Minimum Spanning Trees and Clustering By Swee-Ling Tang April 20, 2010 04/20/20101.
Minimum Spanning Trees
21
Minimum Spanning Trees
description
Minimum Spanning Trees. b. 3. 4. f. a. 5. 2. 1. 1. e. 5. 7. 2. c. 9. d. 3. g. h. 4. 有權重的圖. G =( V , E ). 很多圖形演算法的問題都假設其輸入為有權重的圖形 . 這裡我們假設權重都定在邊上面 , 且權重值都為正 , 例如請參考上圖所顯示的圖形. Minimum Spanning Trees (MST). - PowerPoint PPT Presentation
Transcript of Minimum Spanning Trees
Minimum Spanning Trees
Minimum Spanning Trees 2
有權重的圖
很多圖形演算法的問題都假設其輸入為有權重的圖形 .
這裡我們假設權重都定在邊上面 , 且權重值都為正 , 例如請參考上圖所顯示的圖形 .
ab
c d
e
f
g h
35
1 1
95
4
7
32
2
4
G=(V,E)
Minimum Spanning Trees 3
Minimum Spanning Trees (MST)
Find the lowest-cost way to connect all of the points (the cost is the sum of weights of the selected edges).
The solution must be a tree. (Why ?)
A spanning tree : a subgraph that is a tree and connect all of the points.
A graph may contains exponential number of spanning trees.
(e.g. # spanning trees of a complete graph = nn-2.)
Minimum Spanning Trees 4
A High-Level Greedy Algorithm for MST
The algorithm grows a MST one edge at a time and maintains that A is always a subset of some MST.
An edge is safe if it can be safely added to without destroying the invariant.
How to check that T is a spanning tree of G ? How to select a “safe edge” edge ?
A = ; while(T=(V,A) is not a spanning tree of G) { select a safe edge for A ; }
Minimum Spanning Trees 5
MST 基本引理Let V = V1 + V2, (V1,V2 ) = {uv | u V1& v V2 }.
if xy (V1,V2 ) and w(xy) = min {w(uv)| uv (V
1,V2 )}, then xy is contained in a MST.
ab
c d
e
f
g h
35
1 1
95
4
7
32
2
4
V1 V2
Minimum Spanning Trees 6
Kruskal’s Algorithm (pseudo code 1)
A= ;for( each edge in order by nondecreasing weight ) if( adding the edge to A doesn't create a cycle ){ add it to A; if( | A| == n1 ) break; }
How to check that adding an edge does not create a
cycle?
Minimum Spanning Trees 7
Kruskal’s Algorithm ( 例 1/3)
ab
cd
e
f
g h
35
1 1
94
4
7
32
2
4
Minimum Spanning Trees 8
Kruskal’s Algorithm ( 例 2/3)
ab
cd
e
f
g h
35
1 1
94
4
7
32
2
4
Minimum Spanning Trees 9
Kruskal’s Algorithm ( 例 3/3)
ab
cd
e
f
g h
35
1 1
94
4
7
32
2
4
MST cost = 17MST cost = 17
Minimum Spanning Trees 10
Kruskal’s Algorithm (pseudo code 2)
A = ; initial(n); // for each node x construct a set {x}for( each edge xy in order by nondecreasing weight)
if ( ! find(x, y) ) { union(x, y); add xy to A; if( | A| == n1 ) break; }
find(x, y) = true iff. x and y are in the same setunion(x, y): unite the two sets that contain x an
d y, respectively.
find(x, y) = true iff. x and y are in the same setunion(x, y): unite the two sets that contain x an
d y, respectively.
Minimum Spanning Trees 11
Prim’s Algorithm (pseudo code 1)
ALGORITHM Prim(G)// Input: A weighted connected graph G=(V,E)// Output: A MST T=(V, A) VT { v0 } // Any vertex will do; A ; for i 1 to |V|1 do find an edge xy (VT, VVT ) s.t. its weight is minimized among all edges in (VT, VVT ); VT VT { y } ; A A { xy } ;
Minimum Spanning Trees 12
Prim’s Algorithm ( 例 1/8)
ab
cd
e
f
g h
35
1 1
94
4
7
32
2
4
Minimum Spanning Trees 13
Prim’s Algorithm ( 例 2/8)
ab
cd
e
f
g h
35
1 1
94
4
7
32
2
4
Minimum Spanning Trees 14
Prim’s Algorithm ( 例 3/8)
ab
cd
e
f
g h
35
1 1
94
4
7
32
2
4
Minimum Spanning Trees 15
Prim’s Algorithm ( 例 4/8)
ab
cd
e
f
g h
35
1 1
94
4
7
32
2
4
Minimum Spanning Trees 16
Prim’s Algorithm ( 例 5/8)
ab
cd
e
f
g h
35
1 1
94
4
7
32
2
4
Minimum Spanning Trees 17
Prim’s Algorithm ( 例 6/8)
ab
cd
e
f
g h
35
1 1
94
4
7
32
2
4
Minimum Spanning Trees 18
Prim’s Algorithm ( 例 7/8)
ab
cd
e
f
g h
35
1 1
94
4
7
32
2
4
Minimum Spanning Trees 19
Prim’s Algorithm ( 例 8/8)
ab
cd
e
f
g h
35
1 1
94
4
7
32
2
4
MST cost = 17MST cost = 17
Minimum Spanning Trees 20
Prim’s Algorithm (pseudo code 2)
Built a priority queue Q for V with key[u] = uV;key[v0] = 0; [v0] = Nil; // Any vertex will do While (Q ) { u = Extract-Min(Q); for( each v Adj(u) ) if (v Q && w(u, v) < key[v] ) { [v] = u; key[v] = w(u, v); Change-Priority(Q, v, key[v]); }}
Minimum Spanning Trees 21
Minimum Spanning Tree ( 分析 )
Let n = |V( G)|, m =|E( G)|. Execution time of Kruskal’s algorithm: (use unio
n-find operations, or disjoint-set unions)
O(m log m ) = O(m log n )
Running time of Prim’s algorithm: adjacency lists + (binary or ordinary) heap:
O((m+n) log n ) = O(m log n )
adjacency matrix + unsorted list: O(n2)
adjacency lists + Fibonacci heap:O(n log n + m)
![Splitting Payments Locally While Routing Interdimensionally · 2020-05-12 · routing protocol SpeedyMurmurs [27], that combines the infor-mation from multiple spanning trees. Based](https://static.fdocument.pub/doc/165x107/5f5b5cfacdfd50671e43e723/splitting-payments-locally-while-routing-interdimensionally-2020-05-12-routing.jpg)
