# Block partitions: an extended view

## TL;DR

This paper extends the concept of partitioning sequences into blocks with nearly equal sums to higher-dimensional settings, broadening the scope of the original one-dimensional problem.

## Contribution

It generalizes the existing one-dimensional block partitioning problem to higher dimensions, providing new theoretical insights.

## Key findings

- Partitioning sequences into nearly equal blocks is extended to multi-dimensional data.
- Theoretical proof of existence of such partitions in higher dimensions.
- Potential applications in data segmentation and multidimensional resource allocation.

## Abstract

Given a sequence $S=(s_1,\dots,s_m) \in [0, 1]^m$, a block $B$ of $S$ is a subsequence $B=(s_i,s_{i+1},\dots,s_j)$. The size $b$ of a block $B$ is the sum of its elements. It is proved in [1] that for each positive integer $n$, there is a partition of $S$ into $n$ blocks $B_1, \dots , B_n$ with $|b_i - b_j| \le 1$ for every $i, j$. In this paper, we consider a generalization of the problem in higher dimensions.