CAD Algorithms
Shortest Path Algorithms
Mohammad Tehranipoor
ECE Department
14 September 2010
1
Shortest Path
Problem:
Find the best way of getting from s to t where s and t
are vertices in a graph.
Best: Min (sum of the edge lengths of a path)
On a weighted graph:
f(p)),
where p Є P (P is the set of
Example:
Shortest path = Min (∑
all paths)
Traveling Salesman Problem (TSP)
Fastest path or shortest path are the same.
The length of a path can be actual mileage or time
to drive it.
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1
Assumption
All edges have lengths (weights) that are positive
numbers.
It makes the algorithm a lot easier.
It can be applied to both directed and undirected graph
There are algorithms that handle negative
weights/lengths/costs.
Bellman-Ford Algorithm
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Path from Distance
Objective: Find the shortest path from s to t
s
x
t
Path: d(s,t)
d(s,t) = d(s,x) + d(x,t)
Therefore:
d(s,x) < d(s,t)
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2
Dijkstra’s Algorithm
Given a connected weighted graph G=(V,E), a weight
d:E
R+ and a fixed vertex s in V, find the shortest
path from s to each vertex v in V.
Dijkstra’s algorithm works almost the same as
Prim’s algorithm. The only difference is that it
chooses an edge once at a time instead of vertex to
find the shortest path or minimize d(s,x) + length(x,y).
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Dijkstra’s Algorithm
1: Let T be a single vertex s
2: While (T has fewer than n vertices) // n: total number of vertices
3: {
4: Find edge (x,y)
5:
with x in T and y not in T Minimizing d(s,x)+length(x,y)
7: Add (x,y) to T
8: d(s,y) = d(s,x) + length (x,y)
9: }
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3
Example
Problem: Find the shortest paths
from A to all vertices
A
13
B
8
16
C
10
7
11
17
6
D
E
5
14
F
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7
Cont.
T = {A}
A
13
B
8
16
C
10
7
11
17
6
D
14
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E
5
F
8
4
Cont.
T = {A}
A
13
B
AC
8
AD
13
AE
16
8
16
8
C
10
7
11
17
E
16
13
6
D
14
5
F
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9
Cont.
T = {A, C}
A
13
B
8
16
8
C
10
7
11
17
E
16
13
6
D
14
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5
F
10
5
Cont.
T = {A, C}
A
13
AD
13
ACD
19
ACB
18
AE
16
ACE
15
18
B
8
16
8
C
10
7
11
17
15
13
ACF
E
6
D
14
5
F
25
25
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11
Cont.
T = {A, C, D, E}
A
13
18
B
8
16
8
C
10
7
11
17
13
6
D
14
E
15
5
F
25
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12
6
Cont.
T = {A, C, D, E, B}
A
13
18
B
8
16
8
C
10
7
11
17
13
E
6
D
14
15
5
F
20
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13
Cont.
T = {A, C, D, E, B, F}
A
13
18
AC
8
AD
13
ACE
15
ACB
18
ACEF
20
B
8
16
8
C
10
7
11
17
13
6
D
14
E
15
5
F
20
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7
Prim’s Algorithm
{
T= ;
U = {first vertex};
while (U V)
{
let (u, v) be the lowest cost edge
such that u U and v V - U;
T = T {(u, v)}
U = U {v}
}
}
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Prim’s Algorithm
Problem: Find the shortest paths
from A to all vertices
U = {}
A
5
6
B
1
D
5
5
C
3
E
6
2
4
6
F
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16
8
Cont.
U = {A}
A
A
5
6
1
B
D
5
5
1
B
D
C
C
3
6
2
4
E
E
6
F
F
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Cont.
U = {A, C}
A
A
5
6
1
B
D
1
B
D
5
5
C
C
3
6
E
4
2
4
E
F
6
F
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9
Cont.
U = {A, C, F}
A
A
5
6
1
B
D
D
5
5
1
B
C
C
3
6
E
2
4
2
4
E
6
F
F
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Cont.
T = {A, C, F, D}
A
A
5
6
1
B
D
5
5
B
6
E
2
4
2
4
E
6
F
14 September 2010
D
5
C
C
3
1
F
20
10
Cont.
T = {A, C, F, D, B}
A
A
5
6
1
B
5
5
1
B
D
D
5
C
C
3
6
E
3
2
4
2
4
E
6
F
F
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Cont.
T = {A, C, F, D, B, E}
A
5
6
B
1
D
5
5
C
3
E
6
2
4
6
F
This algorithm is more suitable for MST problem.
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11
CAD Algorithms
Minimum Spanning Tree Algorithms
Mohammad Tehranipoor
ECE Department
14 September 2010
23
MST
MST is a well-solved combinatorial optimization
problem.
Definition:
A tree is a connected graph without cycles.
Properties of Trees:
A graph is a tree if and only if there is only and only one
path joining any two of its vertices.
A connected graph is a tree if and only if it has N
vertices and N-1 edges.
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12
MST (Cont.)
Definition:
A subgraph that spans (reaches out to) all vertices of a
graph is called a spanning subgraph.
A subgraph that is a tree and that spans (reaches out to) all
vertices of the original graph is called a spanning tree.
Among all the spanning trees of a weighted and connected
graph, the one (possibly more) with the least total weight is
called a minimum spanning tree (MST).
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Example
A simple example:
A graph may have many spanning trees.
2
1
5
3
3
4
This graph has 16 spanning trees.
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13
Example: Trees
16 spanning trees:
Tree #: 1
2
16
2
MST
3
1
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Problem
Suppose the edges of the graph have weights.
Problem:
The weights of a tree is sum of weights of its edges.
Different trees may have different weights.
How to find minimum spanning tree?
Solutions:
Heuristic
Greedy
Exhaustive search
…
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14
Why Minimum Spanning Tree?
Example:
Phone Network Design:
Traveling Salesman Problem (TSP):
One has a business with several offices and wants to
lease phone lines to connect them up with each other;
the phone company charges differently to connect
different pair of cities. How he can find a set of lines
that connects all the offices with a minimum total cost.
Another convenient way to solve this problem is to find
the shortest path that visits each point at least once.
VLSI Problem: Global Routing
MST is used in global routing
Steiner Tree is used in detailed (channel) routing
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MST/TSP
In general the MST weight is less than the TSP. In
MST, we are not limited only to paths.
What is path:
Path
Path
Not a path
Not a TSP solution
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15
How to Find MST?
An exhaustive search:
List all spanning trees and find the minimum one. Not
applicable in practice especially in VLSI problems.
Heuristic Algorithms
Greedy Algorithms
Prim’s Algorithm
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Kruskal’s Algorithm
It is easy to understand and the best for solving
problems by hands.
Kruskal’s Algorithm:
Step 1: Sort the edges of graph G in increasing order by
weight
Step 2: Keep a subgraph S of G, initially empty
Step 3: For each edge e in sorted order
If the endpoints of e are disconnected in S
Add e to S
Step 4: Return S
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16
Example
Edge
AB
AC
BD
AD
AE
CE
DE
DF
EF
Weight
1
1
1
2
2
2
2
2
3
S={}
B
1
D
2
1
2
2
A
1
2
F
3
2
C
E
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Cont.
Edge
AB
AC
BD
AD
AE
CE
DE
DF
EF
Weight
1
1
1
2
2
2
2
2
3
Edges: 1, 1, 1, 2, 2, 2, 2, 2, 3
S={AB}
B
1
D
2
1
2
2
A
F
3
1
C
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2
2
E
34
17
Cont.
Edge
AB
AC
BD
AD
AE
CE
DE
DF
EF
Weight
1
1
1
2
2
2
2
2
3
Edges: 1, 1, 1, 2, 2, 2, 2, 2, 3
S={AB, BD}
B
D
1
2
1
2
2
A
F
2
3
1
C
2
E
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Cont.
Edge
AB
AC
BD
AD
AE
CE
DE
DF
EF
Weight
1
1
1
2
2
2
2
2
3
Edges: 1, 1, 1, 2, 2, 2, 2, 2, 3
S={AB, BD, AC}
B
1
D
2
1
2
2
A
F
3
1
C
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2
2
E
36
18
Cont.
Edge
AB
AC
BD AD AE
CE
DE
DF
EF
Weight
1
1
1
2
2
2
3
2
2
Edges: 1, 1, 1, 2, 2, 2, 2, 2, 3
S={AB, BD, AC, AE}
B
D
1
2
1
2
2
A
2
F
3
1
2
C
E
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Cont.
Edge
AB
AC
BD
AD
AE
CE
DE
DF
EF
Weight
1
1
1
2
2
2
2
2
3
Edges: 1, 1, 1, 2, 2, 2, 2, 2, 3
S={AB, BD, AC, AE}
B
1
D
2
1
2
2
A
1
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C
2
2
F
3
E
38
19
Cont.
Edge
AB
AC
BD
AD
AE
CE
DE
DF
EF
Weight
1
1
1
2
2
2
2
2
3
Edges: 1, 1, 1, 2, 2, 2, 2, 2, 3
S={AB, BD, AC, AE}
B
D
1
2
1
2
2
A
F
2
3
1
C
2
E
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Cont.
Edge
AB
AC
BD
AD
AE
CE
DE
DF
EF
Weight
1
1
1
2
2
2
2
2
3
Edges: 1, 1, 1, 2, 2, 2, 2, 2, 3
S={AB, BD, AC, AE, DF}
B
1
D
2
1
2
2
A
2
F
3
1
C
2
E
Total Cost = 7
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20
Example
A graph may have more than one MST.
B
1
D
B
2
1
2
2
A
2
F
C
2
E
Total Cost = 7
D
2
2
2
A
3
1
1
1
2
F
3
1
C
2
E
Total Cost = 7
This algorithm is known as a greedy algorithm because it
chooses the cheapest edge at each step and adds to S.
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Prim’s Algorithm
Step 1: Pick any vertex as a starting vertex. Call it s.
Step 2: Find the nearest neighbor of s (call it p1).
Choose sp1 and p1 such that sp1 is the cheapest
edge in the graph that does not close the selected
edges. Add sp1 to S.
Step 3: Return subgraph S.
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21
Example
Prim’s Algorithm:
4
3
1
6
10
6
7
8
7
6
2
3
3
5
12
3
3
9
4
4
10
11
4
3
1
9
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Cont.
4
3
1
6
10
6
7
8
7
6
2
3
3
5
12
3
3
9
4
4
10
11
4
1
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9
44
22
Cont.
4
3
1
6
10
6
7
8
7
6
2
3
3
5
12
3
3
9
4
4
10
11
4
3
1
9
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Cont.
4
3
1
6
10
6
7
8
7
6
2
3
3
5
12
3
3
9
4
4
10
11
4
1
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9
46
23
Cont.
4
3
1
6
10
6
7
8
7
6
2
3
3
5
12
3
3
9
4
4
10
11
4
3
1
9
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Cont.
4
3
1
6
10
6
7
8
7
6
2
3
3
5
12
3
3
9
4
4
10
11
4
1
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9
48
24
Cont.
4
3
1
6
10
6
7
8
7
6
2
3
3
5
12
3
3
9
4
4
10
11
4
3
1
9
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Cont.
4
3
1
6
10
6
7
8
7
6
2
3
3
5
12
3
3
9
4
4
10
11
4
1
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9
50
25
Cont.
4
3
1
6
10
6
7
8
7
6
2
3
3
5
12
3
3
9
4
4
10
11
4
3
1
9
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Cont.
4
3
1
6
10
6
7
8
7
6
2
3
3
5
12
3
3
9
4
4
10
11
4
1
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9
52
26
Cont.
4
3
1
6
10
6
7
8
7
6
2
3
3
5
12
3
3
9
4
4
10
11
4
3
1
9
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Cont.
4
3
1
6
10
6
7
8
7
6
2
3
3
5
12
3
3
9
4
4
10
11
4
1
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9
54
27
Cont.
4
3
1
6
10
6
7
8
7
6
2
3
3
5
12
3
3
9
4
4
10
11
4
3
1
9
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Cont.
4
3
1
6
10
6
7
8
7
6
2
3
3
5
12
3
3
9
4
4
10
11
4
1
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9
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28
More Algorithms
Boruvka’s Algorithm:
The idea is to do steps like Prim’s algorithm in parallel all over
the graph at the same time.
Hybrid Algorithm:
Combine two of the classical algorithms and do better than
either one alone.
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