Trees: A Mathematical Tool for All Seasons
Posted January 2006.
I hope to convince you that mathematical trees are no less lovely than their biological counterparts...
Trees have been the inspiration of poets and no one doubts the beauty of many wood sculptures nor the ubiquitous uses for lumber. The leaves of trees are beautiful and varied. Many find the spectacular fall colors of trees an inspiration and travel great distances to see the fall foliage. No wonder that mathematicians use the suggestive term "tree" for the special class of structures sampled below, where two rather different drawings of the same tree structure are shown.
A powerful way to represent relationships between objects in visual form can be done using a mathematics structure called a graph. One uses dots called vertices to represent objects and line segments which join the dots, called edges, to represent some relationship between the object which the dots represent. For example, a chemist might draw the diagram below to represent a methane molecule:
Here the diagram shows a person's ancestors.
Minimum cost spanning trees
One example of the power of using trees as a tool appears in a problem which often arises in operations research. Here is one applications setting. Consider the graph with weights as pictured below.
History of algorithms to find minimum cost spanning trees
The history of the minimum cost spanning tree problem is rather interesting and complex. Until recently, most textbook discussions of this problem made reference to the work of two American mathematicians, then employees at Bell Laboratories, who each developed an algorithm for solving the minimum cost spanning tree problem. They are Joseph B. Kruskal who published his work in 1956,
(Photo courtesy of Pieter Kroonenberg (U. of Leiden), who took the photo.)
(This photo, courtesy of Robert Prim, dates from 1971.)
Minimum cost spanning tree algorithms
In what follows we will use the following example (Figure 1) to illustrate the way that the various algorithms in which we are interested work. You can think of this graph as giving 6 sites that must be connected by a high transmission electrical system or a cable system of some kind. The sites that have no line segments connecting them are too expensive to connect. Note that the edges all have distinct costs (lengths), which makes the initial exposition a trifle simpler. However, the discussion below can be modified to deal with the situation in which some edges have equal weights. When there are ties for the weights of the edges, the cost associated with a minimum cost spanning tree is the same for all trees which achieve minimum cost. When the edges all have distinct weights there is a unique tree which solves the problem.
There are nine edges in the above graph. If we sort the weights of these edges in increasing order, we get: 4, 5, 9, 10, 11, 12, 13, 20, 24. If we try to get a cheap connecting system by adding edges in order of increasing cost, we would first insert the edges of cost 4 and 5 as shown in the diagram below:
The next cheapest edge would be 9 but its insertion would create a cycle. To send electricity from vertex 0 to vertex 2 does not require the link from 0 to 2 because it can instead be sent from 0 to 1 and then from 1 to 2. Hence, we need not add the link from 0 to 2. Thus, the next edges we would add would be the edges 3 to 4 and 4 to 5 because these are the cheapest at the next two decision stages. After adding these links we get the following situation:
Note that at this stage, the edges selected do not form a connected subgraph of the original edges. Thus, Kruskal's algorithm does not form a tree at every stage of the algorithm. However, by the time the algorithm terminates, the edges will form a tree. The next cheapest edge, having cost 12, would also form a cycle with previously chosen edges, so it is not added to the collection of links. However, the edge with cost 13 can be added. We now have a collection of edges which forms no cycle and which includes all the vertices of the original graph. Thus, we have found a collection of links which makes it possible to transmit electricity from any of the locations to any of the others, using relays if necessary. Since we are seeking a tree as the final collection of edges (shown in red in Figure 4), we can use the fact that all trees with the same number of vertices have the same number of edges to determine how many edges must be selected before Kruskal's algorithm, or that of Prim or Borůvka, will terminate. Specifically, we know that for a tree, the number of vertices and edges are related by the equation:
So that in applying Kruskal's algorithm, when we have selected a connected number of edges to put into our linking network that is one less than the number of vertices to be linked, we know that we have exactly the right number of linking edges!
Prim's algorithm is also a greedy algorithm, in the sense that it repeatedly makes a best choice in a sequence of stages. However, one difference is that Prim's algorithm always results in a tree at each stage of the procedure, producing a spanning tree at the stage where the algorithm terminates. The idea behind Prim's algorithm is to add a cheapest edge which links a new neighboring vertex (a "super-vertex") to a previously selected collection of vertices. We can choose to initiate the algorithm at any vertex of the dot/line diagram. Thus, if we start the procedure at vertex 3, we have three edges to choose from: edge 13 costing 13, edge 34 costing 10 and edge 35 costing 12. Since edge 34 is cheapest, we select this one. At this stage we have the situation shown in Figure 5:
We now think of shrinking the edge from 3 to 4 to a single "super-vertex." This super-vertex has neighboring vertices connected to it by edges. These are the edges 04, 13, 35, and 45 (with costs 24, 13, 12, and 11, respectively). The cheapest of these in cost is the edge 45, so we select this edge next.
At this stage we can think of vertices 3, 4, and 5 as a "super-vertex." The neighbors of this super-vertex are the edges 04, 13, and 25. Note that we do not consider the edge 35 any longer as a neighbor because it joins two vertices which are contained "within" the super-vertex. Since the edge 13, coincidentally with cost 13, is the cheapest of the neighbors of the super-vertex, we next add the edge 13 to the growing collection of links.
We can now continue in this way until our super-vertex consists of all of the vertices of the original graph. Here is the order in which the vertices are added to the super-vertex starting with the initial vertex 3: 4, 5, 1, 2, 0. Thus the edges added are: 34, 45, 13, 12, 10, which gives rise to exactly the same final collection of connecting edges as previously, as shown in red in Figure 4.
If the blue edges in Figure 8 formed a tree, we would be finished. However, since they do not, we form a collection of super-vertices which arise from the sets of connected vertices which have currently formed with selected edges. We then carry out another "grabbing" stage of the algorithm. In our current example, there are two super-vertices (one consisting of vertices 0, 1, and 2 and the other consisting of vertices 3, 4, and 5). These super-vertices are linked with edges of weight 13, 20, and 24. If we call the super-vertices A and B respectively, A grabs the edge AB of weight 13, and B graphs the edge BA (which is, of course, the same as AB) of weight 13. This new edge, when changed to color blue, together with the blue edges in Figure 8 gives rise to the connecting edges shown in red in Figure 4. Again, we have found the minimum cost spanning tree.
In Figure 10, we have shown three copies of the graph in Figure 9, each with a minimum cost spanning tree of cost 102. This illustrates the fact that a graph which has edges with equal weights can have many minimum cost spanning trees, but that one can prove that all of the minimum cost spanning trees have the same cost. Also note that in each case the edge with maximum cost in the original graph can be part of a minimum cost spanning tree. However, in any cycle in a graph for which one seeks a minimum cost spanning tree, if the edges of this cycle have different costs, then the edge of maximum cost in this cycle can not be part of a minimum cost spanning tree.
The idea behind one proof of Kruskal's algorithm is to assume there is a tree T with a cost less than or equal to that of a tree K that is generated by Kruskal's procedure. Construct the list L of the edges of the tree K in the order in which Kruskal's algorithm inserted them into K. If T is not the same tree as K, find the first edge e in list L which is an edge of K but not of T. When e is added to T, because of a property of trees mentioned at the start, a unique cycle is formed. This cycle must contain at least one edge e' not in K since K, being a tree, has no cycles. Now form the tree T' which consists of adding e to T and removing e'. The cost of this new tree T' is the cost of T plus the weight on e minus the weight on e'. Since T is a cheapest tree and since Kruskal's algorithm chooses edges in order of cheapness subject to not forming a cycle, the weight on e and e' must be equal. This means that T' and T have the same cost, and, thus, K and T' agree on one more edge than K and T in terms of cost. Continuing in this way we see that we may have different trees that achieve minimum cost (these arise from different ways of breaking ties when Kruskal's algorithm is applied) but none that can be less costly than the tree that Kruskal's method produces.
Bazlamacci C. and K. Hindi, Minimum -weight spanning tree algorithms: A survey and empirical study, Computers & Operations Research, 28 (2001) 767-785.
NOTE: Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal, which also provides bibliographic services.
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