Owen Murphy

Nearest neighbor pattern classification perceptrons

A three-layer perceptron that uses the nearest-neighbor pattern classification rule is presented. This neural network is of interest because it is designed specifically for the set of training patterns, and incorporating of the training of the network into the design eliminates the need for the use of training algorithms. The technique therefore provides an alternative to the limitations and unpredictability (such as having too many, too few, or inappropriate training patterns) of the known training techniques. Since the nearest-neighbor classification rule is used, the network is capable of forming arbitrarily complex decision regions. The design and training of the network can be completed in polynomial time, whereas it has been shown that training a neural network is an NP-complete problem. >

Computing independent sets in graphs with large girth

It is shown that the well-known independent set problem remains NP-complete even when restricted to graphs which contain no cycles of length less than cnk where n is the number of vertices in the graph, and c and k are constants satisfying c>0 and 0≤k<1. A polynomial time approximation algorithm for this problem is also examined.

Designing storage efficient decision trees

The problem of designing storage-efficient decision trees from decision tables is examined. It is shown that for most cases, the construction of the storage optimal decision tree is an NP-complete problem, and therefore a heuristic approach to the problem is necessary. A systematic procedure analogous to the information-theoretic heuristic is developed. The algorithm has low computational complexity and performs well experimentally. >

Lower bounds on the stability number of graphs computed in terms of degrees

Abstract Wei discovered that the stability number, α( G ), of a graph, G , with degree sequence d 1 , d 2 ,…, d n is at least w(G) = Σ i=1 n 1 d i +1 It is shown that this bound can be replaced by a function b ( G ), computable from d 1 , d 2 ,…, d n using only O( n ) additions and comparisons. For all graphs, b ( G ) ⩾ ⌜ w ( G )⌝, the inequality sometimes holding as strict. In addition, it is shown that Weis bound can be increased by ( w ( G ) − 1)⧸ Δ ( Δ + 1) when G is connected, by w ( G ) k ⧸2 Δ ( Δ + 1) when G is k -connected but not complete, and by ( w ( G ) + m − n )⧸ Δ ( Δ + 1) when G is triangle-free; in each case, Δ, n , and m denote the largest degree of a vertex in G , the number of vertices of G , and the number of edges of G , respectively.

Graph theoretic algorithms for the PLA folding problem

Graph theoretic properties of the PLA folding problem which not only give insight into the various folding problems but also provide efficient algorithms for solving the problems are presented. This work is based on the transformation of the PLA into graphs where cliques (completely connected subgraphs) in the graphs correspond to PLA folding sets. A simple heuristic technique known as the greedy algorithm can often identify near-maximum cliques in polynomial time. Experimental data show that this technique is extremely effective when applied to the graphs which generally arise in folding. Variations of the general folding problem such as bipartite folding and constrained folding are addressed. >

The efficiency of using k-d trees for finding nearest neighbors in discrete space

Abstract In this paper, we examine the efficiency of the k-d tree for retrieving from a file of fixed-length binary key records the best match to a given input word. We provide guidelines for determining if the search of the tree will provide any savings when compared with an exhaustive search.

Edge density and independence ratio in triangle-free graphs with maximum degree three

Abstract A simple polynomial-time algorithm is presented which computes independent sets of guaranteed size in connected triangle-free noncubic graphs with maximum degree 3. Let n and m denote the number of vertices and edges, respectively, and let c = m/n denote the edge density where c c

An information theoretic design and training algorithm for neural networks

An algorithmic approach to designing feedforward neural networks for pattern classification is presented. The technique computes a single hidden layer of nodes by adding one node at a time until the desired classification has been achieved. At each iteration, a node that maximizes an information theoretic measure is selected from a collection of candidates. The methodology is heuristic in nature, intending to solve the NP-hard problem of constructing a neural network with a minimum number of nodes. Two strategies for computing a collection of candidate nodes are presented and some experimental results obtained by using the strategies are reported. >

A unifying framework for trie design heuristics

Abstract A general framework for the design of tries with various cost functions is presented. A systematic procedure is proposed which is analogous to the information theoretic heuristic in the use of uncertainty-based information measures. For any file of records to be retrieved by means of keys arranged in a trie, an uncertainty measure must constitute a lower bound on the cost of any trie associated with the file. A trie can be constructed by selecting as the next key to be tested the key that maximizes the information per unit cost based on the uncertainty measure chosen. This stepwise heuristic for constructing tries can be applied to arbitrary cost functions, and it is shown that these information measures possess properties similar to those from information theory. The technique is applied to the problem of constructing tries for the r -error digital search problem which deals with locating the binary word in a file that matches a given input word, while taking into account that as many as r digits of the input may be erroneous.

Computing nearest neighbor pattern classification perceptrons

A procedure for computing three-layer neural networks that classify patterns using the nearest neighbor rule is presented. The procedure constructs a neural network with expected O(n) neurons while requiring expected O(n) computation time for n uniformly distributed training patterns.