SUPPLEMENT TO CHAPTER 6 MINIMUM SPANNING-TREE PROBLEMS

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1 S h 6- SUPPLMNT TO HPTR 6 MINIMUM SPNNIN-TR PROLMS hapter 6 focuses on network optimization problems. These are problems that can be described in terms of a complete network that has both nodes and links (or arcs) and the objective is to optimize the operation of the network. We now turn our attention to a different kind of problem where the objective is to design the network. The nodes are given, but we must decide which links to give to the network. Specifically, each potential link has a cost (different for different links) for inserting it into the network. We are required to provide enough links to provide a path between every pair of nodes. The objective is to do this in a way that minimizes the total cost of the links. Such a problem is referred to as a minimum spanning-tree problem, as illustrated by the following example. n xample: The Modern orp. Problem Management of the Modern orporation has decided to have a state-of-the-art fiber-optic network installed to provide high-speed communications (data, voice, and video) between its major centers. The nodes in igure show the geographical layout of the corporation's major centers (which include corporate headquarters, a supercomputer facility, and a research park, as well as production and distribution centers). The dashed lines are the potential locations of fiber-optic cables. (Other cables between pairs of centers also are possible but have been ruled out as uneconomical.) The number next to each dashed line gives the cost (in millions of dollars) if that particular cable is chosen as one to be installed. igure display of Modern orp.'s major centers (the nodes), the possible locations for fiber-optic cables (the dashed lines), and the cost in millions of dollars for those cables (the numbers). ny pair of centers does not need to have a cable directly connecting them in order to take full advantage of the fiber-optic technology for high-speed communications between these centers. ll that is necessary is to have a series of cables that connect the centers.

2 S h 6- The problem is to determine which cables should be installed to minimize the total cost of providing high-speed communications between every pair of centers. This is, in fact, a minimum spanning-tree problem. The optimal solution for this problem is shown in igure, where the links in this network correspond to the possible cables in igure that should be chosen for installation. (Note that there is indeed a path between every pair of centers.) The resulting cost of this fiber-optic network is Total cost = = ($ million). ny other design of the network that connects every pair of centers would cost at least $ million more. igure The fiber-optic network that provides the optimal solution for Modern's minimum spanning-tree problem. What is the reason for the strange name, minimum spanning-tree problem? Here is the explanation. In the terminology of network theory, the network in igure is a tree because it does not have any paths that begin and end at the same node without backtracking (i.e., no paths that cycle). It also is a spanning tree because it is a tree that provides a path between every pair of nodes (so it "spans" all the nodes). inally, it is a minimum spanning tree because it minimizes the total cost among all spanning trees. eneral haracteristics Just as for Modern's problem, every minimum spanning-tree problem satisfies the following assumptions.

3 S h 6- ssumptions Of Minimum Spanning-Tree Problem. You are given the nodes of a network but not the links. Instead, you are given the potential links and the positive cost (or a similar measure) for each if it is inserted into the network.. You wish to design the network by inserting enough links to satisfy the requirement that there be a path between every pair of nodes.. The objective is to satisfy this requirement in a way that minimizes the total cost of doing so. n optimal solution for this problem always is a spanning tree. Here is an easy way to recognize a spanning tree. The number of links in a spanning tree always is one less than the number of nodes. urthermore, each node is directly connected by a single link to at least one other node. See that this description fits the spanning tree in igure, where there are six links and seven nodes (all directly connected to at least one other node). Remove any one of these links and assumption above would be violated (no spanning tree). (heck this.) Incur the needless extra cost of adding another link instead (without removing one) and you again no longer have a spanning tree. (heck that adding any unused link from igure into igure would create a path that begins and ends at the same node without backtracking, which violates the definition of a tree.) inally, we should point out that, in contrast to transportation, assignment, maximum flow, and shortest path problems, a minimum spanning tree-problem is not a special type of minimum cost flow problem. (It is not even a special type of linear programming problem.) urthermore, it cannot be solved easily by Solver. That is the bad news. The good news is that you can solve it very easily by the algorithm described below without even using a computer. Remarkably Simple lgorithm Starting with no links in the network, each step of the algorithm selects one new link to insert from the list of potential links. s described below, the algorithm continues in this way until every node is touched by a link, at which point the selected nodes form a minimum spanning tree. lgorithm for a Minimum Spanning-Tree Problem. hoice of the first link: Select the cheapest potential link.. hoice of the next link: Select the cheapest potential link between a node that already is touched by a link and a node that does not yet have such a link.. Repeat step over and over until every node is touched by a link (perhaps more than one). t that point, an optimal solution (a minimum spanning tree) has been obtained. [Tie breaking: Ties for the cheapest potential link may be broken arbitrarily without affecting the optimality of the final solution. However, ties in step signal that there may also be (but need not be) other optimal solutions that would be obtained by breaking ties in another way.] pplication of the lgorithm to the Modern orp. Problem Now let us apply this algorithm to Modern's minimum spanning-tree problem as displayed in igure. mong all the potential links (the dashed lines), the one between node and node ties with the one between node and node as the cheapest (a cost of ). Therefore, for step, we

4 S h 6- need to select one of these two potential links to be the first link inserted into the network. reaking the tie arbitrarily, let us select the one between node and node (the other will be chosen later), as shown below. Next, we apply step for the first time. The two nodes that are touched by a link are nodes and, so we need to compare the costs of the potential links between either of these nodes and a node that does not yet have a touching link. These potential links and their costs are : ost = : ost = : ost = : ost = : ost = : ost = Since the cheapest of these is the one between node and node, with a cost of, it is selected to be the next link inserted into the network, as displayed below.

5 S h 6- Now, nodes,, and each are touched by a link (or two links in the case of node ), so the next execution of step requires comparing the costs of the potential links between one of these nodes and one of the others. : ost = : ost = : ost = : ost = : ost = : ost = : ost = The cheapest of these is the potential link between node and node, so it becomes the next link added to the network.

6 S h 6-6 Nodes,,, and now all have touching links, so we next compare the costs of the potential links between one of these nodes and one of the others. (ctually, none of these potential links involve node, since it does not have any potential links that go to a node that is not yet touched by a link.) : ost = : ost = : ost = : ost = The cheapest is the potential link between node and node, so it is added next. ll but nodes and now are touched by a link. Therefore, the only potential links that need to be considered next are between either node or and one of the other nodes.

7 S h 6- : ost = : ost = : ost = : ost = The cheapest by far is the potential link between node and node, so it finally gets inserted into the network. (Remember that this potential link was tied to be the initial link in step.) Since node now is the only node untouched by a link, the only potential links to consider next are those between this node and the others. : ost = : ost = The cheaper one is the potential link between node and node, so we insert it into the network.

8 S h 6-8 very node now is touched by a link, so the algorithm is done and this is our optimal solution. ll the links that have been inserted into the network form a minimum spanning tree with a total cost of = ($ million). ll the remaining potential links (dashed lines) are rejected because the inserted links provide a path between every pair of nodes. Notice that this optimal solution is the same as the one given in igure. (There is only one optimal solution for this particular problem.) What would have happened if the tie had been broken the other way in step by selecting the potential link between node and node to be the initial link inserted into the network instead of the potential link between node and node? o ahead and check this out by cranking through the algorithm from this point. You will find that exactly the same links get selected, but in a different order from before. This algorithm is referred to as a greedy algorithm because it simply grabs the most favorable choice (the cheapest potential link) at each step without worrying about the effect of this choice on subsequent decisions. It is remarkable that such a quick and simple-minded procedure still is guaranteed to find an optimal solution. Rejoice this time, but beware. reedy algorithms normally will not necessarily find optimal solutions for other management science problems.

9 S h 6-9 Some pplications In this age of the information superhighway, applications similar to the Modern orp. example have become increasingly important. However, minimum spanning-tree problems have several other types of applications as well. Here is a list of some key types of applications.. esign of telecommunication networks (computer networks, leased-line telephone networks, cable television networks, etc.). esign of a lightly used transportation network to minimize the total cost of providing the links (rail lines, roads, etc.).. esign of a network of high-voltage electrical power transmission lines.. esign of a network of wiring on electrical equipment (e.g., a digital computer system) to minimize the total length of the wire.. esign of a network of pipelines to connect a number of locations. RVIW QUSTIONS. In a minimum spanning tree problem, what part of the network is given and what part remains to be designed?. What kind of network is being designed in the Modern orp. example?. In the terminology of network theory, what is a tree? spanning tree? minimum spanning tree?. What is an easy way to recognize a spanning tree?. What is the objective of a minimum spanning tree problem? 6. Is a minimum spanning tree problem a special type of minimum cost flow problem?. What kind of algorithm will solve a minimum spanning tree problem (but very few other management science problems)? 8. What are a few types of applications of minimum spanning tree problems? lossary reedy algorithm: n algorithm that simply grabs the most favorable choice at each step without worrying about the effect of this choice on subsequent decisions. Minimum spanning tree: One among all spanning trees that minimizes total cost. Spanning tree: tree that provides a path between every pair of nodes. Tree: network that does not have any paths that begin and end at the same node without backtracking. Problems 6s.. Reconsider the Modern orp. problem. When the algorithm for a minimum spanning-tree problem was applied to this problem, there was a tie at step for choosing the first link. This tie was broken arbitrarily by selecting the potential link between node and node. Now break the tie the other way by selecting the potential link between node and node to be the first link and then reapply the rest of the algorithm. Show each step. (You again should obtain the minimum spanning tree shown in igure.)

10 S h 6-0 6s.. Use the greedy algorithm to find a minimum spanning tree for a network with the following nodes and with the links still to be chosen. The dashed lines between pairs of nodes represent potential links and the number next to each dashed line represents the cost (in thousands of dollars) of inserting that link into the network. 6s.. Use the greedy algorithm to find a minimum spanning tree for a network with the following nodes and with the links still to be chosen. The dashed lines between pairs of nodes represent potential nodes and the number next to each dashed line represents the cost (in millions of dollars) of inserting that link into the network.

11 S h 6-6s. The Wirehouse Lumber ompany will soon begin logging eight groves of trees in the same general area. Therefore, it must develop a system of dirt roads that makes each grove accessible from every other grove. The distance (in miles) between every pair of groves is as follows: istance between Pairs of roves rove Management now wants to determine between which pairs of groves the roads should be constructed to connect all groves with a minimum total length of road. a. escribe how this problem fits the network description of a minimum spanning tree problem. b. Use the greedy algorithm to solve the problem. 6s.. The Premiere ank soon will be hooking up computer terminals at each of its branch offices to the computer at its main office, using special phone lines with telecommunications devices. The phone line from a branch office need not be connected directly to the main office. It can be connected indirectly by being connected to another branch office that is connected (directly or indirectly) to the main office. The only requirement is that every branch office be connected by some route to the main office. The charge for the special phone lines is $00 times the number of miles involved, where the distance (in miles) between every pair of offices is as follows: istance between Pairs of Offices Main..... Main Office ranch ranch ranch ranch ranch Management wishes to determine which pairs of offices should be directly connected by special phone lines in order to connect every branch office (directly or indirectly) to the main office at a minimum total cost.

12 S h 6- a. escribe how this problem fits the network description of a minimum spanning tree problem. b. Use the greedy algorithm to solve the problem. What is the total cost for the special phone lines? 6s.6. Reconsider ase 6-. Suppose now that the following scenario arises with this case. ven before all merican troops and supplies had reached Saint Petersburg, Moscow, and Rostov, infighting among ommander Votachev s troops about whether to make the next attack against Saint Petersburg or against Moscow split the revolutionaries. Troops from Moscow easily overcame the vulnerable revolutionaries. ommander Votachev was imprisoned, and the next step became rebuilding the seven cities razed by his armies. The President s top priority is to help the Russian government to re-establish communications between the seven Russian cities and Moscow at minimum cost. The price of installing communication lines between any two Russian cities varies given the cost of shipping wire to the area, the level of destruction in the area, and the roughness of the terrain. Luckily, a city is able to communicate with all others if it is connected only indirectly to every other city. Saint Petersburg and Rostov are already connected to Moscow, so if any of the seven cities is connected to Saint Petersburg or Rostov, it will also be connected to Moscow. The cost of replacing communication lines between two given cities for which this is possible is shown below. etween ost to Re-establish ommunication Lines Saint Petersburg and Kazan $0,000 Saint Petersburg and Perm $8,000 Saint Petersburg and Ufa $,000 Moscow and Ufa $0,000 Moscow and Samara $9,000 Moscow and Orenburg $0,000 Moscow and Saratov $0,000 Rostov and Saratov $00,000 Rostov and Orenburg $0,000 Kazan and Perm $0,000 Kazan and Ufa $0,000 Kazan and Samara $9,000 Perm and Yekaterinburg $8,000 Perm and Ufa $,000 Yekaterinburg and Ufa $,000 Ufa and Samara $00,000 Ufa and Orenburg $,000 Saratov and Samara $00,000 Saratov and Orenburg $9,000 Where should communication lines be installed to minimize the total cost of re-establishing communications between Moscow and all seven Russian cities?