# Computing the homology of basic semialgebraic sets in weak exponential time

## TL;DR

This paper presents an algorithm for computing the homology of basic semialgebraic sets that operates in weak exponential time, improving upon the doubly exponential complexity of previous methods for most data.

## Contribution

It introduces a weak exponential time algorithm for homology computation of semialgebraic sets, reducing complexity for almost all data compared to prior approaches.

## Key findings

- Algorithm works in weak exponential time for most data
- Previous algorithms had doubly exponential complexity
- Significant complexity reduction for typical cases

## Abstract

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets which works in weak exponential time. That is, out of a set of exponentially small measure in the space of data the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity which is doubly exponential (and this is so for almost all data).