Simplifying inclusion-exclusion formulas

Let $\mathcal{F}=\{F_1,F_2, \ldots,F_n\}$ be a family of $n$ sets on a ground set $S$, such as a family of balls in $\mathbb{R}^d$. For every finite measure $\mu$ on $S$, such that the sets of $\mathcal{F}$ are measurable, the classical inclusion-exclusion formula asserts that $\mu(F_1\cup F_2\cup\cdots\cup F_n)=\sum_{I:\emptyset\ne I\subseteq[n]} (-1)^{|I|+1}\mu\Bigl(\bigcap_{i\in I} F_i\Bigr)$; that is, the measure of the union is expressed using measures of various intersections. The number of terms in this formula is exponential in $n$, and a significant amount of research, originating in applied areas, has been devoted to constructing simpler formulas for particular families $\mathcal{F}$. We provide an upper bound valid for an arbitrary $\mathcal{F}$: we show that every system $\mathcal{F}$ of $n$ sets with $m$ nonempty fields in the Venn diagram admits an inclusion-exclusion formula with $m^{O(\log^2n)}$ terms and with $\pm1$ coefficients, and that such a formula can be computed in $m^{O(\log^2n)}$ expected time. For every $\varepsilon>0$ we also construct systems with Venn diagram of size $m$ for which every valid inclusion-exclusion formula has the sum of absolute values of the coefficients at least $\Omega(m^{2-\varepsilon})$.Comment: 17 pages, 3 figures/tables; improved lower bound in v

Local Cliques in ER-Perturbed Random Geometric Graphs

Random graphs are mathematical models that have applications in a wide range of domains. We study the following model where one adds Erd\H{o}s--R\'enyi (ER) type perturbation to a random geometric graph. More precisely, assume $G_\mathcal{X}^{*}$ is a random geometric graph sampled from a nice measure on a metric space $\mathcal{X} = (X,d)$. The input observed graph $\widehat{G}(p,q)$ is generated by removing each existing edge from $G_\mathcal{X}^*$ with probability $p$, while inserting each non-existent edge to $G_\mathcal{X}^{*}$ with probability $q$. We refer to such random $p$-deletion and $q$-insertion as ER-perturbation. Although these graphs are related to the objects in the continuum percolation theory, our understanding of them is still rather limited. In this paper we consider a localized version of the classical notion of clique number for the aforementioned ER-perturbed random geometric graphs: Specifically, we study the edge clique number for each edge in a graph, defined as the size of the largest clique(s) in the graph containing that edge. The clique number of the graph is simply the largest edge clique number. Interestingly, given a ER-perturbed random geometric graph, we show that the edge clique number presents two fundamentally different types of behaviors, depending on which "type" of randomness it is generated from. As an application of the above results, we show that by using a filtering process based on the edge clique number, we can recover the shortest-path metric of the random geometric graph $G_\mathcal{X}^*$ within a multiplicative factor of $3$, from an ER-perturbed observed graph $\widehat{G}(p,q)$, for a significantly wider range of insertion probability $q$ than in previous work

Efficient construction of the lattice of frequent closed patterns and simultaneous extraction of generic bases of rules

In the last few years, the amount of collected data, in various computer science applications, has grown considerably. These large volumes of data need to be analyzed in order to extract useful hidden knowledge. This work focuses on association rule extraction. This technique is one of the most popular in data mining. Nevertheless, the number of extracted association rules is often very high, and many of them are redundant. In this paper, we propose a new algorithm, called PRINCE. Its main feature is the construction of a partially ordered structure for extracting subsets of association rules, called generic bases. Without loss of information these subsets form representation of the whole association rule set. To reduce the cost of such a construction, the partially ordered structure is built thanks to the minimal generators associated to frequent closed patterns. The closed ones are simultaneously derived with generic bases thanks to a simple bottom-up traversal of the obtained structure. The experimentations we carried out in benchmark and "worst case" contexts showed the efficiency of the proposed algorithm, compared to algorithms like CLOSE, A-CLOSE and TITANIC.Comment: 50 pages, in Frenc

Construction efficace du treillis des motifs fermés fréquents et extraction simultanée des bases génériques de règles

Durant ces dernières années, les quantités de données collectées, dans divers domaines d’application de l’informatique, deviennent de plus en plus importantes. Ces quantités suscitent le besoin d’analyse et d’interprétation afin d’en extraire des connaissances utiles. Dans ce travail, nous nous intéressons à la technique d’extraction des règles d’association à partir de larges contextes. Cette dernière est parmi les techniques les plus fréquemment utilisées en fouille de données. Toutefois, le nombre de règles extraites est généralement important avec en outre la présence de règles redondantes. Dans cet article, nous proposons un nouvel algorithme, appelé PRINCE, dont la principale originalité est de construire une structure partiellement ordonnée (nommée treillis d’Iceberg) dans l’objectif d’extraire des ensembles réduits de règles, appelés bases génériques. Ces bases forment un sous-ensemble, sans perte d’information, des règles d’association. Pour réduire le coût de cette construction, le treillis d’Iceberg est calculé grâce aux générateurs minimaux, associés aux motifs fermés fréquents. Ces derniers sont simultanément dérivés avec les bases génériques grâce à un simple parcours ascendant de la structure construite. Les expérimentations que nous avons réalisées sur des contextes de référence et « pire des cas » ont montré l’efficacité de l’algorithme proposé, comparativement à des algorithmes tels que CLOSE, A-CLOSE et TITANIC.In the last few years, the amount of collected data, in various computer science applications, has grown considerably. These large volumes of data need to be analyzed in order to extract useful hidden knowledge. This work focuses on association rule extraction. This technique is one of the most popular in data mining. Nevertheless, the number of extracted association rules is often very high, and many of them are redundant. In this paper, we propose a new algorithm, called PRINCE. Its main feature is the construction of a partially ordered structure for extracting subsets of association rules, called generic bases. Without loss of information these subsets form representation of the whole association rule set. To reduce the cost of such a construction, the partially ordered structure is built thanks to the minimal generators associated to fréquent closed patterns. The closed ones are simultaneously derived with generic bases thanks to a simple bottom up traversal of the obtained structure. The experimentations we carried out in benchmark and « worst case » contexts showed the efficiency of the proposed algorithm, compared to algorithms like CLOSE, A-CLOSE and TITANIC

Lattice-Based Precoding And Decoding in MIMO Fading Systems

In this thesis, different aspects of lattice-based precoding and decoding for the transmission of digital and analog data over MIMO fading channels are investigated: 1) Lattice-based precoding in MIMO broadcast systems: A new viewpoint for adopting the lattice reduction in communication over MIMO broadcast channels is introduced. Lattice basis reduction helps us to reduce the average transmitted energy by modifying the region which includes the constellation points. The new viewpoint helps us to generalize the idea of lattice-reduction-aided precoding for the case of unequal-rate transmission, and obtain analytic results for the asymptotic behavior of the symbol-error-rate for the lattice-reduction-aided precoding and the perturbation technique. Also, the outage probability for both cases of fixed-rate users and fixed sum-rate is analyzed. It is shown that the lattice-reduction-aided method, using LLL algorithm, achieves the optimum asymptotic slope of symbol-error-rate (called the precoding diversity). 2) Lattice-based decoding in MIMO multiaccess systems and MIMO point-to-point systems: Diversity order and diversity-multiplexing tradeoff are two important measures for the performance of communication systems over MIMO fading channels. For the case of MIMO multiaccess systems (with single-antenna transmitters) or MIMO point-to-point systems with V-BLAST transmission scheme, it is proved that lattice-reduction-aided decoding achieves the maximum receive diversity (which is equal to the number of receive antennas). Also, it is proved that the naive lattice decoding (which discards the out-of-region decoded points) achieves the maximum diversity in V-BLAST systems. On the other hand, the inherent drawbacks of the naive lattice decoding for general MIMO fading systems is investigated. It is shown that using the naive lattice decoding for MIMO systems has considerable deficiencies in terms of the diversity-multiplexing tradeoff. Unlike the case of maximum-likelihood decoding, in this case, even the perfect lattice space-time codes which have the non-vanishing determinant property can not achieve the optimal diversity-multiplexing tradeoff. 3) Lattice-based analog transmission over MIMO fading channels: The problem of finding a delay-limited schemes for sending an analog source over MIMO fading channels is investigated in this part. First, the problem of robust joint source-channel coding over an additive white Gaussian noise channel is investigated. A new scheme is proposed which achieves the optimal slope for the signal-to-distortion-ratio (SDR) curve (unlike the previous known coding schemes). Then, this idea is extended to MIMO channels to construct lattice-based codes for joint source-channel coding over MIMO channels. Also, similar to the diversity-multiplexing tradeoff, the asymptotic performance of MIMO joint source-channel coding schemes is characterized, and a concept called diversity-fidelity tradeoff is introduced in this thesis