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We prove a natural encoding scheme intractable (by showing it UR-complete, a technique which may be used when a problem does not yield to a proof of NP-completeness). This is the first non number-theoretic problem that is UR-complete but not known to be NP-complete. We also redefine UR-completeness (henceforth refered to as PR-completeness) in probabilistic terms thus making the notion conceptually...
T. Parsons proposed and partially analyzed the following pursuit-evasion problem on graphs: A team of searchers traverse the edges of a graph G in pursuit of a fugitive, who moves along the edges of the graph with complete knowledge of the locations of the pursuers. What is the smallest number s(G) of searchers that will suffice for guaranteeing capture of the fugitive? We show that determining whether...
We study a restricted programming language over tree structures. For this language, we give systematic translation rules which map programs into complexity descriptors. The descriptors are in the form of generating functions of average costs. Such a direct approach avoids the recourse to recurrences; it therefore simplifies the task of analyzing algorithms in the class considered and permits analysis...
We investigate two lowest common ancestor (LCA) problems on trees. We give a linear time algorithm for the off-line problem, on a random access machine (RAM). The half-line problem is one in which LCA queries on a fixed tree are arriving on line. We extend our RAM algorithm to answer each individual query in 0(1) time, with 0(n) preprocessing time. Tarjan observed that this result helps to explicate...
We introduce a property of boolean functions, called transitivity which holds of integer, polynomial, and matrix products as well as of many interesting related computational problems. We show that the area of any circuit computing a transitive function grows quadratically with the circuit's maximum data-rate, expressed in bit/second. This result provides a precise analytic expression of an area-time...
The task of eliminating invisible parts when generating an image is one of the central problems in computer graphics. One approach achieves the desired obscuring effect by assigning priority numbers to the faces of an object. Image generation can be speeded up significantly in these priority schemes if priority information of the faces can be partly computed in advance, before a viewing position is...
A formal framework is presented in which to explore the complexity issues of data structures which accommodate various types of range queries. Within this framework, a systematic and reasonably tractable method for assessing inherent complexity is developed. Included among the interesting results are the following: the fact that non-linear lower bounds are readily accessible, and the existence of...
In this note some aspects of the average behavior of the well known sorting algorithm heapsort are investigated. There are two different methods to construct heaps, which are due to Williams, and to Floyd, and for both methods the respective distribution of the heaps is computed in a continuous model. For Floyd's method moreover the expected number of interchanges, and comparisons which are necessary...
It is well known that time bounds for machines correspond closely to size bounds for networks, and that space bounds correspond to depth bounds. It is not known whether simultaneous time and space bounds correspond to simultaneous size and depth bounds. It is shown here that simultaneous time and "reversal" bounds correspond to simultaneous size and depth bounds, and that simultaneous time...
A conjunctive normal form (cnf) is 2-satisfiable, iff any 2 of its clauses are satisfiable. It is shown that every 2-satisfiable cnf s has an interpretation which satisfies at least h??length(s) clauses (h=(√5-1)/2∼0.618). This result is optimal, insofar as the given constant h is maximal. The proof is polynomially constructive, i.e., it yields a polynomial algorithm, which computes an interpretation...
In this paper we will investigate transformations that serve as tools in the design of new data structures. Specifically, we study general methods for converting static structures (in which all elements are known before any searches are performed) to dynamic structures (in which the insertion of a new element can be mixed with searches). We will see three classes of such transformations (each based...
We present a formal proof of correctness of the on-line division algorithm specified in an earlier paper [1]. We also derive two radix 4 on-line division algorithms, with non-redundant and redundant operands respectively.
Let Vk (n) be the minimum average number of pairwise comparisons needed to find the k-th largest of n numbers (k≥2), assuming that all n! orderings are equally likely. D. W. Matula proved that, for some absolute constant c, Vk(n)- n ≤ ck log log n as n → ∞. In the present paper, we show that there exists an absolute constant c′ ≫ 0 such that Vk(n) - n ≥ c′k log log n as n → ∞, proving a conjecture...
We analyze the computational complexity of determining whether F is satisfiable when F is a formula of the classical predicate calculus which obeys certain syntactic restrictions. For example, for the monadic predicate calculus and the Gödel or ∃ ... ∃∀∀∃ ... ∃ prefix class we obtain lower and upper nondeterministic time bounds of the form cn/log n. The main tool in in these proofs is a finite version...
A model for searching algorithms is developed which includes most tree-like searching structures such as lists, binary trees, AVL trees and 2, 3-trees. It is shown that no searching algorithm employing a data structure that is uniquely represented (up to isomorphism) can provide search, insert and delete functions all operating faster than c√n time for every n key tree. The c√n bound is shown to be...
We develop optimal algorithms for forming the intersection of geometric objects in the plane and apply them to such diverse problems as linear programming, hidden-line elimination, and wire layout. Given N line segments in the plane, finding all intersecting pairs requires O(N2) time. We give an O(N log N) algorithm to determine whether any two intersect and use it to detect whether two simple plane...
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