Reachability in High Treewidth Graphs

Reachability is the problem of deciding whether there is a path from one vertex to the other in the graph. Standard graph traversal algorithms such as DFS and BFS take linear time to decide reachability; however, their space complexity is also linear. On the other hand, Savitch\u27s algorithm takes quasipolynomial time although the space bound is O(log^2 n). Here, we study space efficient algorithms for deciding reachability that run in polynomial time. In this paper, we show that given an n vertex directed graph of treewidth w along with its tree decomposition, there exists an algorithm running in polynomial time and O(w log n) space that solves the reachability problem

Reachability in High Treewidth Graphs

Reachability is the problem of deciding whether there is a path from one vertex to the other in the graph. Standard graph traversal algorithms such as DFS and BFS take linear time to decide reachability however their space complexity is also linear. On the other hand, Savitch's algorithm takes quasipolynomial time although the space bound is $O(\log^2 n)$. Here, we study space efficient algorithms for deciding reachability that runs simultaneously in polynomial time. In this paper, we show that given an $n$ vertex directed graph of treewidth $w$ along with its tree decomposition, there exists an algorithm running in polynomial time and $O(w\log n)$ space, that solves reachability in the graph

Challenges for Efficient Query Evaluation on Structured Probabilistic Data

Query answering over probabilistic data is an important task but is generally intractable. However, a new approach for this problem has recently been proposed, based on structural decompositions of input databases, following, e.g., tree decompositions. This paper presents a vision for a database management system for probabilistic data built following this structural approach. We review our existing and ongoing work on this topic and highlight many theoretical and practical challenges that remain to be addressed.Comment: 9 pages, 1 figure, 23 references. Accepted for publication at SUM 201

New Deterministic Algorithms for Solving Parity Games

We study parity games in which one of the two players controls only a small number $k$ of nodes and the other player controls the $n-k$ other nodes of the game. Our main result is a fixed-parameter algorithm that solves bipartite parity games in time $k^{O(\sqrt{k})}\cdot O(n^3)$, and general parity games in time $(p+k)^{O(\sqrt{k})} \cdot O(pnm)$, where $p$ is the number of distinct priorities and $m$ is the number of edges. For all games with $k = o(n)$ this improves the previously fastest algorithm by Jurdzi{\'n}ski, Paterson, and Zwick (SICOMP 2008). We also obtain novel kernelization results and an improved deterministic algorithm for graphs with small average degree

Optimal Dyck reachability for data-dependence and Alias analysis

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 field-sensitive 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(n2) 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 · logn) 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 sub-cubic 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

IST Austria Technical Report

A fundamental algorithmic problem at the heart of static analysis is Dyck reachability. The input is a graphwhere the edges are labeled with different types of opening and closing parentheses, and the reachabilityinformation is computed via paths whose parentheses are properly matched. We present new results for Dyckreachability 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 asfollows:First, we consider Dyck reachability on bidirected graphs, which is the standard way of performing field-sensitive points-to analysis. Given a bidirected graph withnnodes andmedges, we present: (i) an algorithmwith worst-case running timeO(m+n·α(n)), whereα(n)is the inverse Ackermann function, improving thepreviously knownO(n2)time bound; (ii) a matching lower bound that shows that our algorithm is optimalwrt to worst-case complexity; and (iii) an optimal average-case upper bound ofO(m)time, improving thepreviously knownO(m·logn)bound.Second, we consider the problem of context-sensitive data-dependence analysis, where the task is to obtainanalysis summaries of library code in the presence of callbacks. Our algorithm preprocesses libraries in almostlinear 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 sub-cubic bounds cannot be obtained without obtaining sub-cubic combinatorial algorithms for Boolean MatrixMultiplication, which is a long-standing open problem. Thus we establish that the existing combinatorialalgorithms for Dyck reachability are (conditionally) optimal for general graphs. We also show that the samehardness holds for graphs of constant treewidth.Finally, we provide a prototype implementation of our algorithms for both alias analysis and data-dependenceanalysis. Our experimental evaluation demonstrates that the new algorithms significantly outperform allexisting methods on the two problems, over real-world benchmarks