Lecture 10 - Graph part 2.pptx,discrete mathemactics

Ta Quang Hung
FACULTY OF INFORMATION TECHNOLOGY
Discrete Mathematics
Fall, 2023
Lecture 10: Graph – Part 2

Lecture 10 The University of New South Wales
Contents
Page 2
2 Dijkstras Algorithm
3 Bellman-Ford Algorithm
1 Shortest Path Algorithms
4 Summary

Discrete Mathematics Page 3
SHORTEST PATH
ALGORITHMS

Introduction
• Perhaps the most intuitive graph-processing
problem is one that you encounter regularly, when
using a map application or a navigation system to
get directions from one place to another. A graph
model is immediate: vertices correspond to
intersections and edges correspond to roads, with
weights on the edges that model the cost, perhaps
distance or travel time. The possibility of one-way
roads means that we will need to consider edge-
weighted digraphs. In this model, the problem is
easy to formulate:
Find the lowest-cost way to get
from one vertex to another
Page 4

Example: Navigation System
Page 5
Vertices correspond to
intersections and edges
correspond to roads.
Weights on the edges that
model the cost, perhaps
distance or travel time.

Example: Geographic Information System (GIS)
Page 6
Multiple layers are
combined to build GIS

Example: Printed Circuit Board (PCB)
Page 7
Autoroute Mode: Find the
shortest path for connections to
minimize resistances.

Example: PCB layout for DDR3 RAM interface
Page 8

Example: Game
Page 9
Base under attack: Troops must
find the shortest path back home.
Game map:
Look like a
maze.

Example: Uber Taxi
Page 10
Fare Estimate: calculate the minimum fee for the trip.

Properties of Shortest Path
• Paths are directed. A shortest path must respect the
direction of its edges.
• The weights are not necessarily distances. We use
examples where vertices are points in the plane and weights
are Euclidean distances, such as the digraph on the facing
page. But the weights might represent time or cost or an
entirely different variable and do not need to be proportional
to a distance at all. We are emphasizing this point by using
mixed-metaphor terminology where we refer to a shortest
path of minimal weight or cost.
• Not all vertices need be reachable. If t is not reachable
from s, there is no path at all, and therefore there is no
shortest path from s to t. For simplicity, our small running
example is strongly connected (every vertex is reachable
from every other vertex).
Page 11

Properties of Shortest Path (con't)
• Negative weights introduce complications. For the moment, we
assume that edge weights are positive (or zero).
• Shortest paths are normally simple. Algorithms ignore zero-weight
edges that form cycles, so that the shortest paths they find have no
cycles.
• Shortest paths are not necessarily unique. There may be multiple
paths of the lowest weight from one vertex to another; we are content to
find any one of them.
• Parallel edges and self-loops may be present. Only the lowest-weight
among a set of parallel edges will play a role, and no shortest path
contains a self-loop (except possibly one of zero weight, which we
ignore). In the text, we implicitly assume that parallel edges are not
present for convenience in using the notation v->w to refer
unambiguously to the edge from v to w.
Page 12

DIJKSTRAS ALGORITHM

Dijkstras Algorithm
Page 14
• This algorithm can’t work with negative edge
weights.
• Dijkstras algorithm runs in O(|n|+|m|log|m|) where n
vertices and m edges.

Dijkstras Algorithm: Example
Page 15
S = {A}
Find path from A to all other vertices?
Set L(A) = 0
Step 0
A
B
C D
E
G
F
3
6
2
7
1
2
4 5
6
8
9
3
3
Solved node
Unsolved node
Solving node
Denote:
• w(A,B) = weighting coefficient in the
path from A to B. So, from the graph,
we have w(A,B) = 3.
• L(Y) = Label of Y. Minimum cost of Y
or the length of the shortest path
from orignating vertice to Y. L(Y) is a
sum of all weighting coefficient and
the cost of all vertices on the shortest
path.
• S: A set of all vertices that the
shortest path go through.

Page 16
S = {A,G}
Check all paths from A to
adjiacent vertices: F, G, B, C
L(A) + w(A,G) = 1
L(A) + w(A,B) = 3
L(A) + w(A,F) = 2
L(A) + w(A,C) = 7
Next vertice is G, add G to set S
Set L(G) = 1
A
B
C D
E
G
F
Solved node
Unsolved node
3
6
2
7
1
2
4 5
6
8
9
3
3
Solving node
0
Step 1

Page 17
S = {A,G,F}
Check all paths from A, G: F, B, E, D,
C.
L(G) + w(G,F) = 5
L(G) + w(G,E) = 10
L(G) + w(G,D) = 9
L(G) + w(G,B) = 3
Next vertice is F, set L(F)=2, add F
to set S:
A
B
C D
E
G
F
Solved node
Unsolved node
3
6
2
7
1
2
4 5
6
8
9
3
3
Solving node
0
1
L(A) + w(A,B) = 3
L(A) + w(A,F) = 2
L(A) + w(A,C) = 7
Step 2

Page 18
A
B
C D
E
G
F
Solved node
Unsolved node
3
6
2
7
1
2
4 5
6
8
9
3
3
Solving node
0
1
2
S = {A,G,F,B}
Check all paths from A, G, F to
adjiacent vertices: B, E, C, D.
L(G) + w(G,E) = 10
L(G) + w(G,D) = 9
L(G) + w(G,B) = 3
Next vertice is B, set L(B) = 3, add B
to set S:
L(A) + w(A,B) = 3
L(A) + w(A,C) = 7
L(F) + w(F,E) = 7
L(F) + w(F,D) = 5
Step 3

Page 19
S = {A,G,F,B,D}
Check all paths from A, G, F, B to
adjiacent vertices: E, C, D.
L(G) + w(G,E) = 10
L(G) + w(G,D) = 9
Next vertice is D, set L(D) = 5, add D
to set S:
L(A) + w(A,C) = 7
L(F) + w(F,E) = 7
L(F) + w(F,D) = 5
L(B) + w(B,C) = 9
A
B
C D
E
G
F
Solved node
Unsolved node
3
6
2
7
1
2
4 5
6
8
9
3
3
Solving node
0
1
2
3
Step 4

Page 20
S = {A,G,F,B,D,C}
Check all paths from A, G, F, B, D to
adjiacent vertices: E, C.
L(G) + w(G,E) = 10
Next vertice is C, set L(C) = 7, add C
to set S:
L(A) + w(A,C) = 7
L(F) + w(F,E) = 7
L(B) + w(B,C) = 9
A
B
C D
E
G
F
Solved node
Unsolved node
3
6
2
7
1
2
4 5
6
8
9
3
3
Solving node
0
1
2
3
5
L(D) + w(D,C) = 8
L(D) + w(D,E) = 11
Step 5

Page 21
S = {A,G,F,B,D,C,E}
Check all paths from A, G, F, B, D, C
to adjiacent vertices: E.
L(G) + w(G,E) = 10
Next vertice is E, set L(E) = 7, add E
to set S:
L(F) + w(F,E) = 7
L(D) + w(D,E) = 11
A
B
C D
E
G
F
Solved node
Unsolved node
3
6
2
7
1
2
4 5
6
8
9
3
3
Solving node
0
1
2
3
5
7
Step 6

Page 22
n A G F B D C E Shortest Path
0 0       A
1 0 (1,A) (2,A) (3,A)  (7,A)  A,G
2 0 (1,A) (2,A) (3,A) (9,G) (7,A) (10,G) A,G,F
3 0 (1,A) (2,A) (3,A) (5,F) (7,A) (7,F) A,G,F,B
4 0 (1,A) (2,A) (3,A) (5,F) (7,A) (7,F) A,G,F,B,D
5 0 (1,A) (2,A) (3,A) (5,F) (7,A) (7,F) A,G,F,B,D,C
6 0 (1,A) (2,A) (3,A) (5,F) (7,A) (7,F) A,G,F,B,D,C,E
0 (1,A) (2,A) (3,A) (5,F) (7,A) (7,F) A,G,F,B,D,C,E
Denote:
• (L,P): where L is label of the vertice, and P is preceding vertice in the path.

BELLMAN-FORD
ALGORITHM

Bellman-Ford Algorithm
Page 24
• This algorithm can work with negative edge weights.
• Can detect a negative cycle.
• Bellman-Ford runs in O(|n|-|m|) where n vertices and
m edges.

Bellman-Ford Algorithm: Example
Page 25
Find path from Z to all other vertices?
U
Z
X Y
V
6
7
8
2
5
-3
-4
7
9
Solved node
Unsolved node
Solving node
Denote:
• w(A,B) = weighting coefficient in the
path from A to B. So, from the graph,
we have w(A,B) = 3.
-2

Page 26
Step 0
U
Z
X Y
V
6
7
8
2
5
-3
-4
7
9
Solved node
Unsolved node
Solving node
-2
 


n Z U X V Y
0 0 ,- ,- ,- ,-
0

Page 27
Step 1
U
Z
X Y
V
6
7
8
2
5
-3
-4
7
9
Solved node
Unsolved node
Solving node
-2
6,Z 

7,Z
n Z U X V Y
0 0 ,- ,- ,- ,-
1 0 6,Z 7,Z ,- ,-
From Z, can go to U, and X. Then assign
labels to U, X by adding cost of Z
(in previous step – step 0) to w(Z,U) and
w(Z,X). We denote
0
L1(U) = 6; P1(U) = Z
L1(X) = 7; P1(X) = Z

Page 28
Step 2
U
Z
X Y
V
6
7
8
2
5
-3
-4
7
9
Solved node
Unsolved node
Solving node
-2
6,Z 4,X
2,U
7,Z
n Z U X V Y
0 0 ,- ,- ,- ,-
1 0 6,Z 7,Z ,- ,-
2 0 6,Z 7,Z 4,X 2,U
0
Recalculate U,X,V,Y by (L,P) in step 2:
L2(V) = min{L1(U)+w(U,V),L1(X)+w(X,V)}
= min{6+5,7-3} = 4, P2(V) = X
L2(Y) = min{L1(U)+w(U,Y), L1(X)+w(X,Y),
= min{6-4,7+9} = 2; P2(Y) = U
L2(U) = min{L1(Z)+w(Z,U),L1(V)+w(V,U)}
= min{0+6, } = 6; P2(U) = Z
L2(X) = min{L1(U)+w(U,X), L1(Z)+w(Z,X),
= min{6+8,0+7} = 7; P2(X) = Z

Page 29
Step 3
U
Z
X Y
V
6
7
8
2
5
-3
-4
7
9
Solved node
Unsolved node
Solving node
-2
2,V 4,X
2,U
7,Z
n Z U X V Y
0 0 ,- ,- ,- ,-
1 0 6,Z 7,Z ,- ,-
2 0 6,Z 7,Z 4,X 2,U
3 0 2,V 7,Z 4,X 2,U
0
= min{6+5,7-3} = 4, P3(V) = 4,X
= min{6-4,7+9} = 2; P3(Y) = 2,U
= min{0+6,4-2} = 2; P3(U) = 2,V
= min{6+8,0+7} = 7; P3(X) = 7,Z

Page 30
Step 4
U
Z
X Y
V
6
7
8
2
5
-3
-4
7
9
Solved node
Unsolved node
Solving node
-2
2,V 4,X
2,U
7,Z
n Z U X V Y
0 0 ,- ,- ,- ,-
1 0 6,Z 7,Z ,- ,-
2 0 6,Z 7,Z 4,X 2,U
3 0 2,V 7,Z 4,X 2,U
4 0 2,V 7,Z 4,X -2,U
0
= min{2+5,7-3} = 4, P4(V) = X
= min{2-4,7+9} = -2; P4(Y) = U
= min{0+6,4-2} = 2; P4(U) = V
= min{2+8,0+7} = 7; P4(X) = Z

Summary
Page 31
2 Dijkstras Algorithm
3 Bellman-Ford Algorithm
1 Shortest Path Algorithms

Lecture 10 - Graph part 2.pptx,discrete mathemactics

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