# Fragile Complexity of Comparison-Based Algorithms

## TL;DR

This paper introduces the concept of fragile complexity in comparison-based algorithms, analyzing the maximum comparisons any element undergoes, and provides bounds and separations between deterministic and randomized approaches for fundamental problems.

## Contribution

It defines fragile complexity, establishes bounds for core problems, and demonstrates differences between deterministic and randomized algorithms in this context.

## Key findings

- Fragile complexity differs from comparator network depth.
- Randomization can reduce fragile complexity.
- Bounds are established for Minimum, Selection, Sorting, and Heap Construction.

## Abstract

We initiate a study of algorithms with a focus on the computational complexity of individual elements, and introduce the fragile complexity of comparison-based algorithms as the maximal number of comparisons any individual element takes part in. We give a number of upper and lower bounds on the fragile complexity for fundamental problems, including Minimum, Selection, Sorting and Heap Construction. The results include both deterministic and randomized upper and lower bounds, and demonstrate a separation between the two settings for a number of problems. The depth of a comparator network is a straight-forward upper bound on the worst case fragile complexity of the corresponding fragile algorithm. We prove that fragile complexity is a different and strictly easier property than the depth of comparator networks, in the sense that for some problems a fragile complexity equal to the best network depth can be achieved with less total work and that with randomization, even a lower fragile complexity is possible.