346 research outputs found
An Efficient Local Search for Partial Latin Square Extension Problem
A partial Latin square (PLS) is a partial assignment of n symbols to an nxn
grid such that, in each row and in each column, each symbol appears at most
once. The partial Latin square extension problem is an NP-hard problem that
asks for a largest extension of a given PLS. In this paper we propose an
efficient local search for this problem. We focus on the local search such that
the neighborhood is defined by (p,q)-swap, i.e., removing exactly p symbols and
then assigning symbols to at most q empty cells. For p in {1,2,3}, our
neighborhood search algorithm finds an improved solution or concludes that no
such solution exists in O(n^{p+1}) time. We also propose a novel swap
operation, Trellis-swap, which is a generalization of (1,q)-swap and
(2,q)-swap. Our Trellis-neighborhood search algorithm takes O(n^{3.5}) time to
do the same thing. Using these neighborhood search algorithms, we design a
prototype iterated local search algorithm and show its effectiveness in
comparison with state-of-the-art optimization solvers such as IBM ILOG CPLEX
and LocalSolver.Comment: 17 pages, 2 figure
Region-based memory management for Mercury programs
Region-based memory management (RBMM) is a form of compile time memory
management, well-known from the functional programming world. In this paper we
describe our work on implementing RBMM for the logic programming language
Mercury. One interesting point about Mercury is that it is designed with strong
type, mode, and determinism systems. These systems not only provide Mercury
programmers with several direct software engineering benefits, such as
self-documenting code and clear program logic, but also give language
implementors a large amount of information that is useful for program analyses.
In this work, we make use of this information to develop program analyses that
determine the distribution of data into regions and transform Mercury programs
by inserting into them the necessary region operations. We prove the
correctness of our program analyses and transformation. To execute the
annotated programs, we have implemented runtime support that tackles the two
main challenges posed by backtracking. First, backtracking can require regions
removed during forward execution to be "resurrected"; and second, any memory
allocated during a computation that has been backtracked over must be recovered
promptly and without waiting for the regions involved to come to the end of
their life. We describe in detail our solution of both these problems. We study
in detail how our RBMM system performs on a selection of benchmark programs,
including some well-known difficult cases for RBMM. Even with these difficult
cases, our RBMM-enabled Mercury system obtains clearly faster runtimes for 15
out of 18 benchmarks compared to the base Mercury system with its Boehm runtime
garbage collector, with an average runtime speedup of 24%, and an average
reduction in memory requirements of 95%. In fact, our system achieves optimal
memory consumption in some programs.Comment: 74 pages, 23 figures, 11 tables. A shorter version of this paper,
without proofs, is to appear in the journal Theory and Practice of Logic
Programming (TPLP
The linear system for Sudoku and a fractional completion threshold
We study a system of linear equations associated with Sudoku latin squares.
The coefficient matrix of the normal system has various symmetries arising
from Sudoku. From this, we find the eigenvalues and eigenvectors of , and
compute a generalized inverse. Then, using linear perturbation methods, we
obtain a fractional completion guarantee for sufficiently large and sparse
rectangular-box Sudoku puzzles
Maths Games without Frontiers
The use of games in higher education significantly benefits cognitive, motivational, affective,
and sociocultural perspectives. This paper describes a mathematical challenge for first-year
STEM students at the âCollege of Merit Camplusâ, located in different cities in Italy from South
to Nord, during the autumn of 2020 and 2021. Students in pairs play different puzzler games
to reinforce mathematical prerequisites and basic knowledge. Due to the pandemic situation,
non-digital games were forced to adjust to the remote environment. On the one hand, moving
online was a challenge both from the organisational side and for the students that sometimes
need to cooperate from different locations. On the other end, approaching the games remotely
allowed students from colleges all around Italy to participate. The work describes and comments
on each game in detail, considering the studentsâ performance. In general, it can be stated that
students liked the playful experience, although they found themselves not wholly confident
with some topics and the related time restriction. These games, first-of-all helped review and
train the basic concepts; therefore, the students have a better approach to studying the first
Mathematical course at the University. They found the gamesâ dynamics helpful in highlighting
some simple tricks and common mistakes
Virtual Reality Games for Motor Rehabilitation
This paper presents a fuzzy logic based method to track user satisfaction without the need for devices to monitor users physiological conditions. User satisfaction is the key to any productâs acceptance; computer applications and video games provide a unique opportunity to provide a tailored environment for each user to better suit their needs. We have implemented a non-adaptive fuzzy logic model of emotion, based on the emotional component of the Fuzzy Logic Adaptive Model of Emotion (FLAME) proposed by El-Nasr, to estimate player emotion in UnrealTournament 2004. In this paper we describe the implementation of this system and present the results of one of several play tests. Our research contradicts the current literature that suggests physiological measurements are needed. We show that it is possible to use a software only method to estimate user emotion
Minimal Ramsey graphs, orthogonal Latin squares, and hyperplane coverings
This thesis consists of three independent parts.
The first part of the thesis is concerned with Ramsey theory. Given an integer , a graph is said to be \emph{-Ramsey} for another graph if in any -edge-coloring of there exists a monochromatic copy of . The central line of research in this area investigates the smallest number of vertices in a -Ramsey graph for a given . In this thesis, we explore two different directions. First, we will be interested in the smallest possible minimum degree of a minimal (with respect to subgraph inclusion) -Ramsey graph for a given . This line of research was initiated by Burr, ErdĆs, and LovĂĄsz in the 1970s. We study the minimum degree of a minimal Ramsey graph for a random graph and investigate how many vertices of small degree a minimal Ramsey graph for a given can contain. We also consider the minimum degree problem in a more general asymmetric setting. Second, it is interesting to ask how small modifications to the graph affect the corresponding collection of -Ramsey graphs. Building upon the work of Fox, Grinshpun, Liebenau, Person, and SzabĂł and Rödl and Siggers, we prove that adding even a single pendent edge to the complete graph changes the collection of 2-Ramsey graphs significantly.
The second part of the thesis deals with orthogonal Latin squares. A {\em Latin square of order } is an array with entries in such that each integer appears exactly once in every row and every column. Two Latin squares and are said to be {\em orthogonal} if, for all , there is a unique pair such that and ; a system of {\em mutually orthogonal Latin squares}, or a {\em -MOLS}, is a set of pairwise orthogonal Latin squares. Motivated by a well-known result determining the number of different Latin squares of order log-asymptotically, we study the number of -MOLS of order . Earlier results on this problem were obtained by Donovan and Grannell and Keevash and Luria. We establish new upper bounds for a wide range of values of . We also prove a new, log-asymptotically tight, bound on the maximum number of other squares a single Latin square can be orthogonal to.
The third part of the thesis is concerned with grid coverings with multiplicities. In particular, we study the minimum number of hyperplanes necessary to cover all points but one of a given finite grid at least times, while covering the remaining point fewer times. We study this problem for the grid , determining the number exactly when one of the parameters and is much larger than the other and asymptotically in all other cases. This generalizes a classic result of Jamison for . Additionally, motivated by the recent work of Clifton and Huang and Sauermann and Wigderson for the hypercube , we study hyperplane coverings for different grids over , under the stricter condition that the remaining point is omitted completely. We focus on two-dimensional real grids, showing a variety of results and demonstrating that already this setting offers a range of possible behaviors.Diese Dissertation besteht aus drei unabh\"angigen Teilen.
Der erste Teil beschĂ€ftigt sich mit Ramseytheorie. FĂŒr eine ganze Zahl nennt man einen Graphen \emph{-Ramsey} f\"ur einen anderen Graphen , wenn jede Kantenf\"arbung mit Farben einen einfarbigen Teilgraphen enthĂ€lt, der isomorph zu ist. Das zentrale Problem in diesem Gebiet ist die minimale Anzahl von Knoten in einem solchen Graphen zu bestimmen. In dieser Dissertation betrachten wir zwei verschiedene Varianten. Als erstes, beschĂ€ftigen wir uns mit dem kleinstm\"oglichen Minimalgrad eines minimalen (bezĂŒglich Teilgraphen) -Ramsey-Graphen f\"ur einen gegebenen Graphen . Diese Frage wurde zuerst von Burr, Erd\H{o}s und Lov\'asz in den 1970er-Jahren studiert. Wir betrachten dieses Problem f\"ur einen Zufallsgraphen und untersuchen, wie viele Knoten kleinen Grades ein Ramsey-Graph f\"ur gegebenes enthalten kann. Wir untersuchen auch eine asymmetrische Verallgemeinerung des Minimalgradproblems. Als zweites betrachten wir die Frage, wie sich die Menge aller -Ramsey-Graphen f\"ur verĂ€ndert, wenn wir den Graphen modifizieren. Aufbauend auf den Arbeiten von Fox, Grinshpun, Liebenau, Person und SzabĂł und Rödl und Siggers beweisen wir, dass bereits der Graph, der aus mit einer h\"angenden Kante besteht, eine sehr unterschiedliche Menge von 2-Ramsey-Graphen besitzt im Vergleich zu .
Im zweiten Teil geht es um orthogonale lateinische Quadrate. Ein \emph{lateinisches Quadrat der Ordnung } ist eine -Matrix, gef\"ullt mit den Zahlen aus , in der jede Zahl genau einmal pro Zeile und einmal pro Spalte auftritt. Zwei lateinische Quadrate sind \emph{orthogonal} zueinander, wenn f\"ur alle genau ein Paar existiert, sodass es und gilt. Ein \emph{k-MOLS der Ordnung } ist eine Menge von lateinischen Quadraten, die paarweise orthogonal sind. Motiviert von einem bekannten Resultat, welches die Anzahl von lateinischen Quadraten der Ordnung log-asymptotisch bestimmt, untersuchen wir die Frage, wie viele -MOLS der Ordnung es gibt. Dies wurde bereits von Donovan und Grannell und Keevash und Luria studiert. Wir verbessern die beste obere Schranke f\"ur einen breiten Bereich von Parametern . ZusÀtzlich bestimmen wir log-asymptotisch zu wie viele anderen lateinischen Quadraten ein lateinisches Quadrat orthogonal sein kann.
Im dritten Teil studieren wir, wie viele Hyperebenen notwendig sind, um die Punkte eines endlichen Gitters zu ĂŒberdecken, sodass ein bestimmter Punkt maximal -mal bedeckt ist und alle andere mindestens -mal. Wir untersuchen diese Anzahl f\"ur das Gitter asymptotisch und sogar genau, wenn eins von und viel gröĂer als das andere ist. Dies verallgemeinert ein Ergebnis von Jamison fĂŒr den Fall . Au{\ss}erdem betrachten wir dieses Problem f\"ur Gitter im reellen Vektorraum, wenn der spezielle Punkt ĂŒberhaupt nicht bedeckt ist. Dies ist durch die Arbeiten von Clifton und Huang und Sauermann und Wigderson motiviert, die den HyperwĂŒrfel untersucht haben. Wir konzentrieren uns auf zwei-dimensionale Gitter und zeigen, dass schon diese sich sehr unterschiedlich verhalten können
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