Nicolas Schabanel

The data broadcast problem with non-uniform transmission times

AbstractThe Data Broadcast Problem consists of finding an infinite schedule to broadcast a given set of messages so as to minimize a linear combination of the average service time to clients requesting messages, and of the cost of the broadcast. This problem also models the Maintenance Scheduling Problem and the Multi-Item Replenishment Problem. Previous work concentrated on a discrete-time restriction where all messages have transmission time equal to 1. Here, we study a generalization of the model to a setting of continuous time and messages of non-uniform transmission times. We prove that the Data Broadcast Problem is strongly NP -hard, even if the broadcast costs are all zero, and give 3-approximation algorithms.

Progresses in the analysis of stochastic 2D cellular automata: A study of asynchronous 2D minority

Cellular automata are often used to model systems in physics, social sciences, biology that are inherently asynchronous. Over the past 20 years, studies have demonstrated that the behavior of cellular automata drastically changes under asynchronous updates. Still, the few mathematical analyses of asynchronism focus on 1D probabilistic cellular automata, either on single examples or on specific classes. As for other classic dynamical systems in physics, extending known methods from 1D to 2D systems is a long lasting challenging problem. In this paper, we address the problem of analyzing an apparently simple 2D asynchronous cellular automaton: 2D Minority where each cell, when fired, updates to the minority state of its neighborhood. Our simulations reveal that in spite of its simplicity, the minority rule exhibits a quite complex response to asynchronism. By focusing on the fully asynchronous regime, we are however able to describe completely the asymptotic behavior of this dynamics as long as the initial configuration satisfies some natural constraints. Besides these technical results, we have strong reasons to believe that our techniques relying on defining an energy function from the transition table of the automaton may be extended to the wider class of threshold automata.

Close to optimal decentralized routing in long-range contact networks

In order to explain the ability of individuals to find short paths to route messages to an unknown destination, based only on their own local view of a social network (the small world phenomenon), Kleinberg [The small-world phenomenon: an algorithmic perspective, Proc. 32nd ACM Symp. on Theory of Computing, 2000, pp. 163-170] proposed a network model based on a d-dimensional lattice of size n augmented with k long-range directed links per node. Individuals behavior is modeled by a greedy algorithm that, given a source and destination, forwards a message to the neighbor of the current holder, which is the closest to the destination. This algorithm computes paths of expected length Θ(log2 n/k) between any pair of nodes. Other topologies have been proposed later on to improve greedy algorithm performance. But, Aspnes et al. [Fault-tolerant routing in peer-to-peer systems, in: Proc. of ACM 3st Symp. on Princ. of Distr. Comp. (PODC 2002), Vol. 31, 2002, pp. 223-232] shows that for a wide class of long-range link distributions, the expected length of the path computed by this algorithm is always Ω(log2 n/(k2 log log n)).We design and analyze a new decentralized routing algorithm, in which nodes consult their neighbors near by, before deciding to whom forward the message. Our algorithm uses similar amount of computational resources as Kleinbergs greedy algorithm: it is easy to implement, visits O(log2 n/log2 (1 +k)) nodes on expectation and requires only Θ(log2 n/log(1 +k)) bits of memory--note that [G.S. Manku, M. Naor, U. Wieder, Know thy neighbors neighbor: the power of lookahead in randomized P2P networks, in: Proc. of 36th ACM STOC 2004, 2004, to appear], shows that any decentralized algorithm visits at least Ω(log2 n/k) on expectation. Our algorithm computes however a path of expected length O(log n (log log n)2/log2 (1 + k)) between any pair of nodes. Our algorithm might fit better some human social behaviors (such as web browsing) and may also have successful applications to peer-to-peer networks where the length of the path along which the files are downloaded, is a critical parameter of the network performance.

Towards small world emergence

We investigate the problem of optimizing the routing performance of a virtual network by adding extra random links. Our asynchronous and distributed algorithm ensures, by adding a single extra link per node, that the resulting network is a navigable small world, i.e., in which greedy routing, using the distance in the original network, computes paths of polylogarithmic length between any pair of nodes with probability 1-<i>O</i>(1/<i>n</i>). Previously known small world augmentation processes require the global knowledge of the network and centralized computations, which is unrealistic for large decentralized networks. Our algorithm, based on a careful multi-layer sampling of the nodes and the construction of a light overlay network, bypasses these limitations. For bounded growth graphs, i.e., graphs where, for any node <i>u</i> and any radius <i>r</i> the number of nodes within distance 2<i>r</i> from <i>u</i> is at most a constant times the number of nodes within distance <i>r</i>, our augmentation process proceeds with high probability in <i>O</i>(log <i>n</i> log <i>D</i>) communication rounds, with <i>O</i>(log <i>n</i> log <i>D</i>) messages of size <i>O</i>(log <i>n</i>) bits sent per node and requiring only <i>O</i>(log <i>n</i> log <i>D</i>) bit space in each node, where <i>n</i> is the number of nodes, and <i>D</i> the diameter. In particular, with the only knowledge of original distances, greedy routing computes, between any pair of nodes in the augmented network, a path of length at most <i>O</i>(log² <i>n</i> log² <i>D</i>) with probability 1 - <i>O</i>(1/<i>n</i>), and of expected length <i>O</i>(log <i>n</i> log² <i>D</i>). Hence, we provide a distributed scheme to augment any bounded growth graph into a small world with high probability in polylogarithmic time while requiring polylogarithmic memory. We consider that the existence of such a lightweight process might be a first step towards the definition of a more general construction process that would validate Kleinbergs model as a plausible explanation for the small world phenomenon in large real interaction networks.

The Data Broadcast Problem with Preemption

The data-broadcast problem consists in finding an infinite schedule to broadcast a given set of messages so as to minimize the average response time to clients requesting messages, and the cost of the broadcast. This is an efficient means of disseminating data to clients, designed for environments, such as satellites, cable TV, mobile phones, where there is a much larger capacity from the information source to the clients than in the reverse direction. Previous work concentrated on scheduling indivisible messages. Here, we studied a generalization of the model where the messages can be preempted. We show that this problem is NP-hard, even in the simple setting where the broadcast costs are zero, and give some practical 2-approximation algorithms for broadcasting messages. We also show that preemption can improve the quality of the broadcast by an arbitrary factor.

Progresses in the analysis of stochastic 2D cellular automata: a study of asynchronous 2D minority

Cellular automata are often used to model systems in physics, social sciences, biology that are inherently asynchronous. Over the past 20 years, studies have demonstrated that the behavior of cellular automata drastically changed under asynchronous updates. Still, the few mathematical analyses of asynchronism focus on one-dimensional probabilistic cellular automata, either on single examples or on specific classes. As for other classic dynamical systems in physics, extending known methods from one- to two-dimensional systems is a long lasting challenging problem. In this paper, we address the problem of analysing an apparently simple 2D asynchronous cellular automaton: 2D Minority where each cell, when fired, updates to the minority state of its neighborhood. Our experiments reveal that in spite of its simplicity, the minority rule exhibits a quite complex response to asynchronism. By focusing on the fully asynchronous regime, we are however able to describe completely the asymptotic behavior of this dynamics as long as the initial configuration satisfies some natural constraints. Besides these technical results, we have strong reasons to believe that our techniques relying on defining an energy function from the transition table of the automaton may be extended to the wider class of threshold automata.

Programming Biomolecules That Fold Greedily During Transcription

We introduce and study the computational power of Oritatami, a theoretical model to explore greedy molecular folding, by which a molecule begins to fold before awaiting the end of its production. This model is inspired by a recent experimental work demonstrating the construction of shapes at the nanoscale by folding an RNA molecule during its transcription from an engineered sequence of synthetic DNA. An important challenge of this model, also encountered in experiments, is to get a single sequence to fold into different shapes, depending on the surrounding molecules. Another big challenge is that not all parts of the sequence are meaningful for all possible inputs. Hence, to prevent them from interfering with subsequent operations in the Oritatami folding pathway we must structure the unused portions of the sequence depending on the context in which it folds. Next, we introduce general design techniques to solve these challenges and program molecules. Our main result in this direction is an algorithm that is time linear in the sequence length that finds a rule for folding the sequence deterministically into a prescribed set of shapes, dependent on its local environment. This shows that the corresponding problem is fixed-parameter tractable, although we also prove it NP-complete in the number of possible environments.

Small alliances in graphs

Let G = (V,E) be a graph. A nonempty subset S ⊆ V is a (strong defensive) alliance of G if every node in S has at least as many neighbors in S than in V \S. This work is motivated by the following observation: when G is a locally structured graph its nodes typically belong to small alliances. Despite the fact that finding the smallest alliance in a graph is NP-hard, we can at least compute in polynomial time depthG(v), the minimum distance one has to move away from an arbitrary node v in order to find an alliance containing v. We define depth(G) as the sum of depthG(v) taken over v ⊆ V. We prove that depth(G) can be at most 1/4(3n2 - 2n + 3) and it can be computed in time O(n3). Intuitively, the value depth(G) should be small for clustered graphs. This is the case for the plane grid, which has a depth of 2n. We generalize the previous for bridgeless planar regular graphs of degree 3 and 4. The idea that clustered graphs are those having a lot of small alliances leads us to analyze the value of rp(G) = IP{S contains an alliance}, with S ⊆ V randomly chosen. This probability goes to 1 for planar regular graphs of degree 3 and 4. Finally, we generalize an already known result by proving that if the minimum degree of the graph is logarithmically lower bounded and if S is a large random set (roughly |S| > n/2), then also rp(G) → 1 as n→8.

Concurrent rebalancing of AVL trees: A fine-grained approach

We address the concurrent rebalancing of almost balanced binary search trees (AVL trees). Such a rebalancing may for instance be necessary after successive insertions and deletions of keys. We show that this problem can be studied through the self-reorganization of distributed systems of nodes controlled by local evolution rules in the line of the approach of Dijkstra and Scholten. This yields a much simpler algorithm that the ones previously known. As a by-product, this solves in a very general setting an old question raised by H.T. Kung and P.L. Lehman: where should rotations take place to rebalance arbitrary search trees?

On the Analysis of Simple 2D Stochastic Cellular Automata

Cellular automata are usually associated with synchronous deterministic dynamics, and their asynchronous or stochastic versions have been far less studied although significant for modeling purposes. This paper analyzes the dynamics of a two-dimensional cellular automaton, 2D Minority, for the Moore neighborhood (eight closest neighbors of each cell) under fully asynchronous dynamics (where one single random cell updates at each time step). 2D Minority may appear as a simple rule, but It is known from the experience of Ising models and Hopfield nets that 2D models with negative feedback are hard to study. This automaton actually presents a rich variety of behaviors, even more complex that what has been observed and analyzed in a previous work on 2D Minority for the von Neumann neighborhood (four neighbors to each cell) (2007) This paper confirms the relevance of the later approach (definition of energy functions and identification of competing regions) Switching to the Moot e neighborhood however strongly complicates the description of intermediate configurations. New phenomena appear (particles, wider range of stable configurations) Nevertheless our methods allow to analyze different stages of the dynamics It suggests that predicting the behavior of this automaton although difficult is possible, opening the way to the analysis of the whole class of totalistic automata