Ivan Rapaport

Self-assemblying classes of shapes with a minimum number of tiles, and in optimal time

In this paper we construct fixed finite tile systems that assemble into particular classes of shapes. Moreover, given an arbitrary n, we show how to calculate the tile concentrations in order to ensure that the expected size of the produced shape is n. For rectangles and squares our constructions are optimal (with respect to the size of the systems). We also introduce the notion of parallel time, which is a good approximation of the classical asynchronous time. We prove that our tile systems produce the rectangles and squares in linear parallel time (with respect to the diameter). Those results are optimal. Finally, we introduce the class of diamonds. For these shapes we construct a non trivial tile system having also a linear parallel time complexity.

Cellular automata and communication complexity

The model of cellular automata is fascinating because very simple local rules can generate complex global behaviors. The relationship between local and global function is subject of many studies. We tackle this question by using results on communication complexity theory and, as a by-product, we provide (yet another) classification of cellular automata.

Tiling with bars under tomographic constraints

We wish to tile a rectangle or a torus with only vertical and horizontal bars of a given length, such that the number of bars in every column and row equals given numbers. We present results for particular instances and for a more general problem, while leaving open the initial problem.

Distributed computing of efficient routing schemes in generalized chordal graphs

Efficient algorithms for computing routing tables should take advantage of particular properties arising in large scale networks. Two of them are of special interest: low (logarithmic) diameter and high clustering coefficient. High clustering coefficient implies the existence of few large induced cycles. Considering this fact, we propose here a routing scheme that computes short routes in the class of k-chordal graphs, i.e., graphs with no induced cycles of length more than k. In the class of k-chordal graphs, our routing scheme achieves an additive stretch of at most k-1, i.e., for all pairs of nodes, the length of the route never exceeds their distance plus k-1. In order to compute the routing tables of any n-node graph with diameter D we propose a distributed algorithm which uses O(logn)-bit messages and takes O(D) time. The corresponding routing scheme achieves the stretch of k-1 on k-chordal graphs. We then propose a routing scheme that achieves a better additive stretch of 1 in chordal graphs (notice that chordal graphs are 3-chordal graphs). In this case, distributed computation of routing tables takes O(min{@DD,n}) time, where @D is the maximum degree of the graph. Our routing schemes use addresses of size logn bits and local memory of size 2(d-1)logn bits per node of degree d.

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.

AT-free graphs: linear bounds for the oriented diameter

Let G be a bridgeless connected undirected (b.c.u.) graph. The oriented diameter of G, OD(G), is given by OD(G)=min{diam(H): H is an orientation of G}, where diam(H) is the maximum length computed over the lengths of all the shortest directed paths in H. This work starts with a result stating that, for every b.c.u, graph G, its oriented diameter OD(G) and its domination number γ(G) are linearly related as follows: OD(G)≤ 9γ(G)-5.Since-as shown by Corneil et al. (SIAM J. Discrete Math. 10 (1997) 399)-γ(G)≤ diam(G) for every AT-free graph G, it follows that OD(G) ≤ 9 diam(G)-5 for every b.c.u. AT-free graph G. Our main result is the improvement of the previous linear upper bound. We show that OD(G) ≤ 2 diam(G)+11 for every b.c.u. AT-free graph G. For some subclasses we obtain better bounds: OD(G) ≤ 3/2 diam(G)+25/2 for every interval b.c.u. graph G, and OD(G) ≤ 5/4 diam(G)+ 29/2 for every 2-connected interval b.c.u. graph G. We prove that, for the class of b.c.u. AT-free graphs and its previously mentioned subclasses, all our bounds are optimal (up to additive constants).

The Simultaneous Number-in-Hand Communication Model for Networks: Private Coins, Public Coins and Determinism

We study the multiparty communication model where players are the nodes of a network and each of these players knows his/her own identifier together with the identifiers of his/her neighbors. The players simultaneously send a unique message to a referee who must decide a graph property. The goal of this article is to separate, from the point of view of message size complexity, three different settings: deterministic protocols, randomized protocols with private coins and randomized protocols with public coins. For this purpose we introduce the boolean function Twins. This boolean function returns 1 if and only if there are two nodes with the same neighborhood.

Modeling heterocyst pattern formation in cyanobacteria

BackgroundTo allow the survival of the population in the absence of nitrogen, some cyanobacteria strains have developed the capability of differentiating into nitrogen fixing cells, forming a characteristic pattern. In this paper, the process by which cyanobacteria differentiates from vegetative cells into heterocysts in the absence of nitrogen and the elements of the gene network involved that allow the formation of such a pattern are investigated.MethodsA simple gene network model, which represents the complexity of the differentiation process, and the role of all variables involved in this cellular process is proposed. Specific characteristics and details of the systems behavior such as transcript profiles for ntcA, hetR and patS between consecutive heterocysts were studied.ResultsThe proposed model is able to capture one of the most distinctive features of this system: a characteristic distance of 10 cells between two heterocysts, with a small standard deviation according to experimental variability. The systems response to knock-out and over-expression of patS and hetR was simulated in order to validate the proposed model against experimental observations. In all cases, simulations show good agreement with reported experimental results.ConclusionA simple evolution mathematical model based on the gene network involved in heterocyst differentiation was proposed. The behavior of the biological system naturally emerges from the network and the model is able to capture the spacing pattern observed in heterocyst differentiation, as well as the effect of external perturbations such as nitrogen deprivation, gene knock-out and over-expression without specific parameter fitting.

Brief Announcement: A Hierarchy of Congested Clique Models, from Broadcast to Unicast

The CONGEST model is a synchronous, message-passing model of distributed computation in which each node can send (possibly different) messages of O(log n) bits along each of its incident communication links in each round, where n is the number of computing nodes in the system. In the particular case where the communication network is a complete graph, we have the unicast congested clique model. On the other end is the broadcast version of the congested clique model, in which each node can only broadcast a single message over all its links in each round. In this paper we explore the space, in terms of round complexity, that lies between these two congested clique models. Hence, we parametrize the congested clique model with the range r, the maximum number of different messages a node can send over its incident links in one round. Additionally, we study the effect of the bandwidth b, the maximum size in bits of these messages. We show that the space between the unicast and broadcast congested clique models is very rich and interesting. For instance, we show that a problem (especially designed for this work) takes Ω(n/ log n) rounds in the broadcast model (r = 1), while it can be solved in two rounds if two messages can be sent (r = 2). Other gaps are found in other parts of the spectrum of values of r. We do this by providing techniques to simulate protocols with different parameters. Therefore, we conclude that, with respect to their power to solve certain problems, there is a strict hierarchy of congested clique models.

Solving the Induced Subgraph Problem in the Randomized Multiparty Simultaneous Messages Model

We study the message size complexity of recognizing, under the broadcast congested clique model, whether a fixed graph H appears in a given graph G as a minor, as a subgraph or as an induced subgraph. The n nodes of the input graph G are the players, and each player only knows the identities of its immediate neighbors. We are mostly interested in the one-round, simultaneous setup where each player sends a message of size