The Fine-Grained and Parallel Complexity of Andersen's Pointer Analysis

TL;DR

This paper thoroughly analyzes the computational complexity of Andersen's Pointer Analysis, establishing tight bounds and demonstrating the problem's inherent cubic bottleneck, while also exploring conditions for more efficient solutions and parallelizability.

Contribution

The paper provides the first comprehensive fine-grained and parallel complexity landscape of Andersen's Pointer Analysis, including tight bounds and optimal algorithms under various restrictions.

Findings

Established an $O(n^3)$ upper bound for general APA.

Proved an $ ilde{O}(n^{ ext{omega}})$ algorithm under mild restrictions, approaching quadratic time.

Demonstrated the problem's P-completeness, indicating limited parallelizability in general.

Abstract

Pointer analysis is one of the fundamental problems in static program analysis. Given a set of pointers, the task is to produce a useful over-approximation of the memory locations that each pointer may point-to at runtime. The most common formulation is Andersen's Pointer Analysis (APA), defined as an inclusion-based set of $m$ pointer constraints over a set of $n$ pointers. Existing algorithms solve APA in $O(n^2\cdot m)$ time, while it has been conjectured that the problem has no truly sub-cubic algorithm, with a proof so far having remained elusive. In this work we draw a rich fine-grained and parallel complexity landscape of APA, and present upper and lower bounds. First, we establish an $O(n^3)$ upper-bound for general APA, improving over $O(n^2\cdot m)$ as $n=O(m)$. Second, we show that even on-demand APA ("may a specific pointer $a$ point to a specific location $b$?") has an $\Omega(n^3)$ (combinatorial) lower bound under standard complexity-theoretic hypotheses. This formally establishes the long-conjectured "cubic bottleneck" of APA, and shows that our $O(n^3)$-time algorithm is optimal. Third, we show that under mild restrictions, APA is solvable in $\tilde{O}(n^{\omega})$ time, where $\omega<2.373$ is the matrix-multiplication exponent. It is believed that $\omega=2+o(1)$, in which case this bound becomes quadratic. Fourth, we show that even under such restrictions, even the on-demand problem has an $\Omega(n^2)$ lower bound under standard complexity-theoretic hypotheses, and hence our algorithm is optimal when $\omega=2+o(1)$. Fifth, we study the parallelizability of APA and establish lower and upper bounds: (i) in general, the problem is P-complete and hence unlikely parallelizable, whereas (ii) under mild restrictions, the problem is parallelizable. Our theoretical treatment formalizes several insights that can lead to practical improvements in the future.