Depth First Search (DFS)

1
Lecture 8
Topics

• Depth First Search (DFS)
• Breadth First Search (BFS)

Reference Introduction to Algorithm by
Cormen Chapter 23 Elementary Graph Algorithm
2
Graph Search (traversal)

• How do we search a graph?
• At a particular vertices, where shall we go next?
  
• Two common framework
• the depth-first search (DFS)
• the breadth-first search (BFS) and
• In DFS, go as far as possible along a single path
  until reach a dead end (a vertex with no edge out
  or no neighbor unexplored) then backtrack
• In BFS, one explore a graph level by level away
  (explore all neighbors first and then move on)

3
Depth-First Search (DFS)

• The basic idea behind this algorithm is that it
  traverses the graph using recursion
• Go as far as possible until you reach a deadend
• Backtrack to the previous path and try the next
  branch
• The graph below, started at node a, would be
  visited in the following order a, b, c, g, h,
  i, e, d, f, j

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DFS Color Scheme

• Vertices initially colored white
• Then colored gray when discovered
• Then black when finished

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DFS Time Stamps

• Discover time du when u is first discovered
• Finish time fu when backtrack from u
• du lt fu

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DFS Example
sourcevertex
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DFS Example
sourcevertex
d f
1







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DFS Example
sourcevertex
d f
1


2




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DFS Example
sourcevertex
d f
1


2



3
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DFS Example
sourcevertex
d f
1


2



3 4
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DFS Example
sourcevertex
d f
1


2


5
3 4
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DFS Example
sourcevertex
d f
1


2


5 6
3 4
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DFS Example
sourcevertex
d f
1


2 7


5 6
3 4
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DFS Example
sourcevertex
d f
1
8

2 7


5 6
3 4
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DFS Example
sourcevertex
d f
1
8

2 7
9

5 6
3 4
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DFS Example
sourcevertex
d f
1
8

2 7
9 10

5 6
3 4
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DFS Example
sourcevertex
d f
1
8 11

2 7
9 10

5 6
3 4
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DFS Example
sourcevertex
d f
1 12
8 11

2 7
9 10

5 6
3 4
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DFS Example
sourcevertex
d f
1 12
8 11
13
2 7
9 10

5 6
3 4
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DFS Example
sourcevertex
d f
1 12
8 11
13
2 7
9 10
14
5 6
3 4
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DFS Example
sourcevertex
d f
1 12
8 11
13
2 7
9 10
1415
5 6
3 4
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DFS Example
sourcevertex
d f
1 12
8 11
1316
2 7
9 10
1415
5 6
3 4
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DFS Algorithm

• DFS(G)
• for each vertex u in V,
• coloruwhite puNIL
• time0
• for each vertex u in V
• if (coloruwhite)
• DFS-VISIT(u)

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DFS Algorithm (Cont.)

• DFS-VISIT(u)
• colorugray
• time time 1
• du time
• for each v in Adj(u) do
• if (colorv white)
• pv u
• DFS-VISIT(v)
• coloru black
• time time 1 fu time

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DFS Complexity Analysis
Initialization complexity is O(V)
DFS_VISIT is called exactly once for each vertex
And DFS_VISIT scans all the edges which causes
cost of O(E)
Overall complexity is O(V E)
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DFS Application

• Topological Sort
• Strongly Connected Component

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Breadth-first Search (BFS)

• Search for all vertices that are directly
  reachable from the root (called level 1 vertices)
• After mark all these vertices, visit all vertices
  that are directly reachable from any level 1
  vertices (called level 2 vertices), and so on.
• In general, level k vertices are directly
  reachable from a level k 1 vertices

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BFS the Color Scheme

• 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

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An Example
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A
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A
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A
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BFS Algorithm

• BFS(G, s)
• For each vertex u in V s,
• coloru white
• du infinity
• pu NIL
• colors GRAY ds 0 ps
  NIL Q empty queue
• ENQUEUE(Q,s)
• while (Q not empty)
• u DEQUEUE(Q)
• for each v ? Adju
• if colorv WHITE
• then colorv GREY
• dv du 1 pv u
• ENQUEUE(Q, v)
• coloru BLACK

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Example
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Example
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Example
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Example
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Example
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Example
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Example
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Example
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Example
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Example
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BFS Complexity Analysis

• Queuing time is O(V) and scanning all edges
  requires O(E)
• Overhead for initialization is O (V)
• So, total running time is O(VE)

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BFS Application

• Shortest path problem