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Chapter 9 Graph algorithms

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Title: Data Representation Methods Author: Preferred Customer Last modified by: Bala Ravikumar Created Date: 6/17/1995 11:31:02 PM Document presentation format – PowerPoint PPT presentation

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Title: Chapter 9 Graph algorithms


1
Chapter 9 Graph algorithms
2
Sample Graph Problems
  • Path problems.
  • Connectedness problems.
  • Spanning tree problems.

3
Graph Problem An Example Graph coloring Given
a graph G, assign colors to vertices using as few
colors as possible so that if there an edge
between u and v, colors assigned to u and v must
be different.
How many colors do we need for this graph?
4
Graph Problem An Example
How many colors do we need for this graph?
Three colors are enough, but two is not enough.
5
  • Graph Coloring applications
  • Graph coloring Given a graph G, assign colors to
    vertices using as few colors as possible so that
    if there an edge between u and v, colors assigned
    to u and v must be different.
  • Applications This problem has many different
    applications.
  • map coloring this is the direct application.
    Countries are vertices, and adjacent countries
    (that share a boundary) are connected by an edge.
  • In real maps, adjacent countries are assigned
    different colors.

6
Graph coloring Application 2 scheduling
problem. Given is a list of courses, and for
each course the list of students who want to
register for it. The goal is to find a time slot
for each course (using as few slots as possible).
If a student is registered for courses p and q,
add an edge between p and q. If the resulting
class can be colored with k the number of
available slots (MFW 8 -9 , MWF 9 10 etc. are
various slots), then we can find a schedule that
will allow the students to take all the courses
they want to.
7
Another application of graph coloring
  • Traffic signal design At an intersection of
    roads, we want to install traffic signal lights
    which will periodically switch between green and
    red. The goal is to reduce the waiting time for
    cars before they get green signal.
  • This problem can be modeled as a coloring
    problem. Each path that crosses the intersection
    is a node. If two paths intersect each other,
    there is an edge connecting them. Each color
    represents a time slot at which the path gets a
    green light.

8
Application we will study
  • We will discuss in some detail an algorithm
    (known as Dijkstras algorithm) for finding the
    shortest path in a weighted graph.
  • This algorithm can be used to in applications
    like map-quest. (Map-quest actually uses a
    variant of this algorithm.)

9
Path Finding
  • Path between 1 and 8.

Path length is 20.
10
Another Path Between 1 and 8
Path length is 28.
11
Example Of No Path
  • No path between 2 and 9.

12
Connected Graph
  • Undirected graph.
  • There is a path between every pair of vertices.

13
Example of a graph Not Connected
14
Connected Graph Example
15
Connected Components
16
Connected Component
  • A maximal subgraph that is connected.
  • Cannot add vertices and edges from original graph
    and retain connectedness.
  • A connected graph has exactly 1 component.

17
Communication Network
  • Each edge is a link that can be constructed
    (i.e., a feasible link).

18
Communication Network Problems
  • Is the network connected?
  • Can we communicate between every pair of cities?
  • Find the components.
  • Want to construct smallest number of feasible
    links so that resulting network is connected.

19
Cycles And Connectedness
  • Removal of an edge that is on a cycle does not
    affect connectedness.

20
Cycles And Connectedness
2
3
8
1
10
4
5
9
11
6
7
  • Connected subgraph with all vertices and minimum
    number of edges has no cycles.

21
Tree
  • Connected graph that has no cycles.
  • n vertex connected graph with n-1 edges.
  • A connected graph in which removal of any edge
    makes it unconnected.
  • An cyclic graph in which addition of any edges
    introduces a cycle.

22
Spanning Tree
  • Subgraph that includes all vertices of the
    original graph.
  • Subgraph is a tree.
  • If original graph has n vertices, the spanning
    tree has n vertices and n-1 edges.

23
Minimum Cost Spanning Tree
  • Tree cost is sum of edge weights/costs.

24
A Spanning Tree
2
3
4
8
8
1
10
6
2
4
5
4
4
3
5
9
11
8
5
6
2
7
6
7
  • Spanning tree cost 51.

25
Minimum Cost Spanning Tree
2
3
4
8
8
1
10
6
2
4
5
4
4
3
5
9
11
8
5
6
2
7
6
7
  • Spanning tree cost 41.

26
Graph Representation
  • Adjacency Matrix
  • Adjacency Lists
  • Linked Adjacency Lists
  • Array Adjacency Lists

27
Adjacency Matrix
  • 0/1 n x n matrix, where n of vertices
  • Ai,j 1 iff (i,j) is an edge

0
1
0
1
0
1
0
0
0
1
0
0
0
0
1
1
0
0
0
1
0
1
1
1
0
28
Adjacency Matrix Properties
  • Diagonal entries are zero.
  • Adjacency matrix of an undirected graph is
    symmetric.
  • A(i,j) A(j,i) for all i and j.

29
Adjacency Matrix (Digraph)
  • Diagonal entries are zero.
  • Adjacency matrix of a digraph need not be
    symmetric.

30
Adjacency Matrix
  • n2 bits of space
  • For an undirected graph, may store only lower or
    upper triangle (exclude diagonal).
  • (n-1)n/2 bits
  • O(n) time to find vertex degree and/or vertices
    adjacent to a given vertex.
  • O(1) time to determine if there is an edge
    between two given vertices.

31
Adjacency Lists
  • Adjacency list for vertex i is a linear list of
    vertices adjacent from vertex i.
  • An array of n adjacency lists.

aList1 (2,4) aList2 (1,5) aList3
(5) aList4 (5,1) aList5 (2,4,3)
32
Linked Adjacency Lists
  • Each adjacency list is a chain.

Array Length n of chain nodes 2e
(undirected graph) of chain nodes e (digraph)
33
Weighted Graphs
  • Cost adjacency matrix.
  • C(i,j) cost of edge (i,j)
  • Adjacency lists gt each list element is a pair
    (adjacent vertex, edge weight)

34
Single-source Shortest path problem
  • directed, weighted graph is the input
  • specified source s.
  • want to compute the shortest path from s to all
    the vertices.

35
Example
36
  • Dijkstras algorithm
  • Works when there are no negative weight edges.
  • takes time O(e log n) where e number of edges,
    n number of vertices.
  • suppose n 105, e 106, then the number of
    computations 2 x 107

37
  • Data structures needed
  • Adjacency list rep. of graph
  • a heap
  • some additional structures (e.g. array)

.
38
key operation relaxation on edge
For each node v, the algorithm assigns a value
dv which gets updated and will in the end
become the length of the shortest path from s to
v.
39
implementation of relaxation
40
Dijkstras algorithm
41
Dijkstras algorithm Example
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