Maintaining Arrays of Contiguous Objects

TL;DR

This paper studies algorithms for dynamically managing a set of modules in a contiguous array, addressing fragmentation, relocation constraints, and providing bounds and methods for online insertions and deletions.

Contribution

It introduces new algorithmic results for module relocation with contiguous constraints, including bounds on sorting and NP-hardness, and efficient online insertion/deletion methods.

Findings

Bound of Theta(n^2) on physical sorting with large free space

NP-hardness results for arbitrary layouts

O(1) moves per insertion/deletion in online scenarios

Abstract

In this paper we consider methods for dynamically storing a set of different objects ("modules") in a physical array. Each module requires one free contiguous subinterval in order to be placed. Items are inserted or removed, resulting in a fragmented layout that makes it harder to insert further modules. It is possible to relocate modules, one at a time, to another free subinterval that is contiguous and does not overlap with the current location of the module. These constraints clearly distinguish our problem from classical memory allocation. We present a number of algorithmic results, including a bound of Theta(n^2) on physical sorting if there is a sufficiently large free space and sum up NP-hardness results for arbitrary initial layouts. For online scenarios in which modules arrive one at a time, we present a method that requires O(1) moves per insertion or deletion and amortized cost O(m_i log M) per insertion or deletion, where m_i is the module's size, M is the size of the largest module and costs for moves are linear in the size of a module.