Transcript ECE 669 Parallel Computer Architecture Lecture 10 Graph Applications ECE669 L10: Graph Applications March 2, 2004

ECE 669
Parallel Computer Architecture
Lecture 10
Graph Applications
ECE669 L10: Graph Applications
March 2, 2004
Outline
° Many applications can be modeled as graphs
° Graphs require determination of shortest paths,
adjacency, and other values
° Efficient parallel algorithms exist which break up
the search
° Used to model many applications
• VLSI CAD
• Mapping
• Sales planning
ECE669 L10: Graph Applications
March 2, 2004
Graph
°
A graph is a pair (V, E), where
• V is a set of nodes, called vertices
• E is a collection of pairs of vertices, called edges
• Vertices and edges are positions and store elements
°
Example:
• A vertex represents an airport and stores the three-letter airport
code
• An edge represents a flight route between two airports and stores
the mileage of the route
SFO
PVD
ORD
LGA
HNL
ECE669 L10: Graph Applications
LAX
DFW
MIA
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Edge Types
°
Directed edge
• ordered pair of vertices (u,v)
• first vertex u is the origin
• second vertex v is the
destination
• e.g., a flight
°
ORD
flight
AA 1206
PVD
ORD
849
miles
PVD
Undirected edge
• unordered pair of vertices (u,v)
• e.g., a flight route
°
Directed graph
• all the edges are directed
• e.g., route network
°
Undirected graph
• all the edges are undirected
• e.g., flight network
ECE669 L10: Graph Applications
March 2, 2004
Applications
°
Electronic circuits
• Printed circuit board
• Integrated circuit
°
°
°
Transportation networks
• Highway network
• Flight network
Computer networks
• Local area network
• Internet
• Web
cslab1a
math.brown.edu
cs.brown.edu
brown.edu
qwest.net
att.net
Databases
• Entity-relationship diagram
cox.net
John
Paul
ECE669 L10: Graph Applications
cslab1b
David
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Terminology
°
End vertices (or endpoints) of
an edge
• U and V are the endpoints
of a
°
Edges incident on a vertex
• a, d, and b are incident on
V
°
Adjacent vertices
• U and V are adjacent
°
Degree of a vertex
• X has degree 5
°
Parallel edges
• h and i are parallel edges
°
a
U
V
b
d
h
X
c
e
g
f
Y
Self-loop
• j is a self-loop
ECE669 L10: Graph Applications
Z
i
W
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j
Terminology
°
Path
• sequence of alternating
vertices and edges
• begins with a vertex
• ends with a vertex
• each edge is preceded and
followed by its endpoints
°
Simple path
• path such that all its vertices
and edges are distinct
°
Examples
• P1=(V,b,X,h,Z) is a simple
path
• P2=(U,c,W,e,X,g,Y,f,W,d,V) is a
path that is not simple
ECE669 L10: Graph Applications
a
U
c
V
b
d
P2
P1
X
e
W
g
f
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Y
h
Z
Terminology
°
°
°
Cycle
• circular sequence of
alternating vertices and edges
• each edge is preceded and
followed by its endpoints
Simple cycle
• cycle such that all its vertices
and edges are distinct
Examples
• C1=(V,b,X,g,Y,f,W,c,U,a,) is a
simple cycle
• C2=(U,c,W,e,X,g,Y,f,W,d,V,a,)
is a cycle that is not simple
ECE669 L10: Graph Applications
a
U
c
V
b
d
C2
X
e
C1
g
W
f
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h
Y
Z
Edge List Structure
°
Vertex object
• element
• reference to position in
vertex sequence
°
u
a
Edge object
v
• element
c
b
d
w
z
• origin vertex object
• destination vertex
object
• reference to position in
edge sequence
°
z
w
v
Vertex sequence
• sequence of vertex
objects
°
u
a
b
Edge sequence
• sequence of edge
objects
ECE669 L10: Graph Applications
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c
d
Traveling Salesman Problem
• Goal is to find the shortest route that starts at one city
and visit each of a list of other cities exactly once before
returning to the first city.
• For n cities there are (n-1)! diferent paths starting and
ending in city 1.
• In practice an exact solution is infeasible except for
small values of n.
• Many approximation algorithms are developed.
ECE669 L10: Graph Applications
March 2, 2004
Backtracking Algorithm
One sequential solution is to examine all feasible paths.
A path is feasible if it is not longer than the best complete path that
has been determined so far.
We start with city 1, there are n-1 possible cities to visit next.
From each of these, there are n-2 possible cities to visit and so on.
1
4
2
3
4
3
4
3
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4
2
3
4
2
3
2
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Backtracking Algorithm
The standard method to examine all the paths in a tree
is to use depth-first search method (DFS).
There is no need to consider a path that is known to be
longer than the shortest completed path that has been
found so far.
1
4
2
3
4
3
4
3
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2
3
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Parallel Solution
• The paths are independent; thus, we could examine all of
them in parallel.
• The approach is to provide a fixed number of slave
processes that share a pool of tasks.
• Each task consists of a partial path, the number of cities
visited on a path, and the path length.
• Each process takes a partial path and extends it with
every city which has not been considered.
• If the length of the path is longer than the length of the
shortest complete path, the path is discarded.
ECE669 L10: Graph Applications
March 2, 2004
Traveling Salesman
° Example
2
•
1
2
4
1
5
•
2
•
3
•4
7
2
6
2
5
3•
•6
1
6
° Find shortest tour: E.g. (1 2 5 6 3 4) and then back
to 1
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Exhaustive Search
1 ,c=0
•
1
6
2
c=5
3
•
c=3
2 ,c=1
•
4
2
4
•
2
7
6 ,c=6
•
3 ,c=4
•
4 ,c=7
•
• 5,c=3
XX
•
° Find lowest c path, •
•
•
X
• But (n - 1) ! tours!!!
2
• Can exploit parallelism though!
ECE669 L10: Graph Applications
X
2
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Branch and Bound Algorithms - Prune!
° Naive
• 1,c=0
1
•
2,c=1
2
•
3,c=2
ECE669 L10: Graph Applications
7
•
4,c=7
6
•
6,c=6
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Branch and Bound Algorithms - Prune!
°
Naive
• 1,c=0
1
•
2,c=1
•
3,c=2
4
2
3
c=5
2
4
c=3
2
7
•
4,c=7
6
•
6,c=6
5
c=3
• (1, 2 ...) Most promising, follow that lead -greedy
ECE669 L10: Graph Applications
March 2, 2004
Branch and Bound Algorithms - Prune!
Naive
2,c=1
•
1
1,c=0
•
2
7
4
2
3
c=5
•
2
6
3,c=2
•
•
4,c=7
4
c=3
5
c=3
4
2•
c=6
6
4
•
c=8
6,c=6
1
5
5
•
c=7
6
•
c=3
• (1, 2 ...) Most promising, follow that lead -greedy
• (1, 3 ...) Most promising
ECE669 L10: Graph Applications
March 2, 2004
Branch and Bound Algorithms - Prune!
2,c=1
•
2
7
4
2
3
c=5
1,c=0
•
1
4
•
c=3
5
2
3,c=2
c=3
4
6
•
•
6,c=6
•
4,c=7
6
5
1
6
• 3 c =9
2•
c=6
•
c=8
4
•
c=7
5
•6
c=3
5
• 5 c =14
2
• 6 c =16
5
•1
c =21
• (1, 2 ...) Most promising, follow that lead -greedy
• (1, 3 ...) Most promising
• Continue ... till a tour is found
ECE669 L10: Graph Applications
March 2, 2004
Branch and Bound Algorithms - Prune!
2,c=1
•
4
3
c=5
4
•
c=3
6
5
• 5 c=14
2
• 6 c=16
5
6
•
•
6,c=6
•
4,c=7
c=3
4
6
2
•
5
•4
c=8
2•
c=6
• 3 c=9
•1
7
2
5
1,c=0
2
3,c=2
2
•
1
(1 2 4 3 5 6)
BOUND: 21
19
1
•5
c=7
•6
c=3
6 c=5
1
•
3 c=6
6
• 4 c=12
c=21
7
• 1 c=19
• Continue ... ‘til a tour is found
ECE669 L10: Graph Applications
March 2, 2004
Representing Graphs
Representing a graph in an algorithm may be accomplished via
two different methods: adjacency matrix and linked list.
1. Consider a graph with n vertices that are numbered 1,2,…, n.
The adjacency matrix of this graph can be defined as n x n
matrix A with the following properties:

1 if (v(I), v(j))
E
A(i, j) =
0 otherwise.
ECE669 L10: Graph Applications
March 2, 2004
Better Algorithm, but basically Branch and Bound
° Little et al.
° Basic Ideas:
2
Cost matrix
4
2
2
1
1
2
•
4
3
4
9
6
1
2
6
9
3
1
6
4
2
2
•
6
•
2
3
1
2
•
4
4
2
•
2
•
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Better Algorithm, but basically Branch and Bound
Little et al.
2
2
1
1
•
2
2
0
4
•
3
4
2
5
9
6
6
•
4
4
2
2
1
2
6
9
3
1
6
4
3
1
2
4
2
•
•
4
2
•
4
2
To
•
Notion of reduction:
2
4
• Subtract same value from each row or column
1
•
•3
9
6
At least
ECE669 L10: Graph Applications
March 2, 2004
4
•
4
Better Algorithm, but basically Branch and Bound
Little et al.
2
2
1
1
•
0
3
4
2
5
4
4
9
4
1
2
3
4
2
1
2
1
1
0
1
•
6
2
2
6
9
•
•
3
1
6
2
2
6
4
4
4
•
2
•
2
To
From 2
•
2
2
4
1
3
1
•
2
4
ECE669 L10: Graph Applications
At least 1
•3
9
6
At least
4
•
4
March 2, 2004
Better Algorithm, but basically Branch and Bound
Little et al.
2
2
1
1
•
0
3
4
2
5
4
4
9
4
1
2
3
4
2
1
2
1
1
0
1
•
6
2
2
6
9
•
•
3
1
6
2
2
6
4
4
4
•
2
•
2
To
From 2
•
2
2
4
1
3
1
•
2
4
ECE669 L10: Graph Applications
At least 1
•3
9
6
At least
4
•
4
March 2, 2004
Better Algorithm
2
1
2
0
•
1
0
21
3
2
5
4
4
6
64
•
4
5
•
0
9
4
2
1
4
3
4
10
0
•
2
1
•
21
42
1
•
20
2
2
9
1
4
2
1
3
1
6
1
2
6
4
To
From 2
•
2
•
2
4
4
1
1
3
•3
9
ECE669 L10: Graph Applications
Reduce all
rows & columns
10 is cost of opt. tour
6
•
4
At least 1
2
2
1+1+2+1+1+4
At least 4
March 2, 2004
Further Analysis
In other words, the Traveling Salesman Problem is stated as:
Given a set of vertices and non-negative cost C(I,J) associated
with each pair of vertices I and J, find a circuit containing
every vertex in the graph so that the cost of the entire path is
minimized.
The problem can be classified as
• Symmetric Salesman Problem – C(I,J)=C(J,I) for all I,J
• Asymmetric Traveling Salesman Problem - C(I,J)
for all I, J.
ECE669 L10: Graph Applications
March 2, 2004

C(J,I)
Another Example
1
3
5
9
8
2
5
5
2
6
9
3
4
3
4
7
4
8
2
5
1
7
6
2
4
6
3
ECE669 L10: Graph Applications
7
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Graph Coloring Problem
Let G be undirected graph and let c be an integer.
Assignment of colors to the vertices or edges such that
no two adjacent vertices are to be similarly colored.
We want to minimize the number of colors used.
The smallest c such that a c-coloring exists is called the
graph’s chromatic number and any such c-coloring is an
optimal coloring.
ECE669 L10: Graph Applications
March 2, 2004
Coloring of Graph
1. The graph coloring optimization problem: find the
minimum number of colors needed to color a graph.
2. The graph coloring decision problem: determine if
there exists a coloring for a given graph which uses
at most m colors.
Two colors
ECE669 L10: Graph Applications
No solution with
two colors
March 2, 2004
Coloring of Graphs
Practical applications: scheduling, time-tabling, register
allocation for compilers, coloring of maps.
A simple graph coloring algorithm - choose a color and an
arbitrary starting vertex and color all the vertices that can be
colored with that color.
Choose next starting vertex and next color and repeat the
coloring until all the vertices are colored.
Four colors
ECE669 L10: Graph Applications
Three colors are enough
March 2, 2004
Summary
° Many applications can be modeled as graphs
° Graphs require determination of shortest paths,
adjacency, and other values
° Efficient parallel algorithms exist which break up
the search
° Used to model many applications
• VLSI CAD
• Mapping
• Sales planning
ECE669 L10: Graph Applications
March 2, 2004