Math 381 Discrete Mathematical Modeling

Transcription

1 Math 381 Discrete Mathematical Modeling Sean Griffin Today: -Equipment Replacement Problem -Min Spanning Tree Problem -Clustering

2 Friday s Plan Intro to LPSolve software Download before class (link on course webpage) Bring laptop to class if you have one!

3 Shortest Path Problem Dijkstra s algorithm Last Time:

4 Application: Equipment Replacement Suppose I want to plan out my car purchases for the next years.

5 Application: Equipment Replacement Suppose I want to plan out my car purchases for the next years. Say a new car costs $1, 000. The cost of maintaining a car during a year depends on its age at the beginning of the year:

6 Application: Equipment Replacement Suppose I want to plan out my car purchases for the next years. Say a new car costs $1, 000. The cost of maintaining a car during a year depends on its age at the beginning of the year: To avoid paying a high maintenance cost for an old car, I may trade in my car and purchase a new car. The trade-in price depends on the age of the car. Age of car Annual Maintenance Trade-in Price

7 Application: Equipment Replacement Suppose I want to plan out my car purchases for the next years. Say a new car costs $1, 000. The cost of maintaining a car during a year depends on its age at the beginning of the year: To avoid paying a high maintenance cost for an old car, I may trade in my car and purchase a new car. The trade-in price depends on the age of the car. Goal: Minimize the net cost = (purchasing cost + maintenance cost money from trade-ins), during the next five years. Age of car Annual Maintenance Trade-in Price

8 V = {1,, 3,,, } Equipment Replacement Graph Node i represents the beginning of year i. E the set of directed edges (i, j) for 1 i < j Edge (i, j) represents buying a car at beginning of year i, and keeping it until beginning of year j. Cost c i,j = Total net cost of purchasing a car in beginning of year i, and trading it in at beginning of year j. = maintenance cost incurred in years i through j 1 + cost of purchasing car at beginning of year i trade-in value received at beginning of year j

9 Equipment Replacement Graph c 1 =, , 000 7, 000 = 7, 000 c 13 = (, 000 +, 000) + 1, 000, 000 = 1, 000 c 1 = (, 000 +, 000 +, 000) + 1, 000, 000 = 1, 000 c 3 = (, 000) + 1, 000 7, 000 = 7, The min cost 1 path tells me when to buy a new car. Homework Exercise 1: Compute the minimum cost path on a smaller version of this graph.

10 Minimum Spanning Tree Problem

11 Motivating Problem Suppose the UW has five supercomputers. The distance between each pair of computers is given by the following graph with edge costs: 1 3 The computers must be interconnected by cable. Question: What is the minimum length of cable required?

12 Min Spanning Tree Problem (setup) Let G be a simple undirected graph, vertices V and edges E. We say G is connected if there is a path between any two vertices in the graph. Connected One connected component. Not Connected Two connected components.

13 Min Spanning Tree Problem (setup) A cycle in G is a path of edges, which starts and ends at the same vertex, and no other vertex is repeated along the way. Formally, it is a sequence of edges e 1, e,..., e k, where e 1 = {v 0, v 1 }, e = {v 1, v },..., e k = {v k 1, v 0 }.

14 Min Spanning Tree Problem (setup) A cycle in G is a path of edges, which starts and ends at the same vertex, and no other vertex is repeated along the way. Formally, it is a sequence of edges e 1, e,..., e k, where e 1 = {v 0, v 1 }, e = {v 1, v },..., e k = {v k 1, v 0 }. e e 1 v 0 v 1 v v 3 e e 3

15 Min Spanning Tree (setup) A collection of edges U E is spanning if every vertex v V is on some edge in U.

16 Min Spanning Tree (setup) A collection of edges U E is spanning if every vertex v V is on some edge in U.

17 Min Spanning Tree (setup) A collection of edges U E is spanning if every vertex v V is on some edge in U. A collection of edges U E is a spanning tree if: (1) U is spanning. () U is connected. (3) U does not contain any cycles.

18 Min Spanning Tree (setup) A collection of edges U E is spanning if every vertex v V is on some edge in U. A collection of edges U E is a spanning tree if: (1) U is spanning. () U is connected. (3) U does not contain any cycles. Spanning, but contains a cycle.

19 Min Spanning Tree (setup) A collection of edges U E is spanning if every vertex v V is on some edge in U. A collection of edges U E is a spanning tree if: (1) U is spanning. () U is connected. (3) U does not contain any cycles. Spanning, but contains a cycle. Spanning Tree

20 Kruskal s Algorithm Minimum Spanning Tree Problem Input: Undirected graph G, costs c e R for all e E. Output: A spanning tree T E minimizing c(t ) := e T c e.

21 Kruskal s Algorithm Minimum Spanning Tree Problem Input: Undirected graph G, costs c e R for all e E. Output: A spanning tree T E minimizing c(t ) := e T c e. The solution is to be... GREEDY!

22 Kruskal s Algorithm Minimum Spanning Tree Problem Input: Undirected graph G, costs c e R for all e E. Output: A spanning tree T E minimizing c(t ) := e T c e. Kruskal s Algorithm Input: A connected graph G with costs c e R for all e E. Output: A minimum spanning tree T of G. (1) Sort the edges so that c e1 c e c em () Set T =. (3) For i = 1,..., m, do: If T {e i } does not contain a cycle: Update T := T {e i }.

23 Example 1 3

24 Example

25 Example

26 Example

27 Example

28 Example We don t add the edge 3, since it would create a cycle!

29 Example We don t add the edge 3, since it would create a cycle!

30 Example We don t add the edge 3, since it would create a cycle! Nor edge for the same reason! 1 3

31 Example We don t add the edge 3, since it would create a cycle! Nor edge for the same reason! 1 3

32 Example We don t add the edge 3, since it would create a cycle! Nor edge for the same reason! Finally, we have a spanning tree, so we stop. The minimum cost of a spanning tree is thus = 9. So 9 units of cable are required to connect the supercomputers. 1 3

33 Idea for Correctness of Kruskal s Why does Kruskal s algorithm output a minimal spanning tree?

34 Idea for Correctness of Kruskal s Why does Kruskal s algorithm output a minimal spanning tree? The basic idea is to prove that, if T is the final output of the algorithm, then there are no edge swaps that can improve T.

35 Idea for Correctness of Kruskal s Why does Kruskal s algorithm output a minimal spanning tree? The basic idea is to prove that, if T is the final output of the algorithm, then there are no edge swaps that can improve T. In other words, if you try to swap a more expensive edge for a less expensive edge, T will no longer be a minimal spanning tree.

36 Idea for Correctness of Kruskal s Why does Kruskal s algorithm output a minimal spanning tree? The basic idea is to prove that, if T is the final output of the algorithm, then there are no edge swaps that can improve T. In other words, if you try to swap a more expensive edge for a less expensive edge, T will no longer be a minimal spanning tree. 1 3

37 Idea for Correctness of Kruskal s Why does Kruskal s algorithm output a minimal spanning tree? The basic idea is to prove that, if T is the final output of the algorithm, then there are no edge swaps that can improve T. In other words, if you try to swap a more expensive edge for a less expensive edge, T will no longer be a minimal spanning tree. 1 3

38 Application: Clustering Given: n points in space, the data of the pairwise distances between any two of these points, and a number k. Problem: Divide the points into k clusters so that the minimum distance between items in different groups is maximized.

39 Application: Clustering Given: n points in space, the data of the pairwise distances between any two of these points, and a number k. Problem: Divide the points into k clusters so that the minimum distance between items in different groups is maximized.

40 Application: Clustering Given: n points in space, the data of the pairwise distances between any two of these points, and a number k. Problem: Divide the points into k clusters so that the minimum distance between items in different groups is maximized. C 1 a C c C 3 Want min{a, b, c} to be large. Call this number the separation of the clustering. b

41 Idea of Algorithm: Application: Clustering Create the graph whose vertices are the points, edges between any two of the points. Cost is the distance between two points. Run Kruskal s Algorithm until the set of edges T has k connected components, then stop. The k connected components are the clusters.

42 k = 3

43 k = 3

44 k = 3

45 k = 3

46 k = 3

47 k = 3

48 k = 3

49 Proof of Correctness Another perspective: Find the minimum spanning tree, then delete the k 1 edges with the highest costs. C 1 C d C 3 Say d = distance of the smallest of these deleted edges. Then the spacing between any two of the clusters is at least d. If C is a different clustering of the points with k clusters, there must be two points p and q which are in the same cluster in C but in different clusters in C.

50 Proof of Correctness If C is a different clustering of the points with k clusters, there must be two points p and q which are in the same cluster in C but in different clusters in C. C m C l p C i q Say p, q C i. Since C i is a connected component, there is a path between the two points in the min spanning tree There must be some edge on this path which passes between clusters C m and C l. Each edge on the path must be distance d, so the distance between C m and C l is d. Hence, the separation of the clustering C is at most d.

51 Question What is a specific example of a real-world problem you might solve with a clustering algorithm?

52 Question What is a specific example of a real-world problem you might solve with a clustering algorithm? Computer vision

53 Next Time Intro to LPSolve software Download before class (link on course webpage) Bring laptop to class if you have one! Converting graph problems to 0-1 IP s

54 References Old to New car image: Greedy pig: pic with logo/179/ vector-a-vector-illustration-of-greedy-cartoon-pig-holdingarmful-of-money jpg