The Join the Club Interpretation of Some. Graph Algorithms

Transcription

1 The Join the Club Interpretation of Some Graph Algorithms Harold Reiter Isaac Sonin June 8, 2000

2 Abstract Several important tree construction algorithms of graph theory are described and discussed using an easily remembered interpretation. Under this interpretation, vertices are thought of as potential members of a club. A rule for adjoining new vertices to the tree corresponds to a specific rule by which new members join the club.

3 Interest in graph theory has accelerated during the last two decades because of its far-reaching applications in other branches of mathematics, especially operations research, as well as outside of mathematics. There is hardly a discipline cannot lay claim to advances attained by the development of methods and models of graph theory. Computer scientists and economists use communication and transportation networks, sociologists employ graphs to study social mobility and stratification, anthropologists use interval graphs in the dating of artifacts, psychologists use them in clustering and in transactional analysis, and chemists count and study isomers using graphs and tree models. The abundance of real life situations for which graph theory is useful has encouraged many mathematicians and non-mathematicians to learn and use it. An important part of graph theory is concerned with algorithms. Searching and sorting, finding shortest paths, constructing spanning trees with desirable properties, matching and coloring vertices are problems whose solutions are algorithms. Hence, nearly every course in discrete mathematics, combinatorics, and graph theory contains some material on algorithms. Some students find graph algorithms hard to remember. Though most algorithms are based on relatively simple ideas, their formal presentation may be rather long, and understanding them may require substantial mathematical maturity. Another difficulty students sometimes encounter is that often algorithms 1

4 are described using pseudo-code, an approach which lends itself to the construction of computer programs, but which may be very difficult for students to internalize. What is needed is something akin to a mnemonic device to help students remember and understand these algorithms. We describe below an approach to several algorithms of graph theory using ideas with which the typical student is already familiar. Let G = (V, E) be a graph with vertex set V and edge set E. We assume throughout this paper that our graph is connected. Otherwise, we could consider each component separately. Recall that a graph H obtained from another graph G by selecting some vertices of G and some edges of G joining vertices of H is called a subgraph of G. A tree is a connected graph which contains no cycles. A spanning tree of a graph G is a tree which is a subgraph of G containing all the vertices of G. Each of our algorithms results in the construction of a spanning tree T of G. Each spanning tree construction starts at a designated vertex which we call the root of the tree. We interpret V as the people living in a community. An edge between two people means that they are acquainted. There is a club in the community to which everyone wishes to belong. Eventually, everyone in the community will become a member. People join the club one by one. When a non-member u joins the club, he or she does so because a member v with whom u is acquainted recommends him or her. We shall call v the sponsor of u. The 2

5 rules given below specify who can join the club at each stage and who can be the sponsor. We adopt a basic rule for joining the club. This basic rule gives rise to a large class of rooted spanning subtrees. Specific variations of the basic rule result in a Breadth First Search (BFS) spanning tree, a Depth First Search (DFS) spanning tree, a Minimum Total (MT) spanning tree, and a Shortest Path (SP) spanning tree. These spanning trees are useful in solving many problems. For example, a BFS spanning tree can be used to determine the length of a shortest path (i.e., one with the fewest edges) from the root to any other vertex. A DFS spanning tree can be used to locate the bridges in a graph and, when no bridges exist, to establish a strongly connected orientation of the graph. A bridge is an edge whose removal disconnects the graph. An orientation is an assignment of a direction to each edge; an orientation is strongly connected if it is possible to get from any vertex to any other vertex along directed edges without violating the direction. See figure 3. For formal discussions of these algorithms and their applications, see any of the references. We will see that other interesting problems that can be solved using MT and SP spanning trees. We call a club member open if the member knows a non-member in the community. Of course the status of a club member may change from open to non-open at some time during the club s formation, but not vice-versa. Basic Rule. There is one founding member, and all non-members join the 3

6 club, one at a time. At each stage, a non-member joins the club on the recommendation of an open member (until no one else can join the club). The construction process results in a sequence of subgraphs. We begin with the arbitrarily selected root (the founder), and at each stage one vertex (the new member) and one edge (from the new member to his or her sponsor) are added to the graph. At each stage in the formation, each open member can recommend any non-member with whom he or she is acquainted. Notice that at some stages during the formation of the club, it is possible that there are several open members, and that some of them can recommend any of several non-members. In these cases, we can choose the sponsor however we like, and after that we can choose any of that sponsor s nonmember acquaintances as the new member. This convention of breaking ties arbitrarily applies to all the algorithms discussed here. Figures 1, 2, and 3 show just one graph of the few possible. 4

7 u 2 u 5 u 1 u 6 u 0. u 8 u 3. u 9 u 4 u7. Fig. 1 (Applying the Basic Rule) The Spanning tree is marked by bold lines. The subscripts represent the order in which the vertices became members of the club.. Let us show that any subgraph which results from the Basic Rule must be a spanning tree. Notice that at each stage in the process, the set of vertices (club members), together with the edges connecting them with their sponsors, is a connected subgraph of G. We must also show that no cycle is 5

8 ever formed. Suppose at some stage a cycle is formed. Consider the junior member u of this cycle, that is the last member of the cycle to join the club. As a junior, u cannot be a sponsor of either of the adjacent vertices. Hence u must be sponsored by both of them, contrary to the Basic Rule. Therefore, we have an acyclic graph at each stage of the club formation process. Finally, if at some stage there are still non-members, then because G is connected, there must be a non-member acquainted with a member. Hence the process continues until all non-members join the club. Thus we will have a tree that includes all the vertices; that is, a spanning tree. Before discussing particular algorithms, we develop some notation. The club founder will be denoted u 0, and the other members are denoted u 1, u 2,... in the order in which they join the club. A club member u i is called senior to u j if i < j. In that case, we say u j is junior to u i. In other words, u i is senior to u j if u i joined the club before u j. We now provide specific rules for membership which result in the BFS spanning tree and the DFS spanning tree. In fact our rules not only provide the construction of spanning trees, but also the corresponding tree labeling; that is, a labeling of the n vertices with the numbers 0, 1,, n 1 so that on the unique path from the founder to each member, the labels occur in increasing order. Seniority Rule. At each stage, a non-member joins the club on the recommendation of the most senior open member. 6

9 The Seniority Rule is a variation of the Basic Rule in the sense that, again at each stage a non-member joins the club on the recommendation of a member selected from a nonempty set of open members. However, now only one open member (the most senior among the open members) is allowed to be the sponsor. So all the arguments showing that the Basic Rule results in a spanning tree are valid. The same remark is true for each of the other algorithms where again only certain of the open members are allowed to sponsors. In the Juniority Rule, exactly one is allowed and in the Minimum Fee Rule and the Minimum Reimbursement Rule, possibly several, but again the set of sponsors at each non-final stage is non-empty. A spanning tree that results from applying the Seniority Rule is called a BFS spanning tree. Figure 2 shows the spanning tree that results from applying the Seniority Rule. 7

10 u 2 u 5 u 1 u 6 u 0. u 3 u 9. u 7. u 4 u8 Fig. 2 (Applying the Seniority Rule) The Breadth First Search Spanning tree is marked by bold lines. Juniority Rule. At each stage, a non-member joins the club on the recommendation of the most junior open member. The resulting spanning tree is called the DFS spanning tree. 8

11 u 4 u 3 u 1 u 2 u 0. u 5 u 8. u 7. u 9 u6 Fig. 3 (Applying the Juniority Rule) The Depth First Search Spanning tree is marked by bold lines. Figure 3 shows the spanning tree that results from the application of the Juniority Rule. Figure 3 shows more than the DFS spanning tree. Each tree edge is directed from senior member to junior member. Every other edge is directed from junior member to senior member. In general, a graph without bridges has a strongly connected orientation, which can be established by 9

12 first constructing a DFS spanning tree (Juniority Rule) and then orienting as above. Some algorithms apply not only to connected graphs but also to networks, that is, graphs with a non-negative weight (length) f(e) assigned to each edge e. By a shortest path from vertex u to vertex v, we mean a path the sum of whose weights is minimal (that is, no larger than any other such sum). We assume that each new member must pay an initiation fee, and this amount is the weight on the edge between this member and the member s sponsor. Note that a non-member u who can be sponsored by any of several open members might pay a different fee, depending on who sponsors u. Minimum Fee Rule. At each stage, a non-member with the minimum potential fee joins the club. He or she joins through the sponsorship which realizes this fee. The resulting spanning tree T is called a MT spanning tree and its total weight is given by e T f(e). The Minimum Fee Rule is our way of describing the rule used in Prim s algorithm to find a spanning tree with minimum total weight. A proof that T is indeed a MT spanning tree can be found in [8,10]. 10

13 u 1 8 u 7 u u u u 9 5 u u u 2 6 u3 Fig. 4 (Applying the Minimal Fee Rule) The spanning tree with minimal total sum of weights (lengths). To introduce the last algorithm, we associate with each member a reimbursement as follows. Whenever a person joins the club, that person will have a sponsor. This sponsor has a sponsor, and so on, up to the founder. Each new member u pays a reimbursement to the club equal to the sum of fees on these edges; that is, the sum f(e), taken along the unique list of edges from the founder to u. The reimbursement depends on who sponsors 11

14 the non-member. The term potential reimbursement refers to any of these possible reimbursements. Our next rule can be used to produce a spanning tree which describes the shortest paths from the root vertex to any other vertex. Minimum Reimbursement Rule. At each stage, a non-member with the minimum potential reimbursement joins the club. He or she joins through a sponsorship which realizes that minimum. The Minimum Reimbursement Rule is our way of describing the rule used in Dijkstra s algorithm to find a Shortest Path spanning tree. u 1 (4). 8 u 4 (12) u u 7 (14) u 8(23) u 9 (26). 5 u 6 (14) u5 (13) u 2 (5) 6 u 3 (11) 12

15 Fig. 5 (Applying the Minimal Reimbursement Rule) The spanning tree with shortest paths (=reimbursements) between u 0 and any point in the graph. Reimbursements are given in (). The algorithms of this paper can be applied to directed graphs as well, provided a few extra conditions are satisfied. In this case, if (u, v) is a directed edge, we allow u to sponsor v, but not vice-versa. Other algorithms are also interpretable using the join the club paradigm. For example, both the Ford-Fulkerson max-flow min-cut and the Maximum Matching algorithms construct and use spanning trees as part of an iterated process. See [8,10]. Thus, spanning trees are important tools in algorithmic graph theory. Come join the club of those who know and can apply the algorithms of graph theory. References [1] K. P. Bogart Introductory Combinatorics, 2nd. ed., Harcourt Brace Jovanovich, New York, [2] G. Chartrand, and L. Lesniak, Graphs and Digraphs, 2nd ed., Wadsworth & Brooks/Cole, Monterey, California, [3] F. Harary, Graph Theory, Addison-Wesley, Reading, Massachusetts,

16 [4] Robin Wilson, Introduction to Graph Theory, 3rd. ed., Longman, Harlow, Essex, [5] S. Even, Graph Algorithms, Computer Science Press, Potomac, Maryland, [6] J. A. McHugh, Algorithmic Graph Theory, Prentice-Hall, Englewood Cliffs, NJ, [7] F. S. Roberts, Applied Combinatorics, Prentice-Hall, Englewood Cliffs, NJ, [8] Alan Gibbons, Algorithmic Graph Theory, Cambridge University Press, Cambridge,

17 Isaac Sonin Department of Mathematics UNC Charlotte Charlotte, NC Harold Reiter Department of Mathematics UNC Charlotte Charlotte, NC