Combinatorial Spaces And Order Topologies

TL;DR

This paper explores how order topologies can be used to develop efficient search algorithms in combinatorial spaces, achieving logarithmic complexity without sorting, and discusses the theoretical foundations of these methods.

Contribution

It introduces the use of order topologies for combinatorial search, demonstrating logarithmic complexity and the concept of order invariance in search algorithms.

Findings

Order topologies enable efficient combinatorial search algorithms.

Logarithmic complexity is achievable without sorting in certain topological frameworks.

The paper discusses the theoretical basis for order-invariant search methods.

Abstract

An archetypal problem discussed in computer science is the problem of searching for a given number in a given set of numbers. Other than sequential search, the classic solution is to sort the list of numbers and then apply binary search. The binary search problem has a complexity of O(logN) for a list of N numbers while the sorting problem cannot be better than O(N) on any sequential computer following the usual assumptions. Whenever the problem of deciding partial order can be done in O(1), a variation of the problem on some bounded list of numbers is to apply binary search without resorting to sort. The overall complexity of the problem is then O(log R) for some radius R. A logarithmic upper-bound for finite encodings is shown. Also, the topology of orderings can provide efficient algorithms for search problems in combinatorial spaces. The main characteristic of those spaces is that they have typical exponential space complexities. The factorial case describes an order topology that can be illustrated using the combinatorial polytope . When a known order topology can be combined to a given formulation of a search problem, the resulting search problem has a polylogarithmic complexity. This logarithmic complexity can then become useful in combinatorial search by providing a logarithmic break-down. These algorithms can be termed as the class of search algorithms that do not require read and are equivalent to the class of logarithmically recursive functions. Also, the notion of order invariance is discussed.