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Towards universally optimal sorting algorithms

Sandeep Sen

TL;DR

The paper addresses how sorting efficiency can be made universally optimal by conditioning on input-dependent parameters beyond size, challenging the traditional $\Omega(n \log n)$ worst-case benchmark. It develops an oblivious, partition-sort-based approach guided by a new $\alpha$-sorted subsequence measure, achieving a bound $O\left(\sum_{i=1}^{k} n_i \log\left(\frac{n}{n_i}+1\right)\right)$, with a matching lower bound expressed via $H(\cdot)$. By linking to SUS/SMS, multisorting, and universal optimality, the work broadens adaptive sorting theory and provides a framework for input-structure-aware algorithm design. Practically, it proposes methods that adapt to input structure without explicit parameter estimation, suggesting a versatile paradigm for universal optimality applicable to broader computational problems.

Abstract

We formalize a new paradigm for optimality of algorithms, that generalizes worst-case optimality based only on input-size to problem-dependent parameters including implicit ones. We re-visit some existing sorting algorithms from this perspective, and also present a novel measure of sortedness that leads to an optimal algorithm based on partition sort. This paradigm of measuring efficiency of algorithms looks promising for further interesting applications beyond the existing ones.

Towards universally optimal sorting algorithms

TL;DR

The paper addresses how sorting efficiency can be made universally optimal by conditioning on input-dependent parameters beyond size, challenging the traditional worst-case benchmark. It develops an oblivious, partition-sort-based approach guided by a new -sorted subsequence measure, achieving a bound , with a matching lower bound expressed via . By linking to SUS/SMS, multisorting, and universal optimality, the work broadens adaptive sorting theory and provides a framework for input-structure-aware algorithm design. Practically, it proposes methods that adapt to input structure without explicit parameter estimation, suggesting a versatile paradigm for universal optimality applicable to broader computational problems.

Abstract

We formalize a new paradigm for optimality of algorithms, that generalizes worst-case optimality based only on input-size to problem-dependent parameters including implicit ones. We re-visit some existing sorting algorithms from this perspective, and also present a novel measure of sortedness that leads to an optimal algorithm based on partition sort. This paradigm of measuring efficiency of algorithms looks promising for further interesting applications beyond the existing ones.

Paper Structure

This paper contains 10 sections, 4 theorems, 8 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Oblivious algorithms and parameter estimation
  4. Main contributions
  5. A simple algorithm for the distance bound
  6. A natural characterization of sortedness and a simple algorithm
  7. A lower bound for $(n_1 , n_2 \ldots n_k)$-sorted input
  8. A more robust measure of optimality
  9. Concluding remarks and further work
  10. Appendix

Key Result

Lemma 1

If every element is within distance $k$ of its final position, then we can sort the input in $O(n \log k)$ comparisons.

Figures (6)

  • Figure 1: The 2nd row has elements within distance 1 and 3rd and 4th row within distance 8 or $n/2$ for a row with $n$. Further, the number of inversions in 2nd 3rd and 4th row are $n , n^2 /2 , n^2 /2$ respectively. However, there is still a large degree of sortedness in 3rd and 4th row.
  • Figure 2: Figure (a) depicts that the dirty segment of size at most $k$ can span at most two adjacent $k$-blocks. In (b) the odd-even pairs of blocks are sorted and in (c), the even-odd k-block pairs are sorted, following which the 0-1 array is completely sorted.
  • Figure 3: Row 2 is a stable partitioning around 33 but row 3 is not. Note that stable partition around 39 would yield a sorted set
  • Figure 4: In the example above, 23 ,25, 33 , 34 , 39 is a 5-sorted, but not maximal, as 45 can be appended. The above sequence is a 2-6-2-1-1-1-2-1 sequence corresponding to (62,68) (23 ,25, 33 , 34 , 39 , 45) (6,8) (85)(84)(72)(55,56)(4) sub-sequences.
  • Figure 5: Stable-Partition-sort($A$)
  • ...and 1 more figures

Theorems & Definitions (4)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Theorem 2