Elyar E. Gasanov

Estimates of the complexity of a method to solve the problem of inclusive search

We suggest a method to solve the problem of inclusive search, which has three versions depending on the chosen base set: the set of monotone Boolean function, the set of elementary monotone conjunctions, and the set of Boolean variables. For each version, we estimate the Shannon function for the method complexity and the mean value of the complexity; we study the asymptotic behaviour of the Shannon function and of the logarithm of the mean value, and find that the Shannon function for the complexity of the method suggested asymptotically behaves as the Shannon function for the complexity of inclusive search. This research was supported by the Russian Foundation for Basic Research, grant 98-01-00130.

A constant, in the worst case, algorithm to search for identical objects

We suggest an algorithm to search for identical objects, which, using a memory of volume of order k, almost always performs searching over a set of cardinality k in six elementary operations in the worst case. The research was supported by the Russian Foundation for Basic Research, grant 98-01-00130.

Instantly solvable search problems

We introduce a new class of algorithmic problems called the class of instantly solvable search problems. The problems of this class can be solved, in average, in a time needed to list the data forming the answer plus a constant independent of the dimension of the problem. Examples of instantly solvable search problems are given and the algorithms providing instant solutions are described.

The algorithm for identical object searching with bounded worst-case complexity and linear memory

Abstract We propose and investigate new algorithms permitting to find an identical object in the database using the number of operations not depending on the volume of the database. One algorithm requires memory size that depends linearly on the database volume in the average.

On automaton determinisation of sets of superwords

We introduce the concept of a determinising automaton which, for every superword taken from a given set fed into its input, beginning with some step, at any time t yields the value of the input word at time t + 1, that is, predicts the input superword. We find a criterion whether a given set of superwords is determinisable, that is, whether for the set there exists a determinising automaton. We give the best (in some sense) method to design a determinising automaton for an arbitrary determinisable set of superwords. For some determinisable sets we present optimal and asymptotically optimal determinising automata.

On the functional complexity of a two-dimensional interval search problem

Abstract We suggest a modification of the Bentley-Maurer algorithm which solves a twodimensional interval search problem. This modification allows us to decrease the initially logarithmic average search time to constant, retaining the logarithmic worst-case search time. This algorithm depends on a parameter whose change results in variation of the needed memory from Ϭ(k3) to Ϭ(k log k); the average search time (without the time needed to output the answer) varies from Ϭ(1) to Ϭ(log k). In particular, for any ε > 0 and available memory Ϭ(k1+ε) the average search time is Ϭ(1). On the basis of these results, we give upper bounds for the functional complexity of a two-dimensional interval search problem.

A lower bound for the complexity of inclusive search in the class of tree circuits

In the class of tree information networks with a base set of variables we give examples of inclusive search problems such that the order of magnitude of lower bounds for their complexity exceeds the average time of solution output. This research was supported by the Russian Foundation for Basic Research, grant 95-01-00597.

A lower bound for the complexity of information networks for one partial order relation

We study the search problems where the search relation is a partial order relation. We reveal the common property of these problems called the existence of principal chains. Using this property, we obtain a lower bound for the complexity of the search problems with the natural partial order relation given on the Boolean cube, and prove that the lower bound is asymptotically attainable. This work was supported by the Russian Foundation for Basic Research, grant 95-01-00597.

On a one-dimensional interval search problem

We introduce the notion of an information network with switches which is a generalization of the notion of an information network introduced earlier by the author. For this model the problem of searching the points of a fixed finite set which are within a given interval is investigated. Algorithms solving this problem and some lower bounds of their complexity for different finite sets are obtained.