arXiv subjects

Lucas Castro

Publications and source records attributed to Lucas Castro.

3 recordsLinked to original sources

arXiv2025

Implementation and Brief Experimental Analysis of the Duan et al. (2025) Algorithm for Single-Source Shortest Paths

We present an implementation and experimental analysis of the deterministic algorithm proposed by Duan et al. (2025) for the Single-Source Shortest Path (SSSP) problem, which achieves the best-known asymptotic upper bound of $O(m \log^{2/3} n)$. We provide a worst-case C++ implementation of this algorithm utilizing $O(n \log^{1/3} n)$ space, as well as a variant that reduces memory usage to $O(n)$ while maintaining the same time complexity in expectation. We compare these implementations against Dijkstra's algorithm on sparse random graphs, grids, and U.S. road networks with up to 10 million vertices. Our results show that while the implementations adhere to their theoretical complexity bounds, large constant factors hinder their practical utility; Dijkstra's algorithm remains 3 to 4 times faster in all tested scenarios. Furthermore, we estimate that the number of vertices would need to vastly exceed $10^{67}$ for the worst-case implementation to outperform Dijkstra's. These findings suggest that a substantial reduction in constant factors is required before this theoretical breakthrough can displace established methods in practical applications.

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arXiv2025

Retroactive Monotonic Priority Queues via Range Searching

The best-known fully retroactive priority queue costs $O(\log^2 m \log \log m)$ time per operation and uses $O(m \log m)$ space, where $m$ is the number of operations performed on the data structure. In contrast, standard (non-retroactive) priority queues can cost $O(\log m)$ time per operation and use $O(m)$ space. So far, it remains open whether these bounds can be achieved for fully retroactive priority queues. In this work, we study a restricted variant of priority queues known as monotonic priority queues. First, we show that finding the minimum in a retroactive monotonic priority queue is a special case of the range-searching problem. Then, we design a fully retroactive monotonic priority queue that costs $O(\log m)$ time per operation and uses $O(m)$ space, achieving the same bounds as a standard priority queue.

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arXiv2024

Exploring Monotone Priority Queues for Dijkstra Optimization

This paper presents a comprehensive overview of monotone priority queues, focusing on their evolution and application in shortest path algorithms. Monotone priority queues are characterized by the property that their minimum key does not decrease over time, making them particularly effective for label-setting algorithms like Dijkstra's. Some key data structures within this category are explored, emphasizing those derived directly from Dial's algorithm, including variations of multi-level bucket structures and radix heaps. Theoretical complexities and practical considerations of these structures are discussed, with insights into their development and refinement provided through a historical timeline.

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