第07章加权图WeightedGraphs学习资料
相关主题
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
We store with each vertex v a label d(v) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent vertices
At each step
Example:
In a flight route graph, the weight of an edge represents the distance in miles between the endpoint airports
HNL
ORD SFO
LAX
DFW
Weighted Graphs
PVD LGA
Weighted Graphs
MIA
4
Dijkstra’s Algorithm
The distance of a vertex v from a vertex s is the length of a shortest path between s and v
Dijkstra’s algorithm computes the distances of all the vertices from a given start vertex s
Example:
Shortest path between Providence and Honolulu
Applications
Internet packet routing
Flight reservations
Driving directions
SFO
ORD
PVD
LGA
HNL
LAX
DFW
6
Edge Relaxation
Consider an edge e = (u,z)
such that
u is the vertex most recently
added to the cloud
s
z is not in the cloud
The relaxation of edge e updates distance d(z) as follows:
We add to the cloud the vertex u outside the cloud with the smallest distance label, d(u)
We update the labels of the vertices adjacent to u
Weighted Graphs
Weighted Graphs
2
Weighted Graphs
In a weighted graph, each edge has an associated numerical value, called the weight of the edge
Edge weights may represent, distances, costs, etc.
d(z) min{d(z),d(u) + weight(e)}
s
d(u) = 50
u
e
d(z) = 75
z
d(u) = 50
u
e
d(z) = 60
z
Weighted Graphs
7
Example
8
8 B
7
0 A
2 2
C
4
1
4 D
3 2E
9 F5
8
8 B
7
5 2E
0 A
2 2
C
39
4
1
3 D
8 F5
Weighted Graphs
8
8 B
7
5 2E
0 A
2 2
C
39
4
1
3 D
8 F5Fra Baidu bibliotek
Weighted Graphs
1
Outline and Reading
Weighted graphs (§7.1) Dijkstra’s algorithm (§7.1.1) The Bellman-Ford algorithm (§7.1.2) Shortest paths in dags (§7.1.3) All-pairs shortest paths (§7.2.1) Shortest Path via Matrix Multiplication (§7.2.2) – No slide on this section- too mathematical for slides! Minimum Spanning Trees (§7.3) The Prim-Jarnik Algorithm (§7.3.2) Kruskal's Algorithm (§7.3.1) Baruvka's Algorithm (§7.3.3)
MIA
3
Shortest Path Problem
Given a weighted graph and two vertices u and v, we want to find a path of minimum total weight between u and v.
Length of a path is the sum of the weights of its edges.
Assumptions:
the graph is connected
the edges are undirected
the edge weights are nonnegative
We grow a “cloud” of vertices, beginning with s and eventually covering all the vertices
9
Dijkstra’s Algorithm
A priority queue stores the vertices outside the cloud
Key: distance
Element: vertex
Locator-based methods
insert(k,e) returns a locator
8
8 B
7
0 A
2 2
C
53 9 2E
4
1
3 D
8
7 B
7
0 A
2 2
C
11 F5
53 9 2E
Weighted Graphs
4
1
3 D
8 F5
8
Example (cont.)
8
7 B
7
5 2E
0 A
2 2
C
39
4
1
3 D
8 F5
8
7 B
7
5 2E
0 A
2 2
C
39
4
1
3 D
8 F5
Weighted Graphs
At each step
Example:
In a flight route graph, the weight of an edge represents the distance in miles between the endpoint airports
HNL
ORD SFO
LAX
DFW
Weighted Graphs
PVD LGA
Weighted Graphs
MIA
4
Dijkstra’s Algorithm
The distance of a vertex v from a vertex s is the length of a shortest path between s and v
Dijkstra’s algorithm computes the distances of all the vertices from a given start vertex s
Example:
Shortest path between Providence and Honolulu
Applications
Internet packet routing
Flight reservations
Driving directions
SFO
ORD
PVD
LGA
HNL
LAX
DFW
6
Edge Relaxation
Consider an edge e = (u,z)
such that
u is the vertex most recently
added to the cloud
s
z is not in the cloud
The relaxation of edge e updates distance d(z) as follows:
We add to the cloud the vertex u outside the cloud with the smallest distance label, d(u)
We update the labels of the vertices adjacent to u
Weighted Graphs
Weighted Graphs
2
Weighted Graphs
In a weighted graph, each edge has an associated numerical value, called the weight of the edge
Edge weights may represent, distances, costs, etc.
d(z) min{d(z),d(u) + weight(e)}
s
d(u) = 50
u
e
d(z) = 75
z
d(u) = 50
u
e
d(z) = 60
z
Weighted Graphs
7
Example
8
8 B
7
0 A
2 2
C
4
1
4 D
3 2E
9 F5
8
8 B
7
5 2E
0 A
2 2
C
39
4
1
3 D
8 F5
Weighted Graphs
8
8 B
7
5 2E
0 A
2 2
C
39
4
1
3 D
8 F5Fra Baidu bibliotek
Weighted Graphs
1
Outline and Reading
Weighted graphs (§7.1) Dijkstra’s algorithm (§7.1.1) The Bellman-Ford algorithm (§7.1.2) Shortest paths in dags (§7.1.3) All-pairs shortest paths (§7.2.1) Shortest Path via Matrix Multiplication (§7.2.2) – No slide on this section- too mathematical for slides! Minimum Spanning Trees (§7.3) The Prim-Jarnik Algorithm (§7.3.2) Kruskal's Algorithm (§7.3.1) Baruvka's Algorithm (§7.3.3)
MIA
3
Shortest Path Problem
Given a weighted graph and two vertices u and v, we want to find a path of minimum total weight between u and v.
Length of a path is the sum of the weights of its edges.
Assumptions:
the graph is connected
the edges are undirected
the edge weights are nonnegative
We grow a “cloud” of vertices, beginning with s and eventually covering all the vertices
9
Dijkstra’s Algorithm
A priority queue stores the vertices outside the cloud
Key: distance
Element: vertex
Locator-based methods
insert(k,e) returns a locator
8
8 B
7
0 A
2 2
C
53 9 2E
4
1
3 D
8
7 B
7
0 A
2 2
C
11 F5
53 9 2E
Weighted Graphs
4
1
3 D
8 F5
8
Example (cont.)
8
7 B
7
5 2E
0 A
2 2
C
39
4
1
3 D
8 F5
8
7 B
7
5 2E
0 A
2 2
C
39
4
1
3 D
8 F5
Weighted Graphs