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Breadth First Search

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Breadth FirstSearch

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)

- 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={

1

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4

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

- In an undirected graph, (u, v) and (v, u) represents the same edge, adjacency matrix A of an undirected graph is its own transpose
- It can be adapted to represent weighted graphs.

Breadth First Search

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

- 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

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

Q

Except root node, s

For each vertex u V(G)

color [u] WHITE

d[u] ∞

π [s] NIL

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s

Considering s as root node

color[s] GRAY

d[s] 0

π [s] NIL

ENQUEUE (Q, s)

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Q

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r

- 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

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Q

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- 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

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1

0

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Q

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- 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

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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

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u

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Q

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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

r

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u

1

0

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Q

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DEQUEUE v from Q

Adj[v] = r

color [r] ≠ WHITE

color [v] ← BLACK

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u

1

0

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Q

y

DEQUEUE u from Q

Adj[u] = t, x, y

color [t] ≠ WHITE

color [x] ≠ WHITE

color [y] ≠ WHITE

color [u] ← BLACK

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DEQUEUE y from Q

Adj[y] = u, x

color [u] ≠ WHITE

color [x] ≠ WHITE

color [y] ← BLACK

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)

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.

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)

- 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