[Paper Review] Optimal selection on $X+Y$ simplified with layer-ordered heaps

Oliver Serang|arXiv (Cornell University)|Jan 30, 2020

Algorithms and Data Compression4 citations

TL;DR

This paper presents a simplified, optimal algorithm for selection on the Cartesian sum $X+Y$ using layer-ordered heaps, achieving the same theoretical runtime as prior optimal methods but with significantly improved implementability and practical efficiency. The approach replaces complex data structures like soft heaps with a more intuitive and efficient layer-ordered heap design.

ABSTRACT

Selection on the Cartesian sum, $A+B$, is a classic and important problem. Frederickson's 1993 algorithm produced the first algorithm that made possible an optimal runtime. Kaplan \emph{et al.}'s recent 2018 paper descibed an alternative optimal algorithm by using Chazelle's soft heaps. These extant optimal algorithms are very complex; this complexity can lead to difficulty implementing them and to poor performance in practice. Here, a new optimal algorithm is presented, which uses layer-ordered heaps. This new algorithm is both simple to implement and practically efficient.

Motivation & Objective

• To address the long-standing challenge of efficiently solving the selection problem on the Cartesian sum $X+Y$.
• To simplify existing optimal algorithms that rely on complex data structures like soft heaps or intricate priority queues.
• To present a new algorithm that maintains optimal time complexity while being significantly easier to implement and more efficient in practice.
• To demonstrate that layer-ordered heaps can replace complex structures in optimal selection algorithms for $X+Y$.

Proposed method

• The algorithm uses layer-ordered heaps to maintain and extract elements in a structured, hierarchical manner during the selection process.
• Layer-ordered heaps are employed to efficiently manage the candidate set of sums in $X+Y$ by organizing elements into layers based on their values.
• The method performs a selection process that incrementally identifies the $k$-th smallest sum by leveraging the structural properties of layer-ordered heaps.
• It avoids the use of advanced data structures like soft heaps, relying instead on simpler, more cache-friendly heap operations.
• The algorithm maintains a min-heap of candidate sums, where each element is associated with a layer that reflects its relative order in the sum set.
• The selection proceeds by extracting the minimum element from the heap and inserting new candidates in a way that preserves the layer ordering and ensures correctness.

Experimental results

Research questions

• RQ1 Can a simpler data structure replace complex optimal algorithms for $X+Y$ selection?
• RQ2 Does using layer-ordered heaps lead to both optimal runtime and improved practical performance?
• RQ3 Is it possible to achieve optimal time complexity for $X+Y$ selection without relying on advanced data structures like soft heaps?
• RQ4 How does the layer-ordered heap design improve implementability and efficiency compared to prior optimal algorithms?

Key findings

• The proposed algorithm achieves optimal time complexity for selection on $X+Y$, matching the theoretical bounds of prior optimal methods.
• The use of layer-ordered heaps results in a significantly simpler implementation compared to existing optimal algorithms based on soft heaps or complex priority queues.
• The algorithm demonstrates improved practical efficiency due to better cache behavior and reduced constant factors in operations.
• The method maintains correctness and optimality while eliminating the need for sophisticated data structures such as Chazelle’s soft heaps.