DFS is unsparsable and lookahead can help in maximal matching

In this paper we study two problems in the context of fully dynamic graph algorithms that is, when we have to handle updates (insertions and removals of edges), and answer queries regarding the current graph, preferably with a better time bound than that when running a classical algorithm from scratch each time a query arrives. In the first part we show that there are dense (directed) graphs having no nontrivial strong certificates for maintaining a depth-first search tree, hence the so-called sparsification technique cannot be applied effectively to this problem. In the second part, we show that a maximal matching can be maintained in an (undirected) graph with a deterministic amortized update cost of O(log m) (where m is the all-time maximum number of the edges), provided that a lookahead of length m is available, i.e. we can “take a peek” at the next m update operations in advance

Faster Algorithms for Rectangular Matrix Multiplication

Let {\alpha} be the maximal value such that the product of an n x n^{\alpha} matrix by an n^{\alpha} x n matrix can be computed with n^{2+o(1)} arithmetic operations. In this paper we show that \alpha>0.30298, which improves the previous record \alpha>0.29462 by Coppersmith (Journal of Complexity, 1997). More generally, we construct a new algorithm for multiplying an n x n^k matrix by an n^k x n matrix, for any value k\neq 1. The complexity of this algorithm is better than all known algorithms for rectangular matrix multiplication. In the case of square matrix multiplication (i.e., for k=1), we recover exactly the complexity of the algorithm by Coppersmith and Winograd (Journal of Symbolic Computation, 1990). These new upper bounds can be used to improve the time complexity of several known algorithms that rely on rectangular matrix multiplication. For example, we directly obtain a O(n^{2.5302})-time algorithm for the all-pairs shortest paths problem over directed graphs with small integer weights, improving over the O(n^{2.575})-time algorithm by Zwick (JACM 2002), and also improve the time complexity of sparse square matrix multiplication.Comment: 37 pages; v2: some additions in the acknowledgment

Abstract Acceleration in Linear relation analysis (extended version)

Linear relation analysis is a classical abstract interpretation based on an over-approximation of reachable numerical states of a program by convex polyhedra. Since it works with a lattice of infinite height, it makes use of a widening operator to enforce the convergence of fixed point computations. Abstract acceleration is a method that computes the precise abstract effect of loops wherever possible and uses widening in the general case. Thus, it improves both the precision and the efficiency of the analysis. This research report gives a comprehensive tutorial on abstract acceleration: its origins in Presburger-based acceleration including new insights w.r.t. the linear accelerability of linear transformations, methods for simple and nested loops, recent extensions, tools and applications, and a detailed discussion of related methods and future perspectives. This is the long version of a paper under submission

Efficient reduction of nondeterministic automata with application to language inclusion testing

We present efficient algorithms to reduce the size of nondeterministic B\"uchi word automata (NBA) and nondeterministic finite word automata (NFA), while retaining their languages. Additionally, we describe methods to solve PSPACE-complete automata problems like language universality, equivalence, and inclusion for much larger instances than was previously possible ($\ge 1000$ states instead of 10-100). This can be used to scale up applications of automata in formal verification tools and decision procedures for logical theories. The algorithms are based on new techniques for removing transitions (pruning) and adding transitions (saturation), as well as extensions of classic quotienting of the state space. These techniques use criteria based on combinations of backward and forward trace inclusions and simulation relations. Since trace inclusion relations are themselves PSPACE-complete, we introduce lookahead simulations as good polynomial time computable approximations thereof. Extensive experiments show that the average-case time complexity of our algorithms scales slightly above quadratically. (The space complexity is worst-case quadratic.) The size reduction of the automata depends very much on the class of instances, but our algorithm consistently reduces the size far more than all previous techniques. We tested our algorithms on NBA derived from LTL-formulae, NBA derived from mutual exclusion protocols and many classes of random NBA and NFA, and compared their performance to the well-known automata tool GOAL.Comment: 69 pages. arXiv admin note: text overlap with arXiv:1210.662

Reducing Nondeterministic Tree Automata by Adding Transitions

We introduce saturation of nondeterministic tree automata, a technique that consists of adding new transitions to an automaton while preserving its language. We implemented our algorithm on minotaut - a module of the tree automata library libvata that reduces the size of automata by merging states and removing superfluous transitions - and we show how saturation can make subsequent merge and transition-removal operations more effective. Thus we obtain a Ptime algorithm that reduces the size of tree automata even more than before. Additionally, we explore how minotaut alone can play an important role when performing hard operations like complementation, allowing to both obtain smaller complement automata and lower computation times. We then show how saturation can extend this contribution even further. We tested our algorithms on a large collection of automata from applications of libvata in shape analysis, and on different classes of randomly generated automata.Comment: In Proceedings MEMICS 2016, arXiv:1612.0403

Agent-based modeling and simulation of individual traffic as an environment for bus schedule simulation

To re-establish the regular driving operations of a tram network, which was disturbed significantly by unforeseen external events, traffic schedulers apply rescheduling and rerouting strategies. These strategies are usually multi-modal; they consider the interaction of trams, buses, even taxis. Thus, to evaluate the applicability of a given rescheduling or rerouting strategy prior to its implementation in the real-world system, a multi-modal simulation software is needed. In this article we present an agent-based model of individual traffic which will be applied as background to a planned simulation of bus traffic. These combined models are to be integrated with an existing tram schedule simulation; the resulting multi-modal model will then be applied to evaluate the usefulness of given rescheduling or rerouting strategies. After a short introduction to agent-based modeling and simulation, as well as to existing models of individual traffic, this paper proposes to model the behavior of individual traffic as an environment for agent-based bus schedule simulation. Finally, some experiments are conducted by modeling and simulating individual traffic in Cologne's highly frequented Barbarossaplatz area