The fine-grained and parallel complexity of andersen’s pointer analysis

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. Scalability is extremely important, as points-to information is a prerequisite to many other components in the static-analysis pipeline. Existing algorithms solve APA in O(n²· m) time, while it has been conjectured that the problem has no truly sub-cubic algorithm, with a proof so far having remained elusive. It is also well-known that APA can be solved in O(n²) time under certain sparsity conditions that hold naturally in some settings. Besides these simple bounds, the complexity of the problem has remained poorly understood.

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(n³) upper-bound for general APA, improving over O(n²· 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 Ω(n³) (combinatorial) lower bound under standard complexity-theoretic hypotheses. This formally establishes the long-conjectured “cubic bottleneck” of APA, and shows that our O(n³)-time algorithm is optimal. Third, we show that under mild restrictions, APA is solvable in Õ(n^ω) time, where ω<2.373 is the matrix-multiplication exponent. It is believed that ω=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 Ω(n²) lower bound under standard complexity-theoretic hypotheses, and hence our algorithm is optimal when ω=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.