Graph Introduction , Searching

Graph Theory Basics

- Anil Kishore

Graph

• A collections of objects and pair wise relations between them

• A mathematical structure• Objects are called Vertices ( or Nodes )• Relationship is shown using Edges

Graph

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Vertex Set V = { 1, 2, 3, 4, 5, 6}

Edge Set E = { 12, 14, 34, 45, 46, 25, 56 }

We use N and M for the corresponding sizes|V| = N , |E| = M

Storing a Graph

• How to store a Graph ? The two popular representations are– Adjacency Matrix• N x N matrix A of 0s and 1s

– Adjacency List• A list of neighbors for each of the vertices

Adjacency Matrix

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1 2 3 4 5 61 1 1 0 1 0 0

2 1 1 0 0 1 0

3 0 0 1 1 0 0

4 1 0 1 1 1 1

5 0 1 0 1 1 1

6 0 0 0 1 1 1

A

A[u][v] = 1, if vertex u and vertex v are adjacent

= 0, otherwise

Space : O(N2)

Symmetric Matrix

Adjacency List

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1 : { 2, 4 }2 : { 1, 5 }3 : { 4 }4 : { 1, 3, 5, 6 }5 : { 2, 4, 6 }6 : { 4, 5 }

Space : O(N+2M)

Vertex u : list of all neighbors of u

Directions and Weights

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• Edges and Vertices can have weight e.g.: length of the road, toll gate charge in a city.

• Edges can have direction ( like one-way roads )

Path, Cycle

• A Path of length n-1 is a sequence of vertices u1, u2, … , un such that vertices ui and ui+1 are adjacento e.g. 1 – 2 – 5 – 4 – 3 is a path

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• A Cycle of length n is a sequence of vertices u1, u2, … , un such that vertices ui and ui+1 are adjacent and also u1 and un adjacento e.g. 1 – 2 – 5 – 4 is a cycle

Power of Adjacency Matrix

• Number of paths ( possibly cyclic ) of length K from u to v

F(u, v, K) = Sum of ( F(u, w, k-1) * F(w, v, 1) ) ( for all possible intermediate vertices w )

This is similar to the only computation step in Matrix Multiplication

• Note that F(u, v, 1) = A[u][v]

• AK[u][v] = Number of paths of length exactly Kfrom u to v

Connected, Tree, Complete

• A graph is said to be connected if there exists a path between all pairs of vertices– How many minimum edges we need to make a n vertices

graph connected ?– Cycles introduce redundant edges

• A Tree is a connected graph with out cycles ( acyclic )– A tree on n vertices has exactly (n-1) edges

• A Complete Graph has all possible edges present in it– A complete on n vertices (Kn) has nC2 edges

Traversing a graph

• Visit all the vertices in the graph in a particular order– Depth-first Search (DFS)• visit child nodes before visiting sibling nodes

– Breadth-first Search (BFS)• visit sibling nodes before visiting child nodes

Depth-first Searchalgorithm DFS( u )

// start time of uMark u as ‘visited’FOR each node v Adj.List(∈ u)

IF NOT visited(v) THENpar[v] := uDFS(v)

ENDIFENDFOR//end time of u

END-DFS

• Visit an unvisited neighbor, thus recursively traverse along depth of the graph

• par[v] denotes the first preceding vertex from which vertex v was visited, and defines a DFS tree

• Applications :• Checking connectivity• Finding Connected

Components• Topological ordering

many more…

DFS

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DFS

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DFS

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DFS

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DFS

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DFS

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DFS TREE

DFS

• Recursive algorithm• The active nodes in the recursion are Pushed

and Popped, similar to a Stack• Instead of recursion, can implement using a

Stack data structure

• Complexity : O( N + M )

Breadth-first Searchalgorithm BFS( s )

Mark all vertices u ‘unvisited’Create an empty queue QEnQueue(s, Q)mask s as ‘visited’WHILE NOT Empty(Q) DO

u := DeQueue(Q)FOR each v Adj.List(∈ u) DO IF NOT visited(v)

EnQueue(v, Q)mask v as ‘visited’

ENDFORENDWHILE

End-BFS

• Visit the vertices in the order encountered

• Vertices nearer to s are processed before farther ones

• Applications :• Checking connectivity• Finding Connected

Components• Shortest path in

unweighted graphsmany more…

BFS

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Q :

BFS

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Q : 1

BFS

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Q : u = 1

BFS

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Q : 2, 4u = 1

BFS

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Q : 4u = 2

BFS

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Q : 4, 5u = 2

BFS

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Q : 5u = 4

BFS

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Q : 5, 3, 6u = 4

BFS

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Q : 3, 6u = 5

BFS

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Q : 6u = 3

BFS

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Q : u = 6

BFS

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• Visit the nodes level by level ( level order traversal )• All nodes at level k are the ones with shortest path to s equals to k• Complexity : O( N + M )

s

References

• Introduction to Algorithms– Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein

• www.derekroconnor.net/home/MMS406/

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