Topics
Representation of Graphs
Breadth-first Search
Depth-first Search
Topological Sort
Elementary Graph
Algorithms
Source:
Introduction to Algorithms, Second Edition
by Cormen, Leiserson, Rivest & Stein
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Representation of Graphs
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Graphs
Why graphs?
„
Many real world situations have items that are
connected to each other
Example1: Traveling salesman problem
Example2: Electronic circuits
Example3: Control flow in computer programs
Definition: Graph: A graph is a set of vertices
and a set of edges that connect pairs distinct
vertices
Graph, G = (V, E)
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V = Vertices
E = edges
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Vertices and Edges
Types of Graphs
Dense graphs
Vertex:
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Vertex and Node are used interchangeably
Use vertex when discussing graphs
Use node when discussing representations of graph
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Sparse graphs
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Takes the number of vertices and edges and returns a pointer to graph
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Query Operations:
„
edges
„
directed
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Returns the number of vertices
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true if directed, otherwise false
search
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Modifiers:
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insert
remove
sort
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For sparse matrices
Array Adj of lists
Adjacency Matrix
returns the number of edges
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Adjacency Lists
Graph (v, e)
vertices
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Representations of Graphs
Constructors:
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Each edge has an associated weight given by weight function, w
w: E Æ R
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Graph ADT
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Edges have no direction
Weighted graphs
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Edges are one way
Undirected graphs:
A connection between two vertices is named as an
edge, arc, and link
Use edge for graphs and link for data structures
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On the average, the number of edges connected to a vertex is small
|E| << |V|2
Directed Graphs (Digraphs)
Edge:
„
On the average the number of edges connected to a vertex is large
|E| ≈ |V|2
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For dense graphs
Matrix of Boolean values
Columns and Rows represent vertices
If an edge is present then 1 is used, otherwise
0
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Adjacency Lists
Consists of an array Adj of |V| lists
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One list for each vertex
For vertex u, the list is Adj[u]
In each adjacency list
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For each u ∈ V
Contains all vertices immediately adjacent to u
Adjacency list Adj[u] contains all vertices v
edge(u, v) ∈ E
Adj[u] does not follow any order
If weighted graph, w(u, v) is stored with v
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Tutorial
Discuss how you would write a search
algorithm for graphs based on adjacency
lists
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Adjacency Matrices
Some Properties of Adjacency Lists
Graph G = (V, E)
Assume vertices are
numbered 1, 2,…,|V|
Adjacency matrix
representation is a
For a directed graph
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Sum of lengths of all adjacency lists: |E|
If G is an undirected graph
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Sum of lengths of all adjacency lists: 2|E|
For all graphs, memory required is:
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Θ (V + E)
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Matrix A of size |V| x
|V|
A = (aij)
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Breadth First Search
⎧1
a =⎨
ij
⎩0
if (i, j ) ∈ E
otherwise
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Problem
Inputs:
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A graph G = (V, E)
Source vertex s
Output:
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Discover all the vertices in the graph
Methods:
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Breadth-First Search (BFS)
Depth-First Search (DFS)
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BSF: Expanding Discovery
Boundary (1)
BSF: Main Properties
Discovering a vertex:
Explores all the edges of graph G
Computes distance from s to each reachable
vertex
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Undiscovered vertex:
A distance from a vertex to another vertex is the
smallest number of edges
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It produces a breadth-first tree (BFT)
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Root s
The path from s to vertex v in BFT is the shortest path
from s to v in G
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A vertex that has not been visited by the
algorithm
BSF expands frontier between discovered
and undiscovered vertices
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Works on both directed and undirected graphs
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Reaching a vertex for the first time
Discovers all vertices at distance u from s,
before discovering any vertex at distance u+1
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BSF: Expanding Discovery
Boundary (2)
What does breadth first mean?
Using colors:
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Undiscovered vertices – white
Completely discovered – black
Boundary vertices – grey
If a vertex u is black
If v is adjacent to u
Then v is either black or gray
That is all vertices adjacent to a black vertex are
discovered
Vertices change colors in algorithm:
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White Æ Grey Æ Black
How algorithm assigns colors:
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If a vertex u is gray
Initialization – all nodes white
First time discovered – grey
Completely expanded - black
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Expansion Order in BSF
Mechanism for Expanding boundary
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If v is adjacent to u
Then v can be white
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Using Queue for Representing
Boundary
Vertex Fields
Queues hold all gray vertices
We Dequeue the first vertex
Predecessor or Parent (π)
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In BFT, if u is already discovered, and v is in
adjacency list of u
If you discover v, then add it to the tree
u is parent of v
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Expand it Æ new vertices (not black vertices)
Enqueue the new vertices
color
distance, d
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Functional Description of BFS
Initialize all vertices to white color
Initialize queue with source vertex
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color it gray
Dequeue first element
V Å expand it (adjacent vertices)
If white vertex enqueue it
repeat
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Depth First Search
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Topological Sort
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