Non-Clairvoyant Batch Sets Scheduling: Fairness is Fair enough

Scheduling questions arise naturally in many different areas among which operating system design, compiling,... In real life systems, the characteristics of the jobs (such as release time and processing time) are usually unknown and unpredictable beforehand. The system is typically unaware of the remaining work in each job or of the ability of the job to take advantage of more resources. Following these observations, we adopt the job model by Edmonds et al (2000, 2003) in which the jobs go through a sequence of different phases. Each phase consists of a certain quantity of work and a speed-up function that models how it takes advantage of the number of processors it receives. We consider the non-clairvoyant online setting where a collection of jobs arrives at time 0. We consider the metrics setflowtime introduced by Robert et al (2007). The goal is to minimize the sum of the completion time of the sets, where a set is completed when all of its jobs are done. If the input consists of a single set of jobs, this is simply the makespan of the jobs; and if the input consists of a collection of singleton sets, it is simply the flowtime of the jobs. We show that the non-clairvoyant strategy EQUIoEQUI that evenly splits the available processors among the still unserved sets and then evenly splits these processors among the still uncompleted jobs of each unserved set, achieves a competitive ratio (2+\sqrt3+o(1))\frac{ln n}{lnln n} for the setflowtime minimization and that this is asymptotically optimal (up to a constant factor), where n is the size of the largest set. For makespan minimization, we show that the non-clairvoyant strategy EQUI achieves a competitive ratio of (1+o(1))\frac{ln n}{lnln n}, which is again asymptotically optimal.Comment: 12 pages, 1 figur

Clustering Dynamique par Rayon

National audienceComprendre les dynamiques d'évolution de réseaux sociaux et d'infrastructuresest un problème crucial dans les domaines comme l'épidémiologie, l'urbanismeou la marketing viral. Une quantité croissante de données dynamiquessur des réseaux divers est disponible depuis plusieurs années maisles outils pour analyser ces données ne sont pas toujours adaptés.Nous proposons d'utiliser ces données dynamique pour faire des groupesd'individus de comportement similaire restant stables avec le temps.Pour cela nous introduisons une variante dynamique du problème Sum-Radii Clustering, en utilisant le formalisme du problème DynamicFacility Location, avec la distinction que nous cherchons à minimiser le diamètre des groupes aulieu de la somme des distances au centre. Nous étudions deux adaptations naturelles d'algorithmes probabilistes utilisés pour Dynamic Facility Location (marchant respectivement dans le cas général et quand on se restreint à des espaces métriques). Dans le premier cas, l'algorithme atteint la même borne d'approximation et nous proposons une amélioration, aussi valable pour l'algorithme original (faisant passer le facteur d'approximation de $O(\log nT)$ à $O(\log n)$, où $n$ est le nombre de clients et $T$ la durée en nombre de pas de temps).Enfin, nous montrons que dans le cas métrique, les outils actuels ne permettent pas encorede donner un meilleur résultat, et exhibons un contre-exemple pour le deuxième algorithme, prouvant qu'il ne peut pas atteindre une approximation constante

Polynomial-Time Approximation Scheme for Data Broadcast

The data broadcast problem is to find a schedule for broadcasting a given set of messages over multiple channels. The goal is to minimize the cost of the broadcast plus the expected response time to clients who periodically and probabilistically tune in to wait for particular messages. The problem models disseminating data to clients in asymmetric communication environments, where there is a much larger capacity from the information source to the clients than in the reverse direction. Examples include satellites, cable TV, internet broadcast, and mobile phones. Such environments favor the ``push-based'' model where the server broadcasts (pushes) its information on the communication medium and multiple clients simultaneously retrieve the specific information of individual interest. This paper presents the first polynomial-time approximation scheme (PTAS) for data broadcast with O(1) channels and when each message has arbitrary probability, unit length and bounded cost. The best previous polynomial-time approximation algorithm for this case has a performance ratio of 9/8

Ordonnancement non-clairvoyant: petites simplifications et améliorations de l'analyse de la famille d'algorithmes LAPSβ

International audienceEn 1999, Edmonds [Edmonds1999STOC] introduit un modèle très général de tâches qui traversent différentes phases ayant différentes quantités de travail et capacités à être parallélisées. La force du modèle d'Edmonds est qu'il démontra que même si l'ordonnanceur ne connaît strictement rien des caractéristiques des tâches qu'il est en train d'ordonnancer et est seulement informé de leur arrivée à leur arrivée et de leur complétion à leur complétion, EQUI, qui partage de manière égale les processeurs entre les tâches actives, réussit à être compétitif avec l'ordonnancement optimal hors-line clairvoyant, pour peu qu'EQUI dispose d'un peu plus de deux fois plus de ressources que l'optimum. Ceci signifie que l'ordonnanceur EQUI supporte sans diverger toute charge inférieure à $50\%$. Nous [RobertSchabanel2008SODA] avons par la suite étendu l'analyse d'Edmonds au cas où les tâches sont composées d'un DAG de processus traversant des phases arbitraires et démontré que l'algorithme non-clairvoyant EQUIoEQUI supporte dans ce cas également toute charge inférieure à 50%. En 2009, Edmonds et Pruhs [EdmondsPruhs2009SODA] ont proposé une nouvelle famille d'algorithmes LAPS_b, avec 00

Proving the Turing Universality of Oritatami Co-Transcriptional Folding (Full Text)

We study the oritatami model for molecular co-transcriptional folding. In oritatami systems, the transcript (the "molecule") folds as it is synthesized (transcribed), according to a local energy optimisation process, which is similar to how actual biomolecules such as RNA fold into complex shapes and functions as they are transcribed. We prove that there is an oritatami system embedding universal computation in the folding process itself. Our result relies on the development of a generic toolbox, which is easily reusable for future work to design complex functions in oritatami systems. We develop "low-level" tools that allow to easily spread apart the encoding of different "functions" in the transcript, even if they are required to be applied at the same geometrical location in the folding. We build upon these low-level tools, a programming framework with increasing levels of abstraction, from encoding of instructions into the transcript to logical analysis. This framework is similar to the hardware-to-algorithm levels of abstractions in standard algorithm theory. These various levels of abstractions allow to separate the proof of correctness of the global behavior of our system, from the proof of correctness of its implementation. Thanks to this framework, we were able to computerize the proof of correctness of its implementation and produce certificates, in the form of a relatively small number of proof trees, compact and easily readable and checkable by human, while encapsulating huge case enumerations. We believe this particular type of certificates can be generalized to other discrete dynamical systems, where proofs involve large case enumerations as well

A study of stochastic 2D Minority CA : would wearing stripes be a fatality for snob people ?

Cellular automata are usually associated with synchronous deterministic dynamics, and their asynchronous or stochastic versions have been far less studied although relevant for modeling purposes. The study of their asynchronous dynamics is all the more needed that their asynchronous behaviors are drastically different from their synchronous ones. This paper analyzes the dynamics of a two-dimensional cellular automaton, 2D Minority, under fully asynchronous dynamics, where only one random cell updates at each time step. This cellular automaton is of particular interest in computer science, biology or social science for instance, and already presents a rich variety of behaviors although the apparent simplicity of its transition rule. In particular, it captures some important features, like the emergence of striped patterns, which are common, according to experiments, to other important automata, such as Game of Life. In this paper, we present a mathematical analysis of the first steps and the last steps of the asynchronous dynamics of 2D Minority. Our results are based on the definition of an interaction energy and rely on the analysis of the dynamics of the borders between competing regions. Our results are a first step towards a complete analysis of this stochastic cellular automaton. Many questions remain open: in particular describing mathematically the middle part of the evolution of 2D Minority where many regions compete with each other, or defining similar parameters (energy, borders,...) for other automata (such as Game of Life) that present similarities with 2D Minority in their asynchronous behaviors

Dynamic clustering of evolving networks: some results on the line

International audienceUnderstanding the dynamics of evolving social/infrastructure networks is a central challenge in many applied areas such as epidemiology, viral marketing, city planification, etc. During the last decade, a massive amount of data has been collected on such networks that still resist to analysis. In this article, we propose to use the data on the dynamics to find better partitions of the network into groups by requiring the groups to be stable over time. For that purpose, we introduce a dynamic version of the k-clustering problem which includes a cost for every point that moves from one cluster to another. We show that this yields in many realistic situations better fitting solutions than optimizing independently various snapshots of the network. We present a first non-trivial exact algorithm for this problem when the points move along a line; this algorithm runs in polynomial time when k and the time horizon are bounded by a constant. We conclude with a series of surprising results on the complexity of the structure of optimal solutions for the line case

Oritatami Systems Assemble Shapes No Less Complex Than Tile Assembly Model (ATAM)

Different models have been proposed to understand natural phenomena at the molecular scale from a computational point of view. Oritatami systems are a model of molecular co-transcriptional folding: the transcript (the "molecule") folds as it is synthesized according to a local energy optimisation process, in a similar way to how actual biomolecules such as RNA fold into complex shapes and functions. We introduce a new model, called turedo, which is a self-avoiding Turing machine on the plane that evolves by marking visited positions and that can only move to unmarked positions. Any oritatami can be seen as a particular turedo. We show that any turedo with lookup radius 1 can conversely be simulated by an oritatami, using a universal bead type set. Our notion of simulation is strong enough to preserve the geometrical and dynamical features of these models up to a constant spatio-temporal rescaling (as in intrinsic simulation). As a consequence, turedo can be used as a readable oritatami "higher-level" programming language to build readily oritatami "smart robots", using our explicit simulation result as a compiler. As an application of our simulation result, we prove two new complexity results on the (infinite) limit configurations of oritatami systems (and radius-1 turedos), assembled from a finite seed configuration. First, we show that such limit configurations can embed any recursively enumerable set, and are thus exactly as complex as aTAM limit configurations. Second, we characterize the possible densities of occupied positions in such limit configurations: they are exactly the ??-computable numbers between 0 and 1. We also show that all such limit densities can be produced by one single oritatami system, just by changing the finite seed configuration. None of these results is implied by previous constructions of oritatami embedding tag systems or 1D cellular automata, which produce only computable limit configurations with constrained density