Breadth First Search

• 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)

Representations of Graphs

• 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 = {

Adjacency Matrix

Adjacency matrix of undirected graph 1 2 3 4 51 0 1 0 0 12 1 0 1 1 13 0 1 0 1 04 0 1 1 0 15 1 1 0 1 0

1

45

2

3

1 2 3 4 5 61 0 1 0 1 0 02 0 0 0 0 1 03 0 0 0 0 1 14 0 1 0 0 0 05 0 0 0 1 0 06 0 0 0 0 0 1

1

45

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4

2

• 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.

Adjacency Matrix

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

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

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

Breadth First Search

r s t u

v w x y

Q

Breadth First Search

Except root node, s For each vertex u V(G)color [u] WHITEd[u] ∞π [s] NIL

0

r s t u

v w x y

sQ

Breadth First Search

Considering s as root nodecolor[s] GRAYd[s] 0π [s] NILENQUEUE (Q, s)

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r s t u

v w x y

Breadth First Search

DEQUEUE s from QAdj[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

wQ r

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r s t u

v w x y

Breadth First SearchDEQUEUE w from QAdj[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

rQ t x

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r s t u

v w x y

Breadth First Search

DEQUEUE r from QAdj[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] ← BLACKQ t x v

Breadth First Search

DEQUEUE t from QAdj[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] ≠ WHITEcolor [x] ≠ WHITEcolor [t] ← BLACK

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r s t u

v w x y

Q x v u

Breadth First Search

DEQUEUE x from QAdj[x] = t, u, w, y

color [t] ≠ WHITEcolor [u] ≠ WHITEcolor [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

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r s t u

v w x y

Q v u y

Breadth First Search

DEQUEUE v from QAdj[v] = r

color [r] ≠ WHITE

color [v] ← BLACK

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r s t u

v w x y

Q u y

Breadth First Search

DEQUEUE u from QAdj[u] = t, x, y

color [t] ≠ WHITEcolor [x] ≠ WHITEcolor [y] ≠ WHITE

color [u] ← BLACK

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r s t u

v w x y

Q y

Breadth First Search

DEQUEUE y from QAdj[y] = u, x

color [u] ≠ WHITEcolor [x] ≠ WHITE

color [y] ← BLACK

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r s t u

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Q

• 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)

Breadth First Search

• The shortest-path-distance δ (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.

Shortest Paths

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

Shortest Paths

Print Path

PRINT-PATH (G, s, v)1 if v = s2 then print s3 else if π[v] = NIL4 then print “no path from s to v exists5 else PRINT-PATH (G, s, π[v])6 print v