Average case analysis of DJ graphs

AbstractSreedhar et al. [V.C. Sreedhar, G.R. Gao, Y.-F. Lee, A new framework for elimination-based data flow analysis using DJ graphs, ACM Trans. Program. Lang. Syst. 20 (2) (1998) 388–435; V.C. Sreedhar, Efficient program analysis using DJ graphs, PhD thesis, School of Computer Science, McGill University, Montréal, Québec, Canada, 1995] have presented an elimination-based algorithm to solve data flow problems. A thorough analysis of the algorithm shows that the worst-case performance is at least quadratic in the number of nodes of the underlying graph. In contrast, Sreedhar reports a linear time behavior based on some practical applications.In this paper we prove that for goto-free programs, the average case behavior is indeed linear. As a byproduct our result also applies to the average size of the so-called dominance frontier.A thorough average case analysis based on a graph grammar is performed by studying properties of the j-edges in DJ graphs. It appears that this is the first time that a graph grammar is used in order to analyze an algorithm. The average linear time of the algorithm is obtained by classic techniques in the analysis of algorithms and data structures such as singularity analysis of generating functions and transfer lemmas

Random trees in queueing systems with deadlines

AbstractWe survey our research on scheduling aperiodic tasks in real-time systems in order to illustrate the benefits of modelling queueing systems by means of random trees. Relying on a discrete-time single-server queueing system, we investigated deadline meeting properties of several scheduling algorithms employed for servicing probabilistically arriving tasks, characterized by arbitrary arrival and execution time distributions and a constant service time deadline T. Taking a non-queueing theory approach (i.e., without stable-stable assumptions) we found that the probability distribution of the random time sT where such a system operates without violating any task's deadline is approximately exponential with parameter λT = 1μT, with the expectation E[sT] = μT growing exponentially in T. The value μT depends on the particular scheduling algorithm, and its derivation is based on the combinatorial and asymptotic analysis of certain random trees. This paper demonstrates that random trees provide an efficient common framework to deal with different scheduling disciplines and gives an overview of the various combinatorial and asymptotic methods used in the appropriate analysis

Three-speed ballistic annihilation: phase transition and universality

We consider ballistic annihilation, a model for chemical reactions introduced in the 1980's physics literature. In this particle system, initial locations are given by a renewal process on the line, motions are ballistic (i.e. particles are assigned constant velocities, chosen i.i.d.) and collisions between pairs of particles result in mutual annihilation. We focus on the case when velocities are symmetrically distributed among three values, i.e. particles either remain static (with probability $p$) or move at constant velocity uniformly chosen among $\pm1$. We show that this model goes through a phase transition at $p_c=1/4$ between a subcritical regime where each particle eventually annihilates and a supercritical regime where a positive density of static particles is never hit, confirming 1990's predictions of Droz et al.[8] for the particular case of a Poisson process. Our result covers cases where triple collisions can happen; these are resolved by annihilation of one static and one randomly chosen moving particle. Our arguments, of combinatorial nature, show that, although the model is not completely solvable, certain large scale features can be explicitly computed, and are universal, i.e. insensitive to the distribution of the initial point process. In particular, in the critical and subcritical regimes, the asymptotics of the time decay of the densities of each type of particle is universal (among exponentially integrable interdistance distributions) and, in the supercritical regime, the distribution of the "skyline" process, i.e. the process restricted to the last particles to ever visit a location, has a universal description. We also prove that the alternative model from [6], where triple collisions resolve by mutual annihilation of the three particles involved, does not share the same universality as our model, and find numerical bounds on its critical probability.Comment: This version is a substantial update, introducing new results (asymptotics of densities of particles, definition and description of the skyline process), and extending all results to any interdistance distribution, i.e. including cases where triple collisions may happen, for which we introduce a random resolution rul

Analytic combinatorics : functional equations, rational and algebraic functions

This report is part of a series whose aim is to present in a synthetic way the major methods and models in analytic combinatorics. Here, we detail the case of rational and algebraic functions and discuss systematically closure properties, the location of singularities, and consequences regarding combinatorial enumeration. The theory is applied to regular and context-free languages, finite state models, paths in graphs, locally constrained permutati- ons, lattice paths and walks, trees, and planar maps