Antonios Symvonis

Emilio Di Giacomo, Walter Didimo, Eleni Katsanou et al.

Given a finite set $S$ of points in the plane and a finite set $\mathcal{D}$ of directions, a geometric spanning tree~$T$ of~$S$ is $\mathcal{D}$-monotone if every path in $T$ is monotone with respect to some direction in $\mathcal{D}$. We study the problem of computing, for a given point set $S$ and a given set $\mathcal{D}$ of directions, a minimum-length $\mathcal{D}$-monotone spanning tree of~$S$. We present a quadratic-time algorithm for two directions. More generally, we show that the problem belongs to the complexity class XP when parameterized by the number of directions. We further study, for a given positive integer $k$ and point set~$S$, the problem of finding a minimum-length $\mathcal{D}$-monotone spanning tree of $S$ over all possible sets~$\mathcal{D}$ of $k$ directions. We prove that this problem, too, is in XP when parameterized by~$k$, and present two algorithms that run in $O(n^2 \log n)$ and $O(n^6)$ time for $k=1$ and $k=2$, respectively, where $n$ is the number of points in~$S$. Finally, in contrast to the classical Euclidean minimum spanning tree of a set of points, whose vertex degree is bounded by six, we show that for every even integer~$k$, there exists a point set~$S_k$ and a set $\mathcal{D}_k$ of $k$ directions such that any minimum-length $\mathcal{D}_k$-monotone spanning tree of $S_k$ has maximum vertex degree~$2k$.

Evmorfia N. Argyriou, Aikaterini A. Sotiraki, Antonios Symvonis

Occupational fraud affects many companies worldwide causing them economic loss and liability issues towards their customers and other involved entities. Detecting internal fraud in a company requires significant effort and, unfortunately cannot be entirely prevented. The internal auditors have to process a huge amount of data produced by diverse systems, which are in most cases in textual form, with little automated support. In this paper, we exploit the advantages of information visualization and present a system that aims to detect occupational fraud in systems which involve a pair of entities (e.g., an employee and a client) and periodic activity. The main visualization is based on a spiral system on which the events are drawn appropriately according to their time-stamp. Suspicious events are considered those which appear along the same radius or on close radii of the spiral. Before producing the visualization, the system ranks both involved entities according to the specifications of the internal auditor and generates a video file of the activity such that events with strong evidence of fraud appear first in the video. The system is also equipped with several different visualizations and mechanisms in order to meet the requirements of an internal fraud detection system.