Kenneth Williams

A survey of solved problems and applications on bandwidth, edgesum, and profile of graphs

The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. Akiyama, Exoo, and Harary conjectured that [Δ/2] ≤ la(G) ≤ [Δ+1/2] for any simple graph G with maximum degree Δ. The conjecture has been proved to be true for graphs having Δ = 1, 2, 3, 4, 5, 6, 8, 10. Combining these results, we prove in the article that the conjecture is true for planar graphs having Δ(G) ≠ 7. Several related results assuming some conditions on the girth are obtained as well.

On bandwidth and edgesum for the composition of two graphs

Abstract The composition of two graphs G and H, written G[H], is the graph with vertex set V(G) × V(H) and with (u1, v1) adjacent to (u2, v2) if either u1 is adjacent to u2 in G or u1 = u2 and v1 is adjacent to v2 in H. In this paper, we investigate the bandwidth problem for the composition of two graphs and obtain the bandwidth for several classes of graphs of the forms Kn[G] and K1,n[G]. We also provide optimal numberings which solve the graph edgesum problem for Kn[Pm] and Kn[Cm].

A multidimensional approach to syntactic pattern recognition

Abstract This paper describes a syntactic method for representing the primitive parts of a pattern as nodes of a type of directed graph. A linear representation of the digraph can then be presented to a regular unordered tree automaton for classification. Regular unordered tree automata can be simulated by deterministic pushdown automata, so this procedure can be implemented easily. Regular u -tree automata and the corresponding generative systems, regular u -tree grammars are formally defined. Several results are shown which are applicable to all syntactic pattern recognition schemes involving the use of primitives.

On bandwidth for the tensor product of paths and cycles

The tensor product of graphs G1 and G2 is defined to be G = (V, E) where V = V(G1) × V(G2) and edge ((x1, y1), (x2, y2)) ϵ E whenever (x1, x2) ϵ E(G1) and (y1, y2) ϵ E(G2). We use G1(Tp)G2 to denote G. This paper establishes the bandwidth of the tensor product of a path with a path, a cycle with a path, and a cycle with a cycle. Optimal numberings to achieve each of these bandwidths are provided.

A generalized database directory for nondense attributes

Abstract Inverted file directories are known to support query optimization by providing efficient data access paths for nondense attributes. This article describes a generalized index system for nondense attributes which is based on the inverted file structure while providing additional improvements and options. This generalized inverted index is more efficient and flexible than the standard inverted file. A comprehensive set of file directory construction algorithms to implement the generalized inverted index system is presented. Taken as a whole, these algorithms provide the database designer with a great deal of flexibility in optimizing overall space and time efficiency and in best handling the trade-offs involved.

Profile minimization on triangulated triangles

Given graph G=(V,E) on n vertices, the profile minimization problem is to find a one-to-one function f: V → {1,2,.....,n} such that Σv ∈ V(G){f(v)-minx ∈ N[v] f(x)} is as small as possible, where N[v] = {v} ∪ {x:x is adjacent to v} is the closed neighborhood of v in G. The trangulated triangle Tl is the graph whose vertices are the triples of non-negative integers summing to l, with an edge connecting two triples if they agree in one coordinate and differ by 1 in the other two coordinates. This paper provides a polynomial time algorithm to solve the profile minimization problem for trangulated triangles Tl with side-length l.

Uniform organization of inverted files

A range attribute is defined as an attribute that may assume a range of values. Examples might be Age = (1--10, 11--14, 15--16, ...) or Salary = (0-1000, 1001-1500, ...). This paper is concerned with the selection of ranges that will produce reasonably uniform numbers of records in each range. A set of algorithms has been developed to enable the file designer to obtain a set of ranges such that records are distributed uniformly between the ranges. Although in a given case perfect uniformity may not be achievable, the algorithms can find ranges such that for a set of X records in a range, bounds a and b may be given so that a ≤ X ≤ b for all ranges. The algorithms have been tested with a PASCAL program.

Integrating Mobile Computing and Security into a Computer Science Curriculum (Abstract Only)

The poster describes our project of integrating mobile computing and security into the Computer Science program at North Carolina A&T State University. Twelve (12) course modules in mobile computing and security are being developed and integrated into existing Computer Science courses such as computer programming, software development, operating systems, and information assurance courses. Each course module includes learning objectives, a tutorial, presentation slides, hands-on labs and/or case studies, test questions, etc. The course module material we develop and our teaching experiences will be beneficial to computer science educators who are considering including mobile computing and mobile security into their curricula.

Algorithms on Block-Complete Graphs

Graphs whose blocks are complete subgraphs are said to be block-complete graphs. Polynomial time algorithms to solve several problems, problems which are not believed to be polynomial for general graphs, are presented for connected block-complete graphs. These include: finding a minimum vertex cover, finding a minimum dominating set of radius r, and finding a minimum m-centrix radius r augmentation.

Development and implementation of optimization algorithms for small computers: Optimizing subcase solutions for (M, N)-transitivity

One paradigm for optimizing the efficiency of solutions to instances of a difficult problem is to factor the problem into subcases and to optimize the solution for each subcase. Although an optimal solution for the general problem may not be found by this approach, near optimal solutions may be found for many particular instances. This is the approach illustrated by the following problem domain. When designing a network (communication, computing etc.) it may be necessary to impose the following constraint: Given N and M with N < M, for each path from node (u) to node (v) of length M, there must exist a path from (u) to (v), passing through a subset of the same intermediate nodes, of length N. All paths are simple, thus may not pass through a given node more than once. This is essentially the definition of an (M, N)-transitive digraph. A natural question arises: Given a digraph, G, which may not be (M, N)-transitive, how may the minimum (or near minimum) number of edges be added to ensure that it has this property? (i. e. How may the (M, N)-transitive closure be computed?) An algorithm which answers this question for any digraph has been developed and is provided here, but the algorithm requires exponential time complexity in r, the number of nodes of the digraph. However, for many special cases (including, for any M and N, the computation of (M, 1)-transitive closures and all (M, N)-transitive closures on acyclic digraphs) more efficient, polynomial-time, specialized algorithms have been developed and are provided here. All algorithms have been implemented and tested using DEC Ada on a VAX8650. Comparative tests illustrating the time required to run each of the applicible algorithms have been run and the results are summarized.