Faster Algorithms for Quantitative Verification in Constant Treewidth Graphs

Proc. ACM Program. Lang.

Goharshady

Okati

et al. 2019

Self Cite

There is a huge gap between the speeds of modern caches and main memories, and therefore cache misses account for a considerable loss of efficiency in programs. The predominant technique to address this issue has been Data Packing: data elements that are frequently accessed within time proximity are packed into the same cache block, thereby minimizing accesses to the main memory. We consider the algorithmic problem of Data Packing on a two-level memory system. Given a reference sequence R of accesses to data elements, the task is to partition the elements into cache blocks such that the number of cache misses on R is minimized. The problem is notoriously difficult: it is NP-hard even when the cache has size 1, and is hard to approximate for any cache size larger than 4. Therefore, all existing techniques for Data Packing are based on heuristics and lack theoretical guarantees. In this work, we present the first positive theoretical results for Data Packing, along with new and stronger negative results. We consider the problem under the lens of the underlying access hypergraphs, which are hypergraphs of affinities between the data elements, where the order of an access hypergraph corresponds to the size of the affinity group. We study the problem parameterized by the treewidth of access hypergraphs, which is a standard notion in graph theory to measure the closeness of a graph to a tree. Our main results are as follows: we show that there is a number q * depending on the cache parameters such that (a) if the access hypergraph of order q * has constant treewidth, then there is a linear-time algorithm for Data Packing; (b) the Data Packing problem remains NP-hard even if the access hypergraph of order q * −1 has constant treewidth. Thus, we establish a fine-grained dichotomy depending on a single parameter, namely, the highest order among access hypegraphs that have constant treewidth; and establish the optimal value q * of this parameter. Finally, we present an experimental evaluation of a prototype implementation of our algorithm. Our results demonstrate that, in practice, access hypergraphs of many commonly-used algorithms have small treewidth. We compare our approach with several state-of-the-art heuristic-based algorithms and show that our algorithm leads to significantly fewer cache-misses.

Section: Introductionmentioning

confidence: 60%

Efficient parameterized algorithms for data packing

Proc. ACM Program. Lang.

Goharshady

Okati

et al. 2019

Self Cite

“…Quantitative Analysis. In contrast with classical verification, which classifies a program as either correct or incorrect, quantitative analysis assigns a value to every run of the program that quantifies the cost/revenue generated by that run [17]. In case of smart contracts, this value can naturally model financial gains or losses of a party in the contract, or the amount of gas/energy used by the contract.…”

Section: Motivating Examplesmentioning

confidence: 99%

“…Hence, quantitative analysis of smart contracts is a natural and important problem [15]. In [17], it was shown that treewidth can help significantly in speeding up the computation of several major notions of quantitative analysis. Therefore, treewidth boundedness leads to much faster algorithms for analyzing the economic effects of a smart contract.…”

Section: Motivating Examplesmentioning

confidence: 99%

See 1 more Smart Citation

The treewidth of smart contracts

Proceedings of the 34th ACM/SIGAPP Symposium on Applied Computing

Goharshady

2019

Self Cite

Smart contracts are programs that are stored and executed on the Blockchain and can receive, manage and transfer money in the form of cryptocurrency units. Two important problems regarding smart contracts are formal analysis and compiler optimization. Formal analysis is extremely important, because smart contracts hold funds worth billions of dollars and their code is immutable after deployment. Hence, an undetected bug can potentially cause significant financial losses. Compiler optimization is also crucial, because every action of a smart contract has to be executed and verified by every node in the Blockchain network. Hence, optimizations in compiling smart contracts can lead to significant savings of computation, time and energy. Two classical approaches in both program analysis and compiler optimization are intraprocedural and interprocedural analysis. In intraprocedural analysis, each function is analyzed separately, while interprocedural analysis considers the entire program. In both cases, optimization and analysis problems are often reduced to graph problems over the control flow graph (CFG) of the program. However, the resulting graph problems are often computationally expensive. Hence, there has been ample research on exploiting structural properties of CFGs to obtain efficient algorithms for these problems. One well-studied structural property is the treewidth. Treewidth is a measure of tree-likeness of graphs and small treewidth can be exploited for efficient algorithms. It is known that intraprocedural CFGs of structured programs have treewidth at most 6, whereas the interprocedural treewidth cannot be bounded. Bounded treewidth has been used as a basis for many efficient intraprocedural analyses. In this paper, we explore the idea of exploiting the treewidth of smart contracts for formal analysis and compiler optimization. First, similar to classical programs, we show that the intraprocedural treewidth of structured Solidity and Vyper smart contracts is at most 9. Second, for global analysis, we prove that the interprocedural treewidth of structured smart contracts is bounded by 10 and, in sharp contrast with classical programs, treewidth-based algorithms can be easily applied for interprocedural analysis. Finally, we supplement our theoretical results with experiments using a tool we implemented for computing treewidth of smart contracts and show that the treewidth is much lower in practice. We use 36,764 real-world Ethereum smart contracts as benchmarks and find that they have an average treewidth of at most 3.35 for the intraprocedural case and 3.65 for the interprocedural case.

“…The treewidth property has received a lot of attention in algorithm community, for NP-complete problems [Arnborg and Proskurowski 1989;Bern et al 1987;Bodlaender 1988], combinatorial optimization problems [Bertele and Brioschi 1972], graph problems such as shortest path [Chatterjee et al 2016b;Chaudhuri and Zaroliagis 1995]. In algorithmic analysis of programming languages and verification the treewidth property has been exploited in interprocedural analysis [Chatterjee et al 2015b], concurrent intraprocedural analysis [Chatterjee et al 2016a], quantitative verification of finite-state graphs [Chatterjee et al 2015a], etc. To the best of our knowledge the constant-treewidth property has not be considered for data-dependence analysis.…”

Section: Other Related Workmentioning

confidence: 99%

Optimal Dyck reachability for data-dependence and alias analysis

Proc. ACM Program. Lang.

Choudhary

Pavlogiannis

2017

Self Cite

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.