Dynamic Transitive Closure Research Articles

Dynamic Transitive Closure-based Static Analysis through the Lens of Quantum Search

Many existing static analysis algorithms suffer from cubic bottlenecks because of the need to compute a dynamic transitive closure (DTC). For the first time, this article studies the quantum speedups on searching subtasks in DTC-based static analysis algorithms using quantum search (e.g., Grover’s algorithm). We first introduce our oracle implementation in Grover’s algorithm for DTC-based static analysis and illustrate our quantum search subroutine. Then, we take two typical DTC-based analysis algorithms: context-free-language reachability and set constraint-based analysis, and show that our quantum approach can reduce the time complexity of these two algorithms to truly subcubic ( \(O(N^2\sqrt {N}{\it polylog}(N))\) ), yielding better results than the upper bound ( O ( N 3 /log N )) of existing classical algorithms. Finally, we conducted a classical simulation of Grover’s search to validate our theoretical approach, due to the current quantum hardware limitation of lacking a practical, large-scale, noise-free quantum machine. We evaluated the correctness and efficiency of our approach using IBM Qiskit on nine open-source projects and randomly generated edge-labeled graphs/constraints. The results demonstrate the effectiveness of our approach and shed light on the promising direction of applying quantum algorithms to address the general challenges in static analysis.

An experimental study of algorithms for fully dynamic transitive closure

We have conducted an extensive experimental study on algorithms for fully dynamic transitive closure. We have implemented the recent fully dynamic algorithms by King [1999], Roditty [2003], Roditty and Zwick [2002, 2004], and Demetrescu and Italiano [2000, 2005] along with several variants and compared them to pseudo fully dynamic and simple-minded algorithms developed in a previous study [Frigioni et al. 2001]. We tested and compared these implementations on random inputs, synthetic (worst-case) inputs, and on inputs motivated by real-world graphs. Our experiments reveal that some of the dynamic algorithms can really be of practical value in many situations.

A faster and simpler fully dynamic transitive closure

We obtain a new fully dynamic algorithm for maintaining the transitive closure of a directed graph. Our algorithm maintains the transitive closure matrix in a total running time of O ( mn + ( ins + del ) · n 2 ), where ins ( del ) is the number of insert (delete) operations performed. Here n is the number of vertices in the graph and m is the initial number of edges in the graph. Obviously, reachability queries can be answered in constant time. The algorithm uses only O ( n 2 ) time which is essentially optimal for maintaining the transitive closure matrix. Our algorithm can also support path queries. If v is reachable from u , the algorithm can produce a path from u to v in time proportional to the length of the path. The best previously known algorithm for the problem is due to Demetrescu and Italiano [2000]. Their algorithm has a total running time of O ( n 3 + ( ins + del ) · n 2 ). The query time is also constant. In addition, we also present a simple algorithm for directed acyclic graphs (DAGs) with a total running time of O ( mn + ins · n 2 + del ). Our algorithms are obtained by combining some new ideas with techniques of Italiano [1986, 1988], King [1999], King and Thorup [2001] and Frigioni et al. [2001]. We also note that our algorithms are extremely simple and can be easily implemented.

Fast Dynamic Transitive Closure with Lookahead

In this paper we consider the problem of dynamic transitive closure with lookahead. We present a randomized one-sided error algorithm with updates and queries in O(n ω(1,1,e)−e ) time given a lookahead of n e operations, where ω(1,1,e) is the exponent of multiplication of n×n matrix by n×n e matrix. For e≤0.294 we obtain an algorithm with queries and updates in O(n 2−e ) time, whereas for e=1 the time is O(n ω−1). This is essentially optimal as it implies an O(n ω ) algorithm for boolean matrix multiplication. We also consider the offline transitive closure in planar graphs. For this problem, we show an algorithm that requires $O(n^{\frac{\omega}{2}})$time to process $n^{\frac{1}{2}}$operations. We also show a modification of these algorithms that gives faster amortized queries. Finally, we give faster algorithms for restricted type of updates, so called element updates. All of the presented algorithms are randomized with one-sided error. All our algorithms are based on dynamic algorithms with lookahead for matrix inverse, which are of independent interest.

Mantaining Dynamic Matrices for Fully Dynamic Transitive Closure

In this paper we introduce a general framework for casting fully dynamic transitive closure into the problem of reevaluating polynomials over matrices. With this technique, we improve the best known bounds for fully dynamic transitive closure. In particular, we devise a deterministic algorithm for general directed graphs that achieves O(n 2) amortized time for updates, while preserving unit worst-case cost for queries. In case of deletions only, our algorithm performs updates faster in O(n) amortized time. We observe that fully dynamic transitive closure algorithms with O(1) query time maintain explicitly the transitive closure of the input graph, in order to answer each query with exactly one lookup (on its adjacency matrix). Since an update may change as many as Ω(n 2) entries of this matrix, no better bounds are possible for this class of algorithms.

Network analysis using transitive closure: New methods for exploring networks

We describe two methods for analysing a complex network which focus on examining changes in the transitive closure of the network under directed and stochastic attack of its edges. Dynamic transitive closure analysis (DTCA) examines changes in critical transitive path for increasingly stringent edge length tolerance. Context specific DTCA is a fuzzy extension of DTCA, which examines changes in DTCA under randomly perturbed transitive contexts. We find that such strategies allow a researcher to identify highly connected elements within a network and uncover related sub-networks. Application to simulated random graphs demonstrates that these methods can be used to determine local network connections as well as quantify the overall transitive natures of the network. We have developed an adjustable resolution O(N 3) parallel algorithm to carry out these analyses which scales nearly linearly with the number of nodes in the network.

Dynamic shortest paths and transitive closure: Algorithmic techniques and data structures

In this paper, we survey fully dynamic algorithms for path problems on general directed graphs. In particular, we consider two fundamental problems: dynamic transitive closure and dynamic shortest paths. Although research on these problems spans over more than three decades, in the last couple of years many novel algorithmic techniques have been proposed. In this survey, we will make a special effort to abstract some combinatorial and algebraic properties, and some common data-structural tools that are at the base of those techniques. This will help us try to present some of the newest results in a unifying framework so that they can be better understood and deployed also by non-specialists.

Trade-offs for fully dynamic transitive closure on DAGs: breaking through the O ( n 2 barrier

We present an algorithm for directed acyclic graphs that breaks through the O ( n 2 ) barrier on the single-operation complexity of fully dynamic transitive closure, where n is the number of edges in the graph. We can answer queries in O ( n ε ) worst-case time and perform updates in O ( n ω(1,ε,1)−ε + n 1+ε ) worst-case time, for any ε∈[0,1], where ω(1,ε,1) is the exponent of the multiplication of an n × n ε matrix by an n ε × n matrix. The current best bounds on ω(1,ε,1) imply an O ( n 0.575 ) query time and an O ( n 1.575 ) update time in the worst case. Our subquadratic algorithm is randomized, and has one-sided error. As an application of this result, we show how to solve single-source reachability in O ( n 1.575 ) time per update and constant time per query.

The dynamic complexity of transitive closure is in DynTC 0

This paper presents a fully dynamic algorithm for maintaining the transitive closure of a binary relation. All updates and queries can be computed by constant depth threshold circuits of polynomial size (TC 0 circuits). This places dynamic transitive closure in the dynamic complexity class DynTC 0, and implies that transitive closure can be maintained in database systems that include first-order update queries and aggregation operators, using a database with size polynomial in the size of the relation.

Ultra-fast aliasing analysis using CLA

We describe the design and implementation of a system for very fast points-to analysis. On code bases of about million lines of unpreprocessed C code, our system performs field-based Andersen-style points-to analysis in less than a second and uses less than 10MB of memory. Our two main contributions are a database-centric analysis architecture called compile-link-analyze (CLA), and a new algorithm for implementing dynamic transitive closure. Our points-to analysis system is built into a forward data-dependence analysis tool that is deployed within Lucent to help with consistent type modifications to large legacy C code bases.

Breaking through then3 barrier: Faster object type inference

Abadi and Cardelli [1] present a series of type systems for their object calculi, four of which are first-order. Palsberg [22] has shown how typability in each one of these systems can be decided in time O(n3) and space O(n2), where n is the size of an untyped object expression, using an algorithm based on dynamic transitive closure. In this paper we improve each one of the four type inference problems from O(n3) to the following time complexities: [see article pdf to view table] Furthermore, our algorithms improve the space complexity from O(n2) to O(n) in each case. The key ingredient that lets us “beat” the worst-case time and space complexity induced by general dynamic transitive closure or similar algorithmic methods is that object subtyping, in contrast to record subtyping, is invariant: an object type is a subtype of a “shorter” type with a subset of the method names if and only if the common components have equal types. © 1999 John Wiley & Sons, Inc.

On Certificates and Lookahead in Dynamic Graph Problems

Recent work in dynamic graph algorithms has led to efficient algorithms for dynamic undirected graph problems such as connectivity. However, no efficient deterministic algorithms are known for the dynamic versions of fundamental directed graph problems like strong connectivity and transitive closure, as well as some undirected graph problems such as maximum matchings and cuts. We provide some explanation for this lack of success by presenting quadratic lower bounds on the certificate complexity of the seemingly difficult problems, in contrast to the known linear certificate complexity for the problems which have efficient dynamic algorithms. In many applications of dynamic (di)graph problems, a certain form of lookahead is available. Specifically, we consider the problems of assembly planning in robotics and the maintenance of relations in databases. These give rise to dynamic strong connectivity and dynamic transitive closure problems, respectively. We explain why it is reasonable, and indeed natural and desirable, to assume that lookahead is available in these two applications. Exploiting lookahead to circumvent their inherent complexity, we obtain efficient dynamic algorithms for strong connectivity and transitive closure.

Lower bounds for dynamic transitive closure, planar point location, and parantheses matching

We give a number of new lower bounds in the cell probe model with logarithmic cell size, which entails the same bounds on the random access computer with logarithmic word size and unit cost operati...

Lower Bounds for Dynamic Transitive Closure, Planar Point Location, and Parentheses Matching

We give a number of new lower bounds in the cell probe model<br />with logarithmic cell size, which entails the same bounds on the random access computer with logarithmic word size and unit cost operations. We study the signed prefix sum problem: given a string of length n of zeroes and signed ones, compute the sum of its ith prefix during updates. We show a<br />lower bound of Omega(log n/log log n) time per operations, even if the prefix sums are bounded by log n/log log n during all updates. We also show that if the update time is bounded by the product of the worst-case update time and the<br />answer to the query, then the update time must be Omega(sqrt(log n/ log log n)).<br /> These results allow us to prove lower bounds for a variety of seemingly unrelated<br />dynamic problems. We give a lower bound for the dynamic planar point location in monotone subdivisions of <br />Omega(log n/ log log n) per operation. We give<br />a lower bound for the dynamic transitive closure problem on upward planar graphs with one source and one sink of <br />Omega(log n/(log logn)^2) per operation. We give a lower bound of Omega(sqrt(log n/log log n)) for the dynamic membership problem of any Dyck language with two or more letters. This implies the same<br />lower bound for the dynamic word problem for the free group with k generators. We also give lower bounds for the dynamic prefix majority and prefix equality problems.

Fully Dynamic Transitive Closure in Plane Dags with one Source and one Sink

We give an algorithm for the Dynamic Transitive Closure Problem for planar directed acyclic graphs with one source and one sink. The graph can be updated in logarithmic time under arbitrary edge insertions and deletions that preserve the embedding. Queries of the form `is there a directed path from u to v ?' for arbitrary vertices u and v can be answered in logarithmic time. The size of the data structure and the initialisation time are linear in the number of edges.<br /> <br />The result enlarges the class of graphs for which a logarithmic (or even polylogarithmic) time dynamic transitive closure algorithm exists. Previously, the only algorithms within the stated resource bounds put restrictions on the topology of the graph or on the delete operation. To obtain our result, we use a new characterisation of the transitive closure in plane graphs with one source and one sink and introduce new techniques to exploit this characterisation.<br /> <br />We also give a lower bound of Omega(log n/log log n) on the amortised complexity of the problem in the cell probe model with logarithmic word size. This is the first dynamic directed graph problem with almost matching lower and upper bounds.

Speeding up dynamic transitive closure for bounded degree graphs

In this paper we present an algorithm for solving two problems in dynamically maintaining the transitive closure of a digraph: In the first problem a sequence of edge insertions is performed on an initially empty graph, interspersed withp transitive closure queries of the form: "is there a path froma tob in the graph". Our algorithm solves this problem in timeO (dm *+p), whered is the maximum outdegree of the resulting graphG andm * is the number of edges in the transitive closure ofG. In the second problem, a sequence of edge deletions is performed on anacyclic graph, interspersed withp transitive closure queries. Once again we solve this problem in timeO (dm *+p), whered is the maximum outdegree of the initial graphG andm * is the number of edges in the transitive closure ofG. For bounded degree graphs, this improves upon previous results. Our algorithms also work when insertions and deletions to the graph are intermixed. Finally, we show how to implement the operation findpath (x, y) which retrieves some path fromx toy in time proportional to the length of the path.