Midterm Review Dr. Bernard Chen Ph.D. University of Central Arkansas Spring 2011.
Graph Dr. Bernard Chen Ph.D. University of Central Arkansas.
54
Graph Dr. Bernard Chen Ph.D. University of Central Arkansas
Transcript of Graph Dr. Bernard Chen Ph.D. University of Central Arkansas.
Graph
Dr. Bernard Chen Ph.D.University of Central Arkansas
Graph Algorithms
Graphs and Theorems about Graphs
Graph Algorithms minimum spanning tree
What can graphs model?
Cost of wiring electronic components together.
Shortest route between two cities. Finding the shortest distance
between all pairs of cities in a road atlas.
What is a Graph?
Informally a graph is a set of nodes joined by a set of lines or arrows.
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Directed Graph A directed graph is a pair ( V, E ),
where the set V is a finite set and E is a binary relation on V .
The set V is called the vertex set of G and the elements are called vertices.
The set E is called the edge set of G and the elements are edges
Directed Graph
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4 5 6 V = { 1, 2, 3, 4, 5, 6, 7 }| V | = 7
E = { (1,2), (2,2), (2,4), (4,5), (4,1), (5,4),(6,3) }| E | = 7
Self loop
7 Isolated node
Undirected Graph
An undirected graph G = ( V , E ) , but unlike a digraph the edge set E consist of unordered pairs.
We use the notation (a, b ) to refer to a directed edge, and { a, b } for an undirected edge.
Undirected Graph
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D E F
B C V = { A, B, C, D, E, F } |V | = 6
E = { {A, B}, {A,E}, {B,E}, {C,F} } |E | = 4
Some texts use (a, b) also for undirected edges. So ( a, b ) and ( b, a ) refers to the same edge.
Degree
Degree of a Vertex in an undirected graph is the number of edges incident on it.
In a directed graph , the out degree of a vertex is the number of edges leaving it and the in degree is the number of edges entering it.
Degree
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D E F
B C The degree of B is 2.
Degree
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The in degree of 2 is 2 andthe out degree of 2 is 3.
Weighted Graph
A weighted graph is a graph for which each edge has an associated weight, usually given by a weight function w: E R.
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Weighted Graph
Implementation of a Graph
Adjacency-list representation of a graph G = ( V, E ) consists of an array ADJ of |V | lists, one for each vertex in V. For each u V , ADJ [ u ] points to all its adjacent vertices.
Implementation of a Graph
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Implementation of a Graph
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Adjacency lists
Advantage: Saves space for sparse graphs. Most
graphs are sparse.
“Visit” edges that start at v Must traverse linked list of v Size of linked list of v is degree(v) (degree(v))
Adjacency-matrix-representation Adjacency-matrix-
representation of a graph G = ( V, E) is a |V | x |V | matrix A = ( aij )
such that aij = 1 (or some Object) if (i, j ) E 0 (or null) otherwise.
Adjacency-matrix-representation
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01234
0 1 0 0 11 0 1 1 10 1 0 1 0
0 1 1 0 11 1 0 1 0
Adjacency-matrix-representation
0 1 0 0 10 0 1 1 10 0 0 1 0
0 0 0 0 10 0 0 0 0
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01234
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Adjacency-matrix-representation
Advantage: Saves space on pointers for dense
graphs, and on small unweighted graphs using 1 bit
per edge. Check for existence of an edge (v, u)
(adjacency [i] [j]) == true?) So (1)
Graph Algorithms
Graphs and Theorems about Graphs
Graph Algorithms minimum spanning tree
Minimum Spanning Tree
Example of MST
Problem: Laying Telephone Wire
Central office
Wiring: Naïve Approach
Central office
Expensive!
Wiring: Better Approach
Central office
Minimize the total length of wire connecting the customers
Growing an MST: general idea
GENERIC-MST(G,w)
1. A{}2. while A does not form a spanning tree3. do find an edge (u,v) that is safe
for A4. A A U {(u,v)}5. return A
Tricky part
How do you find a safe edge? This safe edge is part of the minimum
spanning tree
Algorithms for MST
Prim’s Grow a MST by adding a single edge at a
time
Kruskal’s Choose a smallest edge and add it to the
forest If an edge is formed a cycle, it is
rejected
Prim’s greedy algorithm Start from some (any) vertex.
Build up spanning tree T, one vertex at a time.
At each step, add to T the lowest-weight edge in G that does not create a cycle.
Stop when all vertices in G are touched
Prim’s MST algorithm
Example
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Min EdgePick a root
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= in heap
Min Edge = 1
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Min Edge = 2
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Min Edge = 2
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Min Edge = 3
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Min Edge = 4
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Min Edge = 3
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Min Edge = 4
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Min Edge = 6
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Example II
Kruskal’s Algorithm
Choose the smallest edge and add it to a forest
Keep connecting components until all vertices connected
If an edge would form a cycle, it is rejected.
Example
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Min Edge = 1
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Min Edge = 2
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Min Edge = 2
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Min Edge = 3
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Now have 2 disjoint components:
ABFG and CH
Min Edge = 3
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Min Edge = 4
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Two components now merged into one.
Min Edge = 4
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Min Edge = 5
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Min Edge = 6
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