A Competitive Strategy for Distance-Aware Online Shape Allocation

TL;DR

This paper introduces an online strategy for shape allocation within a unit square, aiming to minimize the maximum average Manhattan distance among allocated regions, with proven competitive ratios for continuous and discrete cases.

Contribution

The paper proposes a novel online allocation strategy based on space-filling curves, providing competitive ratio bounds for both continuous shapes and discrete point sets.

Findings

Achieves a competitive ratio of 1.8092 for continuous shapes.

Achieves a competitive ratio of 1.7848 for discrete point sets.

Introduces a new approach using space-filling curves for online shape allocation.

Abstract

We consider the following online allocation problem: Given a unit square S, and a sequence of numbers n_i between 0 and 1, with partial sum bounded by 1; at each step i, select a region C_i of previously unassigned area n_i in S. The objective is to make these regions compact in a distance-aware sense: minimize the maximum (normalized) average Manhattan distance between points from the same set C_i. Related location problems have received a considerable amount of attention; in particular, the problem of determining the "optimal shape of a city", i.e., allocating a single n_i has been studied. We present an online strategy, based on an analysis of space-filling curves; for continuous shapes, we prove a factor of 1.8092, and 1.7848 for discrete point sets.