Optimal Constructions of Hybrid Algorithms

TL;DR

This paper develops optimal hybrid algorithms for on-line problem-solving, combining deterministic and randomized strategies to minimize solving time, and resolves several open questions in the field.

Contribution

It introduces the first optimal deterministic and efficient randomized hybrid algorithms for solving problems with multiple basic algorithms, addressing open research questions.

Findings

Constructed optimal deterministic hybrid algorithm.

Designed efficient randomized hybrid algorithm.

Proved the randomized algorithm's optimality for lambda=1.

Abstract

We study on-line strategies for solving problems with hybrid algorithms. There is a problem Q and w basic algorithms for solving Q. For some lambda <= w, we have a computer with lambda disjoint memory areas, each of which can be used to run a basic algorithm and store its intermediate results. In the worst case, only one basic algorithm can solve Q in finite time, and all the other basic algorithms run forever without solving Q. To solve Q with a hybrid algorithm constructed from the basic algorithms, we run a basic algorithm for some time, then switch to another, and continue this process until Q is solved. The goal is to solve Q in the least amount of time. Using competitive ratios to measure the efficiency of a hybrid algorithm, we construct an optimal deterministic hybrid algorithm and an efficient randomized hybrid algorithm. This resolves an open question on searching with multiple robots posed by Baeza-Yates, Culberson and Rawlins. We also prove that our randomized algorithm is optimal for lambda = 1, settling a conjecture of Kao, Reif and Tate.