A Type System Describing Unboundedness

We consider nondeterministic higher-order recursion schemes as recognizers of languages of finite words or finite trees. We propose a type system that allows to solve the simultaneous-unboundedness problem (SUP) for schemes, which asks, given a set of letters A and a scheme G, whether it is the case that for every number n the scheme accepts a word (a tree) in which every letter from A appears at least n times. Using this type system we prove that SUP is (m-1)-EXPTIME-complete for word-recognizing schemes of order m, and m-EXPTIME-complete for tree-recognizing schemes of order m. Moreover, we establish the reflection property for SUP: out of an input scheme G one can create its enhanced version that recognizes the same language but is aware of the answer to SUP

The L(2,1)-labeling of unigraphs

The L(2, 1)-labeling problem consists of assigning colors from the integer set 0 ...., lambda to the nodes of a graph G in such a way that nodes at a distance of at most two get different colors, while adjacent nodes get colors which are at least two apart. The aim of this problem is to minimize lambda and it is in general NP-complete. In this paper the problem of L(2, 1)-labeling unigraphs, i.e. graphs uniquely determined by their own degree sequence up to isomorphism, is addressed and a 3/2-approximate algorithm for L(2, 1)-labeling unigraphs is designed. This algorithm runs in 0(n) time, improving the time of the algorithm based on the greedy technique, requiring 0(m) time, that may be near to Theta (n(2)) for unigraphs. (C) 2011 Elsevier B.V. All rights reserved

Euclidean Distance Matrices: Essential Theory, Algorithms and Applications

Euclidean distance matrices (EDM) are matrices of squared distances between points. The definition is deceivingly simple: thanks to their many useful properties they have found applications in psychometrics, crystallography, machine learning, wireless sensor networks, acoustics, and more. Despite the usefulness of EDMs, they seem to be insufficiently known in the signal processing community. Our goal is to rectify this mishap in a concise tutorial. We review the fundamental properties of EDMs, such as rank or (non)definiteness. We show how various EDM properties can be used to design algorithms for completing and denoising distance data. Along the way, we demonstrate applications to microphone position calibration, ultrasound tomography, room reconstruction from echoes and phase retrieval. By spelling out the essential algorithms, we hope to fast-track the readers in applying EDMs to their own problems. Matlab code for all the described algorithms, and to generate the figures in the paper, is available online. Finally, we suggest directions for further research.Comment: - 17 pages, 12 figures, to appear in IEEE Signal Processing Magazine - change of title in the last revisio

Exact Algorithms for L(2,1)-Labeling of Graphs

International audienceThe notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph $G=(V,E)$ into an interval of integers $\{0, \dots ,k\}$ is an $L(2,1)$-labeling of $G$ of span $k$ if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbor are mapped onto distinct integers. It is known that for any fixed $k\ge 4$, deciding the existence of such a labeling is an NP-complete problem. We present exact exponential time algorithms that are faster than the naive $O^*((k+1)^n)$ algorithm that would try all possible mappings. The improvement is best seen in the first NP-complete case of $k=4$, where the running time of our algorithm is $O(1.3006^n)$. Furthermore we show that dynamic programming can be used to establish an $O(3.8730^n)$ algorithm to compute an optimal $L(2,1)$-labeling

Fraud detection and link analysis in Genertel\u2019s customer network

This thesis describes the development of a fraud detection scheme for car insurance customers, based only on information that is available at the moment of underwriting. It explains how we manipulated raw anonymised data and turned it into a graph, and how we used this graph to assign a fraudulence score to each node. Finally, it evaluates the performance of this score in identifying unknown fraudsters. The results obtained in the thesis have been obtained by means of several ad hoc optimised and parallel algorithms, which have been tested and run on multiple HPC platforms

Practical programming for static average-case analysis: the MOQA investigation

This work considers the static calculation of a program’s average-case time. The number of systems that currently tackle this research problem is quite small due to the difficulties inherent in average-case analysis. While each of these systems make a pertinent contribution, and are individually discussed in this work, only one of them forms the basis of this research. That particular system is known as MOQA. The MOQA system consists of the MOQA language and the MOQA static analysis tool. Its technique for statically determining average-case behaviour centres on maintaining strict control over both the data structure type and the labeling distribution. This research develops and evaluates the MOQA language implementation, and adds to the functions already available in this language. Furthermore, the theory that backs MOQA is generalised and the range of data structures for which the MOQA static analysis tool can determine average-case behaviour is increased. Also, some of the MOQA applications and extensions suggested in other works are logically examined here. For example, the accuracy of classifying the MOQA language as reversible is investigated, along with the feasibility of incorporating duplicate labels into the MOQA theory. Finally, the analyses that take place during the course of this research reveal some of the MOQA strengths and weaknesses. This thesis aims to be pragmatic when evaluating the current MOQA theory, the advancements set forth in the following work and the benefits of MOQA when compared to similar systems. Succinctly, this work’s significant expansion of the MOQA theory is accompanied by a realistic assessment of MOQA’s accomplishments and a serious deliberation of the opportunities available to MOQA in the future

ONTIC: A Knowledge Representation System for Mathematics

Ontic is an interactive system for developing and verifying mathematics. Ontic's verification mechanism is capable of automatically finding and applying information from a library containing hundreds of mathematical facts. Starting with only the axioms of Zermelo-Fraenkel set theory, the Ontic system has been used to build a data base of definitions and lemmas leading to a proof of the Stone representation theorem for Boolean lattices. The Ontic system has been used to explore issues in knowledge representation, automated deduction, and the automatic use of large data bases