Fine-grained I/O complexity via reductions: new lower bounds, faster algorithms, and a time hierarchy

Abstract

© Erik D. Demaine, Andrea Lincoln, Quanquan C. Liu, Jayson Lynch, and Virginia Vassilevska Williams. This paper initiates the study of I/O algorithms (minimizing cache misses) from the perspective of fine-grained complexity (conditional polynomial lower bounds). Specifically, we aim to answer why sparse graph problems are so hard, and why the Longest Common Subsequence problem gets a savings of a factor of the size of cache times the length of a cache line, but no more. We take the reductions and techniques from complexity and fine-grained complexity and apply them to the I/O model to generate new (conditional) lower bounds as well as faster algorithms. We also prove the existence of a time hierarchy for the I/O model, which motivates the fine-grained reductions. Using fine-grained reductions, we give an algorithm for distinguishing 2 vs. 3 diameter and radius that runs in O(|E|2/(MB)) cache misses, which for sparse graphs improves over the previous O(|V |2/B) running time. We give new reductions from radius and diameter to Wiener index and median. These reductions are new in both the RAM and I/O models. We show meaningful reductions between problems that have linear-time solutions in the RAM model. The reductions use low I/O complexity (typically O(n/B)), and thus help to finely capture the relationship between “I/O linear time” (n/B) and RAM linear time (n). We generate new I/O assumptions based on the di culty of improving sparse graph problem running times in the I/O model. We create conjectures that the current best known algorithms for Single Source Shortest Paths (SSSP), diameter, and radius are optimal. From these I/O-model assumptions, we show that many of the known reductions in the word-RAM model can naturally extend to hold in the I/O model as well (e.g., a lower bound on the I/O complexity of Longest Common Subsequence that matches the best known running time). We prove an analog of the Time Hierarchy Theorem in the I/O model, further motivating the study of fine-grained algorithmic di erences.