Richard Edwin Stearns

An Analysis of Several Heuristics for the Traveling Salesman Problem

Several polynomial time algorithms finding “good,” but not necessarily optimal, tours for the traveling salesman problem are considered. We measure the closeness of a tour by the ratio of the obtained tour length to the minimal tour length. For the nearest neighbor method, we show the ratio is bounded above by a logarithmic function of the number of nodes. We also provide a logarithmic lower bound on the worst case. A class of approximation methods we call insertion methods are studied, and these are also shown to have a logarithmic upper bound. For two specific insertion methods, which we call nearest insertion and cheapest insertion, the ratio is shown to have a constant upper bound of 2, and examples are provided that come arbitrarily close to this upper bound. It is also shown that for any n≥8, there are traveling salesman problems with n nodes having tours which cannot be improved by making n/4 edge changes, but for which the ratio is 2(1−1/n).

System level concurrency control for distributed database systems

A distributed database system is one in which the database is spread among several sites and application programs “move” from site to site to access and update the data they need. The concurrency control is that portion of the system that responds to the read and write requests of the application programs. Its job is to maintain the global consistency of the distributed database while ensuring that the termination of the application programs is not prevented by phenomena such as deadlock. We assume each individual site has its own local concurrency control which responds to requests at that site and can only communicate with concurrency controls at other sites when an application program moves from site to site, terminates, or aborts. This paper presents designs for several distributed concurrency controls and demonstrates that they work correctly. It also investigates some of the implications of global consistency of a distributed database and discusses phenomena that can prevent termination of application programs.

Syntax-Directed Transduction

A transduction is a mapping from one set of sequences to another. A syntax-directed transduction is a particular type of transduction which is defined on the grammar of a context-free language and which is meant to be a model of part of the translation process used in many compilers. The transduction is considered from an automata theory viewpoint as specifying the input-output relation of a machine. Special consideration is given to machines called translators which both transduce and recognize. In particular, some special conditions are investigated under which syntax-directed translations can be performed on (deterministic) pushdown machines. In addition, some time bounds for translations on Turing machines are derived.

NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs

We present NC-approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance trade-off as the best known approximation schemes for planar graphs. We also define the concept of ?-precision unit disk graphs and show that for such graphs the approximation schemes have a better time versus performance trade-off than the approximation schemes for arbitrary unit disk graphs. Moreover, compared to unit disk graphs, we show that for ?-precision unit disk graphs many more graph problems have efficient approximation schemes.Our NC-approximation schemes can also be extended to obtain efficient NC-approximation schemes for several PSPACE-hard problems on unit disk graphs specified using a restricted version of the hierarchical specification language of Bentley, Ottmann, and Widmayer. The approximation schemes for hierarchically specified unit disk graphs presented in this paper are among the first approximation schemes in the literature for natural PSPACE-hard optimization problems.

Two-Tape Simulation of Multitape Turing Machines

It has long been known that increasing the number of tapes used by a Turing machine does not provide the ability to compute any new functions. On the other hand, the use of extra tapes does make it possible to speed up the computation of certain functions. It is known that a square factor is sometimes required for a one-tape machine to behave as a two-tape machine and that a square factor is always sufficient. The purpose of this paper is to show that, if a given function requires computation time <italic>T</italic> for a <italic>k</italic>-tape realization, then it requires at most computation time <italic>T</italic> log <italic>T</italic> for a two-tape realization. The proof of this fact is constructive; given any <italic>k</italic>-tape machine, it is shown how to design an equivalent two-tape machine that operates within the stated time bounds. In addition to being interesting in its own right, the trade-off relation between number of tapes and speed of computation can be used in a diagonalization argument to show that if <italic>T</italic>(<italic>n</italic>) and <italic>U</italic>(<italic>n</italic>) are two time functions such that inf <italic>T</italic>(<italic>n</italic>) log <italic>T</italic>(<italic>n</italic>) ÷ <italic>U</italic>(<italic>n</italic>) = 0 then there exists a function that can be computed within the time bound <italic>U</italic>(<italic>n</italic>) but not within the time bound <italic>T</italic>(<italic>n</italic>).

Properties of deterministic top-down grammars

The class of context-free grammars that can be deterministically parsed in a top down manner with a fixed amount of look-ahead is investigated. These grammars, called LL(k) grammars where k is the amount of look-ahead are defined and a procedure is given for determining if a context-free grammar is LL(k) for a given value of k . A procedure is given for eliminating the e-rules from an LL(k) grammar at the cost of increasing k by 1. There exist cases in which this increase is inevitable. A procedure is given for obtaining a deterministic push-down machine to recognize a given LL(k) grammar and it is shown that the equivalence problem is decidable for LL(k) grammars. Additional properties are also given.

On the State Assignment Problem for Sequential Machines II

In this paper, the problem of determining economical state assignments for finite-state sequential machines is studied. The fundamental idea in this study is to find methods for selection of these assignments in which each binary variable describing the new state depends on as few variables of the old state as possible. In general, these variable assignments in which the dependence is reduced yield more economical implementation for the sequential machine than the assignments in which the dependence is not reduced. The main tool used in this study is the partition with the substitution property on the set of states of a sequential machine. It is shown that for a sequential machine the existence of assignments with reduced dependence is very closely connected with the existence of partitions with the substitution property on the set of states of the machine. It is shown how to determine these partitions for a given sequential machine and how they can be used to obtain assignments with reduced dependence.

Convergent transfer schemes for

Introduction. The object of this paper is to describe a transfer scheme that converges to the kernel of a game and another scheme that converges to the bargaining set. Both the bargaining set and the kernel can be described as payoff vectors in which every objection (appropriately defined for each case) by one player against another player has a counter-objection. If, for a given payoff vector, a player i can sustain an objection against another playerj, a reasonable negotiation step might be for j to transfer enough of his payoff to i so that i can no longer sustain his objection. This results in a new payoff vector subject to further negotiation. We will show that any infinite sequence of such transfers causes the payoff vector to converge to a vector in the appropriate bargaining set provided only that maximal transfers are frequently made. Thus negotiation can approach the bargaining set without the players solving the defining conjunctive-disjunctive system of linear inequalities. Also, this scheme can be used as a computational technique to find one and possibly more bargaining points. The desired convergence is established by proving the convergence of a general transfer scheme of which the kernel and bargaining set schemes are special cases. For the purpose of this paper, a game is given by an ordered pair (N, v) where N is an n-element set of players and v, the characteristic function, is a zero-one normalized set function over the subsets of N (i.e. v satisfies v({i}) = 0 for all i in N and v(N)= 1). Let E be the space of real n-vectors indexed by N and let A be the set of all group and individually rational payoff vectors. Under the zero-one normalization, A is the set of x in E such that

N

It is possible to test a deterministic pushdown machine to determine if the language it recognizes is regular.

-person games

The known proofs that the equivalence and containment problems for regular expressions, regular grammars and nondeterministic finite automata are PSPACE-complete [SM] depend upon consideration of highly unambiguous expressions, grammars and automata. Here, we prove that such dependence is inherent.Deterministic polynomial-time algorithms are presented for the equivalence and containment problems for unambiguous regular expressions, unambiguous regular grammars and unambiguous finite automata. The algorithms are then extended to ambiguity bounded by a fixed k. Our algorithms depend upon several elementary observations on the solutions of systems of homogeneous linear difference equations with constant coefficients and their relationship with the number of derivations of strings of a given length n by a regular grammar.