An Optimal Algorithm for Range Search on Multidimensional Points

TL;DR

This paper introduces a new data structure called BITS k$d$-tree that enables efficient range search and fast updates on multidimensional points, improving upon previous algorithms in computational geometry.

Contribution

The paper presents the BITS k$d$-tree, a novel data structure that achieves $ heta(t)$ query time and supports rapid updates, advancing multidimensional range search methods.

Findings

Range search in $ heta(t)$ time using BITS k$d$-tree

Supports insertion in $ heta(1)$ time and deletion in $O(\log n)$ time

Improves upon previous $O(\log^k n + t)$ algorithms

Abstract

This paper proposes an efficient and novel method to address range search on multidimensional points in $\theta(t)$ time, where $t$ is the number of points reported in $\Re^k$ space. This is accomplished by introducing a new data structure, called BITS $k$d-tree. This structure also supports fast updation that takes $\theta(1)$ time for insertion and $O(\log n)$ time for deletion. The earlier best known algorithm for this problem is $O(\log^k n+t)$ time in the pointer machine model.