The Minimum Cost Spanning Tree Problem

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Presentation transcript:

The Minimum Cost Spanning Tree Problem Lecture 8 The Minimum Cost Spanning Tree Problem Obtain a network, and use the same network to illustrate the shortest path problem for communication networks, the max flow problem, the minimum cost flow problem, and the multicommodity flow problem. This will be a very efficient way of introducing the four problems. (Perhaps under 10 minutes of class time.)

a

The Greedy Algorithm animation

The Greedy Algorithm in Action 35 10 30 15 25 40 20 17 8 11 21 1 2 3 4 5 6 7 2 4 6 1 3 5 7

The Greedy Algorithm in Action 10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

The Greedy Algorithm in Action 10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

The Greedy Algorithm in Action 10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

The Greedy Algorithm in Action 10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

The Greedy Algorithm in Action 10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

The Greedy Algorithm in Action 10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

The Greedy Algorithm in Action 10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

The Greedy Algorithm in Action 10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

The Greedy Algorithm in Action 10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

The Greedy Algorithm in Action 10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

The Greedy Algorithm in Action 10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

The Greedy Algorithm in Action 10 10 8 8 2 4 4 6 2 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11

Correctness

Exchange Property

Self-Reducibility

Data Structures “Disjoint Sets”

Kruskal Algorithm with Data Structure

The Greedy Algorithm in Action Node 1 2 3 4 5 6 7 First 1 2 3 4 5 4 7 10 10 8 8 2 4 2 4 6 6 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11 root node 40

The Greedy Algorithm in Action Node 1 2 3 4 5 6 7 First 1 4 3 4 5 4 7 10 10 8 8 2 2 4 2 4 6 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 5 7 15 15 11 41

The Greedy Algorithm in Action Node 1 2 3 4 5 6 7 First 1 4 3 4 5 4 5 10 10 8 8 2 4 2 4 6 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 7 3 5 7 15 15 11 42

The Greedy Algorithm in Action Node 1 2 3 4 5 6 7 First 1 4 5 4 5 4 5 10 10 8 8 2 4 6 2 4 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 3 5 7 7 3 5 7 15 15 11 43

The Greedy Algorithm in Action Node 1 2 3 4 5 6 7 First 1 4 4 4 4 4 4 10 10 8 8 2 4 6 2 4 6 35 35 15 15 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 5 7 3 3 5 5 7 7 3 5 7 15 15 11 44

The Greedy Algorithm in Action Node 1 2 3 4 5 6 7 First 4 4 4 4 4 4 4 10 10 8 8 2 4 6 2 4 6 35 35 15 15 1 1 1 25 25 20 20 30 30 17 17 21 21 40 40 11 3 3 5 7 5 7 3 5 7 15 15 11 45

Prim’s algorithm animation

Correctness The cut-optimality Condition

Prim’s Algorithm in Action 10 8 2 2 4 4 6 6 35 15 1 1 25 20 30 17 21 40 11 3 5 7 3 5 7 15 The minimum cost arc from yellow nodes to green nodes can be found by placing arc values in a priority queue.

Prim’s Algorithm in Action 25 10 10 8 2 2 2 4 4 6 6 35 35 15 1 1 25 20 30 17 21 40 11 3 5 7 3 5 7 15

Prim’s Algorithm in Action 8 20 30 21 8 10 10 10 2 2 2 4 4 4 6 6 35 35 15 1 1 25 25 30 20 17 21 40 11 3 5 7 3 5 7 15

Prim’s Algorithm in Action 8 8 8 10 10 10 2 2 2 4 4 4 6 6 6 35 35 17 15 15 1 1 25 25 30 30 17 20 20 21 21 40 11 3 5 7 3 5 7 15

Prim’s Algorithm in Action 8 8 8 10 10 10 2 2 2 4 4 4 6 6 6 35 35 15 15 15 1 1 25 25 30 30 17 17 20 20 21 21 40 15 11 11 3 5 5 7 3 5 7 15

Prim’s Algorithm in Action 8 8 8 10 10 10 2 2 2 4 4 4 6 6 6 35 35 15 15 15 1 1 25 25 30 30 17 17 20 20 21 21 40 15 11 11 3 5 5 7 3 5 7 15

Prim’s Algorithm in Action 8 8 8 10 10 10 2 2 2 4 4 4 6 6 6 35 35 15 15 15 1 1 25 25 30 30 17 17 20 20 21 21 40 11 11 11 3 5 5 7 7 3 5 7 15 15

Prim’s Algorithm in Action 8 8 8 10 10 10 2 2 2 4 4 4 6 6 6 35 35 15 15 15 1 1 25 25 30 30 17 17 20 20 21 21 40 11 11 11 3 3 5 5 7 7 3 5 7 15 15 15

Prim’s Algorithm in Action 8 8 8 10 10 10 2 2 2 4 4 4 6 6 6 35 35 15 15 15 1 1 25 25 30 30 17 17 20 20 21 21 40 11 11 11 3 3 5 5 7 7 3 5 7 15 15 15

Prim’s Algorithm in Action 8 8 8 10 10 10 2 2 2 4 4 4 6 6 6 35 35 15 15 15 1 1 25 25 30 30 17 17 20 20 21 21 40 11 11 11 3 3 5 5 7 7 3 5 7 15 15 15

Sollin’s Algorithm animation

Correctness Cut-optimality Condition

Sollin’s Algorithm in Action 10 8 2 2 2 4 4 4 6 6 6 35 15 1 1 1 25 20 30 17 21 40 3 3 5 5 7 7 3 5 11 7 15 Treat all nodes as singleton components, and then select the min cost arc leaving the component.

Sollin’s Algorithm in Action 10 8 2 2 4 4 6 6 6 35 15 1 1 25 20 30 17 21 40 3 3 5 5 7 7 3 5 15 11 7 Find the min cost edge out of each component

Appendix: Priority Queue

Priority Queue A priority queue is a data structure for maintaining a set of elements, each with an associated value, called a key. A min-priority queue supports the following operations: Insert(S,x), Minimum(S), Extract-Min(S), Increase-Key(S,x,k). Min-Heap can be used for implementing min-priority queue.