Minimizing Interference in Ultra-Dense Femtocell Networks Using Graph-Based Frequency Reuse Technique

This paper investigates the performance of graph colouring schemes for frequency assignment in Long-Term Evolution (LTE) networks with ultra-dense femtocells. The aim of the study is to minimize interference in such networks while ensuring efficient spectrum use for these femtocells. The three schemes investigated are the conventional greedy graph colouring algorithm, the saturation degree algorithm and our proposed graph-based theory (GBT) algorithm. The process of frequency assignment is similar in the last two except that the proposed GBT partitions the femtocells into independent sets for an efficient frequency re-use. The performance of these three schemes was analyzed through extensive simulations to determine the SINR and network capacity that can be obtained with the deployment of these schemes using the ITU-R P1238-7 path loss model. The outcome of this study showed that with the absence of a dynamic frequency assignment scheme, interference level is increased as the number of femtocell users within a particular coverage is increased, leading to a reduction in the capacity of such networks. Simulation results showed that all three algorithms considered have the ability to allocate frequencies to femtocells and minimize interference in a densely deployed environment, thereby increasing network capacity. The proposed GBT assigned the least sub-band thereby ensuring spectral efficiency while minimizing harmful interference. Results show that the greedy algorithm has a disadvantage of inefficiently assigning sub-bands randomly, while the saturation degree assigns more sub-bands when compared with the GBT scheme.Keywords — Femtocell, graph colouring, frequency assignment, LTE

Convergence Times of Decentralized Graph Coloring Algorithms

Ordinary graph coloring algorithms are nothing without their calculations, memorizations, and inter-vertex communications. We investigate a class of ultra simple algorithms which can find (Delta+1)-colorings despite drastic restrictions. For each procedure, conflicted vertices randomly recolor one at a time until the graph coloring is valid. We provide an array of run time bounds for these processes, including an O(n*log(Delta)) bound for a variant we propose, and an O(n*Delta) bound which applies to even the most adversarial scenarios

A Distribution Evolutionary Algorithm for Graph Coloring

Graph Coloring Problem (GCP) is a classic combinatorial optimization problem that has a wide application in theoretical research and engineering. To address complicated GCPs efficiently, a distribution evolutionary algorithm based on population of probability models (DEA-PPM) is proposed. Based on a novel representation of probability model, DEA-PPM employs a Gaussian orthogonal search strategy to explore the probability space, by which global exploration can be realized using a small population. With assistance of local exploitation on a small solution population, DEA-PPM strikes a good balance between exploration and exploitation. Numerical results demonstrate that DEA-PPM performs well on selected complicated GCPs, which contributes to its competitiveness to the state-of-the-art metaheuristics

Lattice Linear Problems vs Algorithms

Modelling problems using predicates that induce a partial order among global states was introduced as a way to permit asynchronous execution in multiprocessor systems. A key property of such problems is that the predicate induces one lattice in the state space which guarantees that the execution is correct even if nodes execute with old information about their neighbours. Thus, a compiler that is aware of this property can ignore data dependencies and allow the application to continue its execution with the available data rather than waiting for the most recent one. Unfortunately, many interesting problems do not exhibit lattice linearity. This issue was alleviated with the introduction of eventually lattice linear algorithms. Such algorithms induce a partial order in a subset of the state space even though the problem cannot be defined by a predicate under which the states form a partial order. This paper focuses on analyzing and differentiating between lattice linear problems and algorithms. It also introduces a new class of algorithms called (fully) lattice linear algorithms. A characteristic of these algorithms is that the entire reachable state space is partitioned into one or more lattices and the initial state locks into one of these lattices. Thus, under a few additional constraints, the initial state can uniquely determine the final state. For demonstration, we present lattice linear self-stabilizing algorithms for minimal dominating set and graph colouring problems, and a parallel processing 2-approximation algorithm for vertex cover. The algorithm for minimal dominating set converges in n moves, and that for graph colouring converges in n+2m moves. The algorithm for vertex cover is the first lattice linear approximation algorithm for an NP-Hard problem; it converges in n moves. Some part is cut due to 1920 character limit. Please see the pdf for full abstract.Comment: arXiv admin note: text overlap with arXiv:2209.1470

Efficient Heuristic Solutions to Scheduling Online Courses

The demand for efficient algorithms to automate (near-)optimal timetables has motivated many well-studied scheduling problems in operational research. With most of the courses moving online during the recent pandemic, the delivery of quality education has raised many new technical issues, including online course scheduling. This thesis considers the problem of yielding a near-optimal schedule of the real-time courses in an educational institute, taking into account the conflict among courses, the constraint on the simultaneous consumption of the bandwidth at the hosting servers of the courses, and the maximum utilization of the prime time for the lectures. We propose three approaches for solving the online course scheduling problem; Integer Linear Programming technique, Construction Heuristic using Graph Coloring, and a Hybrid approach using Column Generation technique in combination with Dynamic Programming, and K-coloring. The column generation technique is adopted along with the ILP approach to handling the exponentially increasing number of decision variables in the set-covering problem formulation. This empirical study demonstrates the impact of the input parameters on each approach’s efficiency, including internet bandwidth, number of conflicts, number of connected components. Our results prove the Hybrid approach’s scalability with the change in input parameters and confirm its efficiency in producing near-optimal schedules in a reasonable tim

Uso de métricas generalizadas en clasificadores

En inteligencia artificial existen diversos campos de aplicación, entre ellos se encuentran los cla-sificadores, que asignan elementos a una categoría, y son básicamente de dos tipos: supervisados y no supervisados. Este trabajo se centra en los no supervisados, específicamente en un algoritmo clasificador basado en el problema de coloración de gráficas suaves y uno basado en K-medias. El primero, dado un gráfico completo con ponderaciones en sus aristas, minimiza la suma de las penalizaciones entre los vértices con el mismo color. El algoritmo K-medias utilizado, es uno de los clasificadores más comunes, el cual realiza un agrupamiento con base en la distancia más corta entre cada elemento y alguno de los centroides que fueron seleccionados aleatoriamente, éste se actualiza de manera iterativa con los elementos asignados en ese grupo. Las distancias euclidiana y euclidiana cuadrática son las más utilizadas en la mayoría de las investigaciones que usan sistemas clasificadores, pero no significa que sean con las que se obtienen los mejores resultados. En este trabajo se clasifican las instancias más comunes empleando la distancia Minkowski de orden superior, primeramente con valores enteros, a continuación con valores decimales, y una vez encontrados los valores más prometedores, se explora la región con cambiosde una centésima. Finalmente, se realizaron experimentos con una combinación lineal de métricas, es decir un híbrido, todo lo anterior con el objetivo de observar el comportamiento de dos distintos clasificadores con diversas métricas y semimétricas (que satisfacen la definición de una métrica con excepción de la desigualdad del triángulo), y la precisión que alcanzan para cada base de datos

Problemas de coloración de grafos y grafos fuertemente regulares

En este trabajo se profundizará en varias cuestiones de Teoría de Grafos. Por una parte, se estudiarán los clásicos problemas de coloración de grafos: coloración de vértices, de aristas y de caras; y se probará que los dos últimos pueden reducirse al primero mediante los conceptos de grafo de línea y de grafo dual. Por tratarse de problemas NP-completos no tienen una solución sencilla, así que los abordaremos desde diferentes enfoques: estudiando algunas familias importantes de grafos, mediante razonamientos algorítmicos, y acotando el número y el índice cromático. Y por otra parte, se estudiarán los grafos fuertemente regulares, una familia de grafos altamente estructurada , que resultan interesantes por las propiedades que se deducen de dicha estructura y que están estrechamente relacionados con la combinatoria y la teoría algebraica de grafos.Departamento de Algebra, Geometría y TopologíaGrado en Matemática