A Model of Optimal Network Structure for Decentralized Nearest Neighbor Search

TL;DR

This paper introduces a mathematical model to identify optimal network structures for decentralized nearest neighbor search, providing exact and heuristic solutions for different lattice sizes and analyzing various distance metrics.

Contribution

It presents a novel mathematical programming approach to determine optimal network configurations for decentralized search, addressing key questions about efficiency and properties of such networks.

Findings

Exact solution for 4x4 lattice network

Heuristic solutions for 5x5 to 7x7 lattices

Analysis of different distance metrics (L1, L2, L_inf)

Abstract

One of the approaches for the nearest neighbor search problem is to build a network which nodes correspond to the given set of indexed objects. In this case the search of the closest object can be thought as a search of a node in a network. A procedure in a network is called decentralized if it uses only local information about visited nodes and its neighbors. Networks, which structure allows efficient performing the nearest neighbour search by a decentralised search procedure started from any node, are of particular interest especially for pure distributed systems. Several algorithms that construct such networks have been proposed in literature. However, the following questions arise: "Are there network models in which decentralised search can be performed faster?"; "What are the optimal networks for the decentralised search?"; "What are their properties?". In this paper we partially give answers to these questions. We propose a mathematical programming model for the problem of determining an optimal network structure for decentralized nearest neighbor search. We have found an exact solution for a regular lattice of size 4x4 and heuristic solutions for sizes from 5x5 to 7x7. As a distance function we use L1 , L2 and L_inf metrics. We hope that our results and the proposed model will initiate study of optimal network structures for decentralised nearest neighbour search.