Nejib Zaguia

Embedding a demonic semilattice in a relation algebra

We present a refinement ordering between binary relations, viewed as programs or specifications. This ordering induces a complete join semilattice that can be embedded in a relation algebra. This embedding then allows an easy proof of many properties of the refinement semilattice, by making use of the well-known corresponding properties of relation algebras. The operations of the refinement semilattice corresponding to join and composition in the embedding algebra are, respectively, demonic join and demonic composition. The weakest prespecification and postspecification operators of Hoare and He, defined over a relation algebra, also have corresponding operators in the semilattice.

Simplified Bluetooth Scatternet Formation Using Maximal Independent Sets

Bluetooth standard allows the creation of piconets, with one node serving as its master and up to seven nodes serving as slaves. A Bluetooth ad hoc network can also be formed by interconnecting several piconets into a scatternet. Given a set of Bluetooth nodes which are positioned so that their unit disk graph is connected, the Bluetooth scatternet formation (BSF) problem is to select piconets, and master and slave roles in each piconet, so that the obtained scatternet is connected, has some desirable properties and good performance with respect to some metrics. In this article we propose BSF protocol based on maximal independent sets. It is a two iterations protocol. In the first iteration a piconet containing a maximal independent set is constructed for every device, while the second iteration attempts to simplify the scatternet structure and to delete piconets not essential for the connectivity. A major advantage of this novel protocol is its simplicity. Simulations show its advantage over the best competing protocol and especially for moderately dense networks.

Constructing greedy linear extensions by interchanging chains

A natural way to prove that a particular linear extension of an ordered set is ‘optimal’ with respect to the ‘jump number’ is to transform this linear extension ‘canonically’ into one that is ‘optimal’. We treat a ‘greedy chain interchange’ transformation which has applications to ordered sets for which each ‘greedy’ linear extension is ‘optimal’.

Small representations of finite distributive lattices as congruence lattices

A recent result of G. Gratzer, H. Lakser, and E. T. Schmidt states that for any distributive lattice D with n join-irreducible elements, there exists a lattice L with 0(n2) elements, whose congruence lattice ConL is isomorphic to D . We show that this result is best possible.

Broadcasting with seamless transition from static to highly mobile wireless ad hoc, sensor and vehicular networks

In a broadcasting task, source node wants to send the same message to all the other nodes in the network. Existing solutions range from connected dominating set (CDS) based on static networks to blind flooding for moderate mobility and hyper-flooding for highly mobile and frequently partitioned networks. Existing protocols for all scenarios are based on some threshold parameters (e.g. speed, which may be expensive to gather) to locally select between these three solution approaches. Here, we describe a new seamless broadcasting from static to mobile protocol, which adjusts itself to any mobility scenario without using any mobility or density-related parameter. Unlike existing methods for highly mobile scenarios, in the proposed method, two nodes do not transmit every time they discover each other as new neighbours. Each node maintains a list of two hop neighbours by periodically exchanging ‘hello’ messages, and decides whether or not it is in CDS. Upon receiving the first copy of message intended for broadcasting, it selects a waiting timeout and constructs two lists of neighbours: neighbours that received the same message and neighbours that did not receive it. Nodes not in CDS select longer timeouts than nodes in CDS. These lists are updated upon receiving further copies of the same packet. When timeout expires, node retransmits if the list of neighbours in need of message is non-empty. ‘Hello’ messages received while waiting, or after timeout expiration may revise all lists (and CDS status) and consequently the need to retransmit. This provides a seamless transition of protocol behaviour from static to highly mobile scenarios, which is applicable to a variety of multi-hop wireless networks.

Perpendicular orders

We construct pairs of orders which have only the trivial order-preservinig self-maps in common: the identity and the constants.

A generalized permutahedron

With respect to a fixedn-element ordered setP, thegeneralized permutahedronPerm(P) is the set of all ordered setsP ∩L, whereL is any permutation of the elements of the underlyingn-element set. Considered as a subset of the extension lattice of ann-element set,Perm(P) is cover-preserving. We apply this to deduce, for instance, that, in any finite ordered setP, there is a comparability whose removal will not increase the dimension, and there is a comparability whose addition toP will not increase its dimension.We establish further properties about the extension lattice which seem to be of independent interest, leading for example, to the characterization of those ordered setsP for which this generalized permutahedron is itself a lattice.

Greedy linear extensions with constraints

Abstract Loosely speaking, a greedy linear extension of an ordered set is a linear extension obtained by following the rule: “climb as high as you can”. Given an ordered set P and a partial extension Ṕ of P is there a greedy linear extension of P which satisfies all of the inequalities of Ṕ ? We consider special instances of this question. In particular, we impose conditions bearing on the diagram of an ordered set. Our results have applications, to the ‘jump number scheduling problem’ and to the ‘greedy dimension’.

Robust line extraction based on repeated segment directions on image contours

This paper describes a new line segment detection and extraction algorithm for computer vision, image segmentation, and shape recognition applications. This is an important pre processing step in detecting, recognizing and classifying military hardware in images. This algorithm uses a compilation of different image processing steps such as normalization, Gaussian smooth, thresholding, and Laplace edge detection to extract edge contours from colour input images. Contours of each connected component are divided into short segments, which are classified by their orientation into nine discrete categories. Straight lines are recognized as the minimal number of such consecutive short segments with the same direction. This solution gives us a surprisingly more accurate, faster and simpler answer with fewer parameters than the widely used Hough Transform algorithm for detecting lines segments among any orientation and location inside images. Its easy implementation, simplicity, speed, the ability to divide an edge into straight line segments using the actual morphology of objects, inclusion of endpoint information, and the use of the OpenCV library are key features and advantages of this solution procedure. The algorithm was tested on several simple shape images as well as real pictures giving more accuracy than the actual procedures based in Hough Transform. This line detection algorithm is robust to image transformations such as rotation, scaling and translation, and to the selection of parameter values.

Network decontamination with temporal immunity by cellular automata

Network decontamination (or disinfection) is a widely studied problem in distributed computing. Network sites are assumed to be contaminated (e.g., by a virus) and a team of agents is deployed to decontaminate the whole network. In the vast literature a variety of assumptions are made on the power of the agents, which can typically communicate, exchange information, remember the past, etc. In this paper we consider the problem in a much weaker setting; in fact we wish to describe the global disinfection process by a set of cellular automata local rules without the use of active agents. We consider the grid, which is naturally described by a 2-dimensional cellular automata, and we devise disinfection rules both in the common situation where after being disinfected a cell is prone to re-contamination by contact, and in a new setting where disinfection leaves the cells immune to recontamination for a certain amount of time (temporal immunity). We also distinguish between Von Neuman and Moore neighborhood, showing that, not surprisingly, a bigger neighborhood allows for a more efficient disinfection.