Lesson 5.5. Minimum Spanning Trees. Explore This

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1 Lesson. St harles ounty Minimum Spanning Trees s in many of the previous lessons, this lesson focuses on optimization. Problems and applications here center on two types of problems: finding ways of connecting the vertices of a graph with the least number of edges and finding ways of connecting them with the least number of edges that have the smallest total weight. xplore This n making earthquake preparedness plans, the St. harles ounty government needs a design for repairing the county roads in case of an emergency. igure.1 is a map of the towns in the county and the existing major roads between them. evise a plan that repairs the least number of roads but keeps a route open between each pair of towns. igure.1. The towns in St. harles ounty. Wentzville O'allon Peruque Orchard arm St. Peters St. harles New Melle arvester ugusta

2 hapter More raphs, Subgraphs, and Trees xamine your graph. f it connects each of the towns (vertices) and has no cycles, you ve found a spanning tree. spanning tree of a connected graph is a tree that is a subgraph of and contains every vertex of. spanning tree of the graph in igure.1 would model the minimum number of roads (edges) needed to connect each town in case of an emergency. ompare your plan with the other plans developed in your class. They should all contain the same number of edges but not necessarily the same edges. t is possible for a graph to have many different spanning trees. nd as you may have guessed, for a graph that is not connected, no spanning tree is possible. One systematic way to find a spanning tree for a graph is to delete an edge from each cycle in the graph. Unfortunately, this is not an easy procedure for a very large graph. ut there are other ways of finding a spanning tree for a graph if one exists. One such method that can be easily adapted to computers is called the breadth-first search algorithm. readth-irst Search lgorithm for inding Spanning Trees 1. Pick a starting vertex, S, and label it with a 0.. ind all vertices that are adjacent to S and label them with a 1.. or each vertex labeled with a 1, find an edge that connects it with the vertex labeled 0. arken those edges.. Look for unlabeled vertices adjacent to those with the label 1 and label them. or each vertex labeled, find an edge that connects it with a vertex labeled 1. arken that edge. f more than one edge exists, choose one arbitrarily.. ontinue this process until there are no more unlabeled vertices adjacent to labeled ones. f not all vertices of the graph are labeled, then a spanning tree for the graph does not exist. f all vertices are labeled, the vertices and darkened edges are a spanning tree of the graph.

3 Lesson. Minimum Spanning Trees xample Use the breadth-first search algorithm to find a spanning tree for the following graph. J Solution: s shown in the following figure, the algorithm begins by picking a starting vertex, calling it S, and labeling it with a 0. The labeling and darkening of edges then proceed according to steps through of the algorithm. s you probably noticed, this is not a unique solution. t is just one of the graph s many spanning trees. 1 S 0 1 J

4 hapter More raphs, Subgraphs, and Trees Many applications are best modeled with weighted graphs. When this is the case, it is often not sufficient to find just any spanning tree, but to find one with minimal or maximal weight. Return to the earthquake preparedness situation and reconsider the problem when distances between towns are added to the graph (see igure.1). Wentzville 10 1 New Melle O'allon Peruque St. Peters ugusta igure.1. Map of St. harles ounty with mileage shown. Orchard arm St. harles arvester Refer back to your solution of the original problem and find the total number of miles of road that would need to be repaired if your plan were implemented. ompare your plan with others in your class. Which plan or plans yield the minimum number of miles? Wentzville New Melle 1 O'allon igure.1. Spanning tree of minimum weight for the towns in St. harles ounty. or this particular problem, the minimum possible number of miles of road is. spanning tree with that total weight is shown in igure.1. spanning tree of minimal weight is called a minimum spanning tree. One algorithm for finding a minimum spanning tree for a graph is known as Kruskal s algorithm. t was developed in 1 and named after its designer, Joseph. Kruskal, a leading mathematician at ell Laboratories. 1 Peruque St. Peters ugusta 1 Orchard arm St. harles arvester

5 Lesson. Minimum Spanning Trees xample Use Kruskal s algorithm to find a minimum spanning tree for the following graph. Solution: There are five vertices in the graph, so four edges must be chosen. List the edges from shortest to longest. irst on the list is (). arken it. Then darken (). The next shortest edge is, but if picked, it will form a cycle. So pick (). or the last edge there are two edges of length. ither or can be darkened. The darkened edges of the following graph form one of the minimum spanning trees of the graph. t has a minimal weight of =. 10 Kruskal s Minimum Spanning Tree lgorithm 1. xamine the graph. f it is not connected, there will be no minimum spanning tree.. List the edges in order from shortest to longest. Ties are broken arbitrarily.. arken the first edge on the list.. Select the next edge on the list. f it does not form a cycle with the darkened edges, darken it.. or a graph with n vertices, continue step until n 1 edges of the graph have been darkened. The vertices and the darkened edges are a minimum spanning tree for the graph. Notice that both Kruskal s and ijkstra s (Lesson.) algorithms produce spanning trees. ut unlike ijkstra s 10 shortest path algorithm, which gives you a spanning tree of shortest paths, Kruskal s algorithm yields a spanning tree of minimal total weight. List of dges from Shortest to Longest dge Length

6 hapter More raphs, Subgraphs, and Trees xercises n xercises 1 through, find a spanning tree for each graph if one exists raw a spanning tree for a K graph.

7 Lesson. Minimum Spanning Trees. K L J Sid began using the breadth-first search algorithm to try to find a spanning tree for the preceding graph. e began with vertex, labeled it with a 0, and labeled and with 1s. e then darkened edges and, looked for vertices adjacent to the 1s, and selected vertex. e labeled it with a and darkened edge. 1 0 K L 1 J a. ould Sid have darkened instead of? opy Sid s graph and complete the search for Sid by answering the following questions. b. Which vertices receive s for labels? Label these vertices. c. Which edges subsequently are darkened? arken these edges. d. Three vertices should be labeled. Which ones? Label these vertices and 1 darken the appropriate 0 edges. Your graph could K look like the one at right. L ontinue the algorithm until all vertices are labeled. J heck your darkened edges 1 to make sure they form a spanning tree.

8 0 hapter More raphs, Subgraphs, and Trees. Use the breadth-first search algorithm to find a spanning tree for this graph. egin at vertex. J. Use mathematical induction on the number of edges to prove that every connected graph has a spanning tree. 10. The breadth-first search algorithm can be applied to digraphs if slight changes are made. Modify the algorithm on page so that it can be used with digraphs. pply your modified breadth-first search algorithm to the following digraph. Use Kruskal s algorithm to find a minimum spanning tree for the graphs in xercises 11 through 1. What is the minimal weight in each case?

9 Lesson. Minimum Spanning Trees 1 1. J K L M N The computers in each of the offices at Pattonville igh School need to be linked by cable. The following map shows the cost of each link in hundreds of dollars. What is the minimum cost of linking the five offices? Suppose that the cable in xercise 1 was installed by a disreputable firm that used only the most expensive links. What would be the maximum cost for the four links? 1. ow might Kruskal s minimum spanning tree algorithm be modified to make it a maximum spanning tree algorithm?

10 hapter More raphs, Subgraphs, and Trees nother algorithm that can be used to find a minimum spanning tree is attributed to R.. Prim, a mathematician at the Mathematics enter at ell Labs. Prim s Minimum Spanning Tree lgorithm 1. ind the shortest edge of the graph. arken it and circle its two vertices. Ties are broken arbitrarily.. ind the shortest remaining undarkened edge having one circled vertex and one uncircled vertex. arken this edge and circle its uncircled vertex.. Repeat step until all vertices are circled. 1. Use Prim s algorithm to find a minimum spanning tree for the following graph. What is the minimal weight? 1. Use Prim s algorithm to find a minimum spanning tree for the following graph. What is the minimal weight?

11 Lesson. Minimum Spanning Trees 0. When the shortest path algorithm from Lesson. is applied until all vertices of a graph are used, it yields a spanning tree of the graph. s it always a minimum spanning tree? heck your answer to this question by doing the following. a. ind a minimum spanning tree of the graph above using either Kruskal s or Prim s algorithm. What is the total weight of the minimum spanning tree? b. ind the shortest route from to each of the other vertices using the shortest path algorithm (page ). ive the lengths of each of these routes. c. s the shortest route tree from to each of the other vertices a minimum spanning tree of this graph? xplain why or why not. 1. Traveling salesperson problems, shortest route problems, and minimum spanning tree problems are often confused because each type of problem can be solved by finding a subgraph that includes all of the vertices of the graph. ompare and contrast what each type of problem asks and when each type of problem is used. Project. n this lesson, you have applied two of the three classical minimum spanning tree algorithms, Kruskal s and Prim s. The third algorithm of this group was designed by O. orůvka. nvestigate orůvka s algorithm, learn to apply it, and report on how it differs from Kruskal s and Prim s algorithms.

12 hapter More raphs, Subgraphs, and Trees Splitting Terrorist ells Science News Online ow can you tell if enough members of a terrorist cell have been captured or killed so there s a high probability that the cell can no longer carry out an attack? mathematical model of terrorist organizations might provide some clues. The question is what sort of mathematical model would work best. One way to describe a terrorist organization is in terms of a graph. n this model, each node would represent an individual member of a given cell, and a line linking two nodes would indicate direct communication between those two members. n this hypothetical four-member cell, represented by a graph, oromir, eleborn, and oromir enethor communicate directly with each other, but ragorn communicates only with oromir. When a cell ragorn eleborn enethor member is removed, the corresponding node is deleted from the graph. eleting enough nodes leads to disruption. Mathematically, you could ask the question: ow many nodes must you remove from a given graph before it splits into two or more separate pieces? n this seven-member cell, removing would have little effect on the organization. Removing and instead would split the cell into two units that presumably would be less effective on their own. graph model, however, may not be the best one available for representing a typical terrorist organization, mathematician Jonathan. arley of the Massachusetts nstitute of Technology contends. arley has proposed an alternative approach that reflects an organization s hierarchy. My method uses order theory to quantify the degree to which a terrorist network is still able to function, he says. n this case, the relationship of one individual to another in a cell becomes important. Leaders are represented by the topmost nodes in a diagram of the ordered set representing a cell and foot soldiers are nodes at the bottom. isrupting the organization would be equivalent to disrupting the chain of command, which allows orders to pass from leaders to foot soldiers. n this ordered-set representation of a terrorist cell, points represent individual members and lines show communication links. Members,, and are leaders and rank higher than all other members., J, and K have the lowest rank. given ordered set may have several such chains of command that link a leader with a foot soldier. ll of these chains must be broken for a cell (or remnant) to be considered ineffective. arley s model has several shortcomings. Law enforcement often doesn t know how a terrorist cell is organized or even its full membership. Nonetheless, arley contends, this model enables law enforcement to plan its operations in less of an ad hoc fashion than they might be able to do otherwise. J K