Optimal Dyck reachability for data-dependence and alias analysis

Abstract

A fundamental algorithmic problem at the heart of static analysis is Dyck reachability. The input is a graph where the edges are labeled with different types of opening and closing parentheses, and the reachability information is computed via paths whose parentheses are properly matched. We present new results for Dyck reachability problems with applications to alias analysis and data-dependence analysis. Our main contributions, that include improved upper bounds as well as lower bounds that establish optimality guarantees, are as follows. First, we consider Dyck reachability on bidirected graphs, which is the standard way of performing fieldsensitive points-to analysis. Given a bidirected graph with n nodes and m edges, we present: (i) an algorithm with worst-case running time O(m + n · α(n)), where α(n) is the inverse Ackermann function, improving the previously known O(n 2 ) time bound; (ii) a matching lower bound that shows that our algorithm is optimal wrt to worst-case complexity; and (iii) an optimal average-case upper bound of O(m) time, improving the previously known O(m · log n) bound. Second, we consider the problem of context-sensitive data-dependence analysis, where the task is to obtain analysis summaries of library code in the presence of callbacks. Our algorithm preprocesses libraries in almost linear time, after which the contribution of the library in the complexity of the client analysis is only linear, and only wrt the number of call sites. Third, we prove that combinatorial algorithms for Dyck reachability on general graphs with truly subcubic bounds cannot be obtained without obtaining sub-cubic combinatorial algorithms for Boolean Matrix Multiplication, which is a long-standing open problem. Thus we establish that the existing combinatorial algorithms for Dyck reachability are (conditionally) optimal for general graphs. We also show that the same hardness holds for graphs of constant treewidth. Finally, we provide a prototype implementation of our algorithms for both alias analysis and data-dependence analysis. Our experimental evaluation demonstrates that the new algorithms significantly outperform all existing methods on the two problems, over real-world benchmarks.

Optimal and Perfectly Parallel Algorithms for On-demand Data-Flow Analysis

“…Then, if (u, d 1 ) is reachable from (s p , d 3 ) via a same-context valid path, then by adding the edge e, the vertex (v, d 2 ) also becomes accessible from (s p , d 3 ). Hence, it adds the shortcut edge ((s p , d 3 ), (v, d 2 )) to Q, so that it is later added to the graph H. Moreover, if u is the start s p of the procedure p and v is its end e p , then for every call vertex c l calling the procedure p and its respective return-site r l , we can add summary edges that summarize the effect of calling p (lines [14][15][16][17][18][19]. Finally, lines 20-24 computeĜ as discussed above.…”

“…the tree decomposition obtained by removing b l (lines 10-11). Finally, it repeats the reachability computation on H l (lines [12][13][14]. The running time of the algorithm is O(n · |D * | 3 ).…”

“…Hence, the total runtime of this part of the algorithm is O(n · |D * | 2 · log n). Similarly, in line 8, the algorithm considers O(t 2 ) = O(1) combinations of u, v and O(|D * | 2 ) combinations of d 1 , d 2 and performs O(log n) updates for each of them (lines [12][13][14]. Hence, the total runtime of this part and the whole algorithm is O(n · |D * | 3 · log n).…”

“…In addition to the treewidth, our algorithms also exploit specific structural properties of IFDS. In general, small treewidth alone does not improve the complexity of CFL reachability [14].…”

“…Treewidth has important algorithmic implications, as many graph problems that are hard to solve in general admit efficient solutions on graphs of low treewidth. In the context of program analysis, this property has been exploited to develop improvements for register allocation [62,9] (a technique implemented in the Small Device C Compiler [27]), cache management [17], on-demand algebraic path analysis [16], on-demand intraprocedural data-flow analysis of concurrent programs [19] and data-dependence analysis [14].…”