Fast and Simple Sorting Using Partial Information

Abstract: We consider the problem of sorting $n$ items, given the outcomes of $m$ pre-existing comparisons. We present a simple and natural deterministic algorithm that runs in $O(m+\log T)$ time and does $O(\log T)$ comparisons, where $T$ is the number of total orders consistent with the pre-existing comparisons. Our running time and comparison bounds are best possible up to constant factors, thus resolving a problem that has been studied intensely since 1976 (Fredman, Theoretical Computer Science). The best previous algorithm with a bound of $O(\lg T)$ on the number of comparisons has a time bound of $O(n^{2.5})$ and is more complicated. Our algorithm combines three classic algorithms: topological sort, heapsort with the right kind of heap, and efficient search in a sorted list. It outputs the items in sorted order one by one. It can be modified to stop early, thereby solving the important and more general top-$k$ sorting problem: Given $k$ and the outcomes of some pre-existing comparisons, output the smallest $k$ items in sorted order. The modified algorithm solves the top-$k$ sorting problem in minimum time and comparisons, to within constant factors.

• The paper presents a new algorithm that efficiently sorts items using pre-established comparative data with just O(log T) extra comparisons.
• It synergizes topological sorting, a modified heapsort, and efficient insertion to achieve a time complexity of O(m+n+log T).
• The approach advances theoretical optimality and practical applications in scenarios with incomplete order data, resolving long-standing challenges.

Fast and Simple Sorting Using Partial Information

In this paper, the authors address the computational problem of sorting a set of items—specifically, elements of a totally ordered set—by leveraging existing comparison data. This is a recognized challenge under the umbrella of "sorting under partial information" and has been under scrutiny in theoretical computer science for decades. The primary contribution of this paper is a new algorithm that efficiently sorts these elements utilizing a combination of pre-established comparative data and minimal additional comparisons. The algorithm achieves a running time of $O(m+n+\log T)$ for sorting, where $n$ is the number of items, $m$ is the number of initial comparisons, and $T$ represents the number of total orders consistent with the given comparisons.

The proposed solution is notably efficient, performing only $O(\log T)$ additional comparisons, marking a significant improvement in simplicity and execution speed over the state-of-the-art from prior studies. This advancement resolves long-standing questions in the domain, first posed in 1976, about the computational bounds for this variant of sorting.

Algorithmic Composition

The algorithm, termed "topological heapsort," is an elegant synthesis of three established methodologies: topological sorting, heapsort with a modified heap structure, and efficient insertion operations into a sorted list. This synergy allows the algorithm to navigate the constraints posed by pre-given comparisons effectively, refining both time complexity and simplicity of execution compared to previous methods. Importantly, this technique operates without estimating the total orders $T$ during operation, relying instead on this value solely during analysis.

Theoretical Contributions

The paper's algorithm provides theoretical optimality, improving upon the prior best known solution, which, although effective in comparison count $O( T)$, grappled with a time complexity of $O(n^{2.5})$. Such complexity not only increases computational time but also complicates practical implementation. The new algorithm mitigates these issues by introducing a systemic approach grounded in graph-theoretical concepts.

Their approach exploits a graph representation where vertices correspond to elements and arcs represent established comparisons. Ensuring the graph remains acyclic is crucial, with each solution emerging as a possible topological order consistent with the known data. The mathematical rigor provided around the complexity and logical bounds fortifies the claim of the algorithm’s efficiency and effectiveness, notably leveraging graph theory to inform both the design and analysis.

Practical Implications and Future Directions

Practically, the discussed algorithm broadens the applicability of sorting under partial information, facilitating scenarios where one begins with incomplete comparative datasets. This could translate into improved processes in data science and machine learning domains where initial ordinality information might be sporadically gathered or partially computed.

The conclusion includes a consideration of potential lines of inquiry to further improve heap structures or adapt this algorithmic framework to other problems within computer science, such as sampling and counting topological orders. A future direction could explore dynamic systems where comparisons and items are incrementally updated, necessitating continuous re-sorting—an extension of the current algorithm could potentially handle such scenarios.

In summary, this paper presents a formally sound, efficient, and practical solution to sorting with partial information. Its relevance spans both theoretical and computational pursuits within computer science, with promising applications in optimized sorting operations.