Reconstructing Polyatomic Structures from Discrete X-Rays: NP-Completeness Proof for Three Atoms

We address a discrete tomography problem that arises in the study of the atomic structure of crystal lattices. A polyatomic structure T can be defined as an integer lattice in dimension D>=2, whose points may be occupied by $c$ distinct types of atoms. To ``analyze'' T, we conduct ell measurements that we call_discrete X-rays_. A discrete X-ray in direction xi determines the number of atoms of each type on each line parallel to xi. Given ell such non-parallel X-rays, we wish to reconstruct T. The complexity of the problem for c=1 (one atom type) has been completely determined by Gardner, Gritzmann and Prangenberg, who proved that the problem is NP-complete for any dimension D>=2 and ell>=3 non-parallel X-rays, and that it can be solved in polynomial time otherwise. The NP-completeness result above clearly extends to any c>=2, and therefore when studying the polyatomic case we can assume that ell=2. As shown in another article by the same authors, this problem is also NP-complete for c>=6 atoms, even for dimension D=2 and axis-parallel X-rays. They conjecture that the problem remains NP-complete for c=3,4,5, although, as they point out, the proof idea does not seem to extend to c<=5. We resolve the conjecture by proving that the problem is indeed NP-complete for c>=3 in 2D, even for axis-parallel X-rays. Our construction relies heavily on some structure results for the realizations of 0-1 matrices with given row and column sums

The K-Server Dual and Loose Competitiveness for Paging

This paper has two results. The first is based on the surprising observation that the well-known ``least-recently-used'' paging algorithm and the ``balance'' algorithm for weighted caching are linear-programming primal-dual algorithms. This observation leads to a strategy (called ``Greedy-Dual'') that generalizes them both and has an optimal performance guarantee for weighted caching. For the second result, the paper presents empirical studies of paging algorithms, documenting that in practice, on ``typical'' cache sizes and sequences, the performance of paging strategies are much better than their worst-case analyses in the standard model suggest. The paper then presents theoretical results that support and explain this. For example: on any input sequence, with almost all cache sizes, either the performance guarantee of least-recently-used is O(log k) or the fault rate (in an absolute sense) is insignificant. Both of these results are strengthened and generalized in``On-line File Caching'' (1998).Comment: conference version: "On-Line Caching as Cache Size Varies", SODA (1991

Application of Local Information Entropy in Cluster Monte Carlo Algorithms

The chapter refers to a modification of the so-called adding probability used in cluster Monte Carlo algorithms. The modification is based on the fact that in real systems, different properties can influence its clusterization. Finally, an additional factor related to property disorder was introduced into the adding probability, which leads to more effective free energy minimization during MC iteration. As a measure of the disorder, we proposed to use a local information entropy. The proposed approach was tested and compared with the classical methods, showing its high efficiency in simulations of multiphase magnetic systems where magnetic anisotropy was used as the property influencing the system clusterization

A ϕ\phi-Competitive Algorithm for Scheduling Packets with Deadlines

In the online packet scheduling problem with deadlines (PacketScheduling, for short), the goal is to schedule transmissions of packets that arrive over time in a network switch and need to be sent across a link. Each packet has a deadline, representing its urgency, and a non-negative weight, that represents its priority. Only one packet can be transmitted in any time slot, so, if the system is overloaded, some packets will inevitably miss their deadlines and be dropped. In this scenario, the natural objective is to compute a transmission schedule that maximizes the total weight of packets which are successfully transmitted. The problem is inherently online, with the scheduling decisions made without the knowledge of future packet arrivals. The central problem concerning PacketScheduling, that has been a subject of intensive study since 2001, is to determine the optimal competitive ratio of online algorithms, namely the worst-case ratio between the optimum total weight of a schedule (computed by an offline algorithm) and the weight of a schedule computed by a (deterministic) online algorithm. We solve this open problem by presenting a $\phi$-competitive online algorithm for PacketScheduling (where $\phi\approx 1.618$ is the golden ratio), matching the previously established lower bound.Comment: Major revision of the analysis and some other parts of the paper. Another revision will follo

A Note on Tiling under Tomographic Constraints

Given a tiling of a 2D grid with several types of tiles, we can count for every row and column how many tiles of each type it intersects. These numbers are called the_projections_. We are interested in the problem of reconstructing a tiling which has given projections. Some simple variants of this problem, involving tiles that are 1x1 or 1x2 rectangles, have been studied in the past, and were proved to be either solvable in polynomial time or NP-complete. In this note we make progress toward a comprehensive classification of various tiling reconstruction problems, by proving NP-completeness results for several sets of tiles.Comment: added one author and a few theorem

An instructional model for the teaching of physics, based on a meaningful learning theory and class experiences

Practically all research studies concerning the teaching of Physics point out the fact that conventional instructional models fail to achieve their objectives. Many attempts have been done to change this situation, frequently with disappointing results. This work, which is the experimental stage in a research project of a greater scope, represents an effort to change to a model based on a cognitive learning theory, known as the Ausubel-Novak-Gowin theory, making use of the metacognitive tools that emerge from this theory. The results of this work indicate that the students react positively to the goals of meaningful learning, showing substantial understanding of Newtonian Mechanics. An important reduction in the study time required to pass the course has also been reported