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Lecture 27

Steven S. Skiena


A graph G consists of a set of vertices V together with a set E of vertex pairs or edges.

Graphs are important because any binary relation is a graph, so graphs can be used to represent essentially any relationship.

Example: A network of roads, with cities as vertices and roads between cities as edges.

Example: An electronic circuit, with junctions as vertices as components as edges.

To understand many problems, we must think of them in terms of graphs!

The Friendship Graph

Consider a graph where the vertices are people, and there is an edge between two people if and only if they are friends.  

This graph is well-defined on any set of people: SUNY SB, New York, or the world.

What questions might we ask about the friendship graph?

Data Structures for Graphs

There are two main data structures used to represent graphs.

Adjacency MatricesAn adjacency matrix is an tex2html_wrap_inline308 matrix, where M[i,j] = 0 iff there is no edge from vertex i to vertex j  

It takes tex2html_wrap_inline316 time to test if (i,j) is in a graph represented by an adjacency matrix.

Can we save space if (1) the graph is undirected? (2) if the graph is sparse?

Adjacency ListsAn adjacency list consists of a tex2html_wrap_inline320 array of pointers, where the ith element points to a linked list of the edges incident on vertex i.

To test if edge (i,j) is in the graph, we search the ith list for j, which takes tex2html_wrap_inline330 , where tex2html_wrap_inline332 is the degree of the ith vertex.

Note that tex2html_wrap_inline336 can be much less than n when the graph is sparse. If necessary, the two copies of each edge can be linked by a pointer to facilitate deletions.

Tradeoffs Between Adjacency Lists and Adjacency Matrices


Both representations are very useful and have different properties, although adjacency lists are probably better for most problems.

Traversing a Graph

One of the most fundamental graph problems is to traverse every edge and vertex in a graph. Applications include:

For efficiency, we must make sure we visit each edge at most twice.

For correctness, we must do the traversal in a systematic way so that we don't miss anything.

Since a maze is just a graph, such an algorithm must be powerful enough to enable us to get out of an arbitrary maze.

Marking Vertices

The idea in graph traversal is that we must mark each vertex when we first visit it, and keep track of what have not yet completely explored.

For each vertex, we can maintain two flags:

We must also maintain a structure containing all the vertices we have discovered but not yet completely explored.

Initially, only a single start vertex is considered to be discovered.

To completely explore a vertex, we look at each edge going out of it. For each edge which goes to an undiscovered vertex, we mark it discovered and add it to the list of work to do.

Note that regardless of what order we fetch the next vertex to explore, each edge is considered exactly twice, when each of its endpoints are explored.

Correctness of Graph Traversal

Every edge and vertex in the connected component is eventually visited.

Suppose not, ie. there exists a vertex which was unvisited whose neighbor was visited. This neighbor will eventually be explored so we would visit it:

Traversal Orders

The order we explore the vertices depends upon what kind of data structure is used:

The three possible colors of each node reflect if it is unvisited (white), visited but unexplored (grey) or completely explored (black).

Breadth-First Search


for each vertex tex2html_wrap_inline352 do

color[u] = white

tex2html_wrap_inline354 , ie. the distance from s

p[u] = NIL, ie. the parent in the BFS tree

color[u] = grey

d[s] = 0

p[s] = NIL


while tex2html_wrap_inline366 do

u = head[Q]

for each tex2html_wrap_inline370 do

if color[v] = white then

color[v] = gray

d[v] = d[u] + 1

p[v] = u



color[u] = black

Depth-First Search

DFS has a neat recursive implementation which eliminates the need to explicitly use a stack.

Discovery and final times are sometimes a convenience to maintain.


for each vertex tex2html_wrap_inline382 do

color[u] = white

parent[u] = nil

time = 0

for each vertex tex2html_wrap_inline390 do

if color[u] = white then DFS-VISIT[u]

Initialize each vertex in the main routine, then do a search from each connected component. BFS must also start from a vertex in each component to completely visit the graph.


color[u] = grey (*u had been white/undiscovered*)

discover[u] = time

time = time+1

for each tex2html_wrap_inline400 do

if color[v] = white then

parent[v] = u


color[u] = black (*now finished with u*)

finish[u] = time

time = time+1

next up previous
Next: About this document Up: My Home Page

Steve Skiena
Tue Dec 9 21:51:51 EST 1997