

  19. r s t u 1 0 2 3 2 1 2 3 v w x y Q  Breadth First Search DEQUEUE y from Q Adj[y] = u, x color [u] ≠ WHITE color [x] ≠ WHITE color [y] ← BLACK

  20. Breadth First Search • Each vertex is enqueued and dequeued atmost once • Total time devoted to queue operation is O(V) • The sum of lengths of all adjacency lists is  (E) • Total time spent in scanning adjacency lists is O(E) • The overhead for initialization O(V) Total Running Time of BFS = O(V+E)

  21. Shortest Paths • The shortest-path-distanceδ(s, v) from s to v as the minimum number of edges in any path from vertex s to vertex v. • if there is no path from s to v, then δ(s, v) = ∞ • A path of length δ(s, v) from s to v is said to be a shortest path from s to v. • Breadth First search finds the distance to each reachable vertex in the graph G (V, E) from a given source vertex s  V. • The field d, for distance, of each vertex is used.

  22. Shortest Paths • BFS-Shortest-Paths (G, s) • 1  v V • 2 d [v] ← ∞ • 3 d [s] ← 0 • 4 ENQUEUE (Q, s) • 5 whileQ ≠φ • 6 do v ← DEQUEUE(Q) • 7 for each w in Adj[v] • 8 do ifd [w] = ∞ • 9 then d [w] ← d [v] +1 • 10 ENQUEUE (Q, w)

  23. Print Path • PRINT-PATH (G, s, v) • 1 if v= s • 2 then print s • 3 else if π[v] = NIL • 4 then print “no path froms to v exists • 5 else PRINT-PATH (G, s, π[v]) • 6 print v

More Related