# Upper Bounds on Number of Steals in Rooted Trees

## TL;DR

This paper derives tight upper bounds on the number of work steals in parallel computations modeled by rooted trees, relevant for optimizing multithreaded processing efficiency.

## Contribution

It provides a novel mathematical bound on steals in work stealing algorithms for rooted tree models, extending understanding of parallel computation limits.

## Key findings

- Maximum steals are bounded by a sum involving binomial coefficients and tree parameters.
- The bounds are tight for the specified rooted tree configurations.
- Results inform optimal scheduling in parallel computing environments.

## Abstract

Inspired by applications in parallel computing, we analyze the setting of work stealing in multithreaded computations. We obtain tight upper bounds on the number of steals when the computation can be modeled by rooted trees. In particular, we show that if the computation with $n$ processors starts with one processor having a complete $k$-ary tree of height $h$ (and the remaining $n-1$ processors having nothing), the maximum possible number of steals is $\sum_{i=1}^n(k-1)^i\binom{h}{i}$.