Nattee Niparnan. Graph A pair G = (V,E) V = set of vertices (node) E = set of edges (pairs of vertices) V = (1,2,3,4,5,6,7) E = ((1,2),(2,3),(3,5),(1,4),(4,

GRAPH ALGORITHMNattee Niparnan

Graph

A pair G = (V,E) V = set of vertices

(node) E = set of edges

(pairs of vertices)

V = (1,2,3,4,5,6,7) E = ((1,2),(2,3),(3,5),

(1,4),(4,5),(6,7))

1

2

3

45

67

Term you should already know

directed, undirected graph Weighted graph Bipartite graph Tree

Spanning tree Path, simple path Circuit, simple circuit Degree

Representing a Graph

Adjacency Matrix A = |V|x|V| matrix

axy = 1 when there is an edge connecting node x and node y

axy = 0 otherwise

1

2

3

45

0 1 0 1 0

1 0 1 1 0

0 1 0 0 1

1 1 0 0 1

0 0 1 1 0

1 2 3 4 5

123

45

Representing a Graph

Adjacency List Use a list instead of a matrix For each vertex, we have a linked list of

their neighbor

1

2

3

45

1 2 4

2 1 3 4

. . .

Representing a Graph

Incidences Matrix Row represent edge Column represent node

1

2

3

45

1 1 0 0 0

1 0 0 1 0

0 1 0 1 0

0 1 1 0 0

0 0 1 0 1

0 0 0 1 1

1 2 3 4 5

Exploring a Maze

Exploring Problem

Input: A Graph

Maybe as an adjacency matrix A Starting node v

Output: List of node reachable from v

Maybe as an array indexed by a node

Depth-First-Search

procedure explore(G; v)// Input: G = (V;E) is a graph; v V// Output: visited(u) is set to true for all nodes u reachable from v{ visited(v) = true previsit(v) for each edge (v,u) E if not visited(u)

explore(u) postvisit(v)}

Example

Explore(A)

Extend to Graph Traversal

Traversal is walking in the graph We might need to visit each component

in the graph Can be done using explore

Do “explore” on all non-visited node The result is that we will visit every node

What is the difference between just looking into V (the set of vertices?)

Graph Traversal using DFS

procedure dfs(G){

for all v V visited(v) = falsefor all v V if not visited(v)

explore(v)}

Complexity Analysis

Each node is visited once Each edge is visited twice

Why?

O( |V| + |E|)

Another Example

Another Example

Connectivity in Undirected Graph

If A can reach B Then B can reach A

If A can reach B && B can reach C Then A can reach C

Also C can reach A

Divide V into smaller subset of vertices that can reach each other

Connected Component Problem Input:

A graph Output:

Marking in every vertices identify the connected component Let it be an array ccnum, indexed by

vertices

Solution

Define global variable cc

In previsit() ccnum[v] = cc

Before calling each explore cc++

Ordering in Visit

procedure previsit(v)pre[v] = clockclock = clock + 1

procedure postvisit(v)post[v] = clockclock = clock + 1

Ordering in Visit

•The interval for node u is [pre(u),post(u)]•The inverval for u,v is either• Contained • disjointed• Never intersect

DFS in Directed Graph

Type of Edge in Directed Graph

Directed Acyclic Graph (DAG)

A directed Graph without a cycle Has “source”

A node having only “out” edge Has “sink”

A node having only “in” edge

How can we detect that a graph is a DAG What should be the property of “source” and

“sink” ?

Solution

A directed graph is acyclic if and only if it has no back edge

Sink Having lowest post number

Source Having highest post number

Linearization of Graph

for DAG, we can have an ordering of node

Think of an edge as Causality Time-dependency

AB means A has to be done before B

Linearization ordering of node by causality

Linearization

One possible linearizationB,A,D,C,E,F

Order of work that can be done w/o violating the causality constraints

Topological Sorting Problem

Input: A DAG (non-dag cannot be linearized)

Output: A sequence of vertices

If we have a path from A to B A must come before B in the sequence

Topological Sorting

Do DFS List node by post number

(descending)

1,12

2,9

3,8 4,5

6,7

10,11