Conditional Lower Bound for Inclusion-Based Points-to Analysis
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Inclusion-based (i.e., Andersen-style) points-to analysis is a fundamental static analysis problem. The seminal work of Andersen gave a worst-case cubic O(n^3) time points-to analysis algorithm for C, where n is proportional to the number of program variables. An algorithm is truly subcubic if it runs in O(n^3-δ) time for some δ > 0. Despite decades of extensive effort on improving points-to analysis, the cubic bound remains unbeaten. The best combinatorial analysis algorithms have a “slightly subcubic” O(n^3 / log n) complexity. It is an interesting open problem whether points-to analysis can be solved in truly subcubic time. In this paper, we prove that a truly subcubic O(n^3-δ) time combinatorial algorithm for inclusion-based points-to analysis is unlikely: a truly subcubic combinatorial points-to analysis algorithm implies a truly subcubic combinatorial algorithm for Boolean Matrix Multiplication (BMM). BMM is a well-studied problem, and no truly subcubic combinatorial BMM algorithm has been known. The fastest combinatorial BMM algorithms run in time O(n^3/ log^4 n). Our result includes a simplified proof of the BMM-hardness of Dyck-reachability. The reduction is interesting in its own right. First, it is slightly stronger than the existing BMM-hardness results because our reduction only requires one type of parenthesis in Dyck-reachability (D_1-reachability). Second, we formally attribute the “cubic bottleneck” to the need to solve D_1-reachability, which captures the semantics of pointer references/dereferences. This new perspective enables a more general reduction that applies to programs with arbitrary pointer statements types. Last, our reduction based on D_1-reachability shows that demand-driven points-to analysis is as hard as the exhaustive counterpart.
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