## Abstract

The fragile complexity of a comparison-based algorithm is $f(n)$ if each input

element participates in $O(f(n))$ comparisons.

In this paper, we explore the fragile complexity of algorithms adaptive to

various restrictions on the input, i.e., algorithms with a fragile complexity

parameterized by a quantity other than the input size~$n$. We show

that searching for the predecessor in a sorted array has fragile complexity

$\Theta(\log k)$, where $k$ is the rank of the query element, both

in a randomized and a deterministic setting. For predecessor

searches, we also show how to optimally reduce the amortized fragile complexity

of the elements in the array. We also prove the following results:

Selecting the $k$th smallest element has expected fragile complexity

$O(\log\log k)$ for the element selected. Deterministically finding

the minimum element has fragile complexity $\Theta(\log(\INV))$ and

$\Theta(\log(\RUNS))$, where $\INV$ is the number of inversions in a

sequence and $\RUNS$ is the number of increasing runs in a sequence.

Deterministically finding the median has fragile complexity

$O(\log(\RUNS) + \log\log n)$ and $\Theta(\log (\INV))$.

Deterministic sorting has fragile complexity $\Theta(\log (\INV))$ but it has

fragile complexity $\Theta(\log n)$ regardless of the number of runs.

element participates in $O(f(n))$ comparisons.

In this paper, we explore the fragile complexity of algorithms adaptive to

various restrictions on the input, i.e., algorithms with a fragile complexity

parameterized by a quantity other than the input size~$n$. We show

that searching for the predecessor in a sorted array has fragile complexity

$\Theta(\log k)$, where $k$ is the rank of the query element, both

in a randomized and a deterministic setting. For predecessor

searches, we also show how to optimally reduce the amortized fragile complexity

of the elements in the array. We also prove the following results:

Selecting the $k$th smallest element has expected fragile complexity

$O(\log\log k)$ for the element selected. Deterministically finding

the minimum element has fragile complexity $\Theta(\log(\INV))$ and

$\Theta(\log(\RUNS))$, where $\INV$ is the number of inversions in a

sequence and $\RUNS$ is the number of increasing runs in a sequence.

Deterministically finding the median has fragile complexity

$O(\log(\RUNS) + \log\log n)$ and $\Theta(\log (\INV))$.

Deterministic sorting has fragile complexity $\Theta(\log (\INV))$ but it has

fragile complexity $\Theta(\log n)$ regardless of the number of runs.

Original language | English |
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Journal | Theoretical Computer Science |

Volume | 915 |

Pages (from-to) | 92-102 |

Number of pages | 11 |

ISSN | 0304-3975 |

DOIs | |

Publication status | Published - 5 Jun 2022 |

## Keywords

- Algorithms
- Comparison based algorithms
- Fragile complexity