17 research outputs found
Covariogram of non-convex sets
The covariogram of a compact set A contained in R^n is the function that to
each x in R^n associates the volume of A intersected with (A+x). Recently it
has been proved that the covariogram determines any planar convex body, in the
class of all convex bodies. We extend the class of sets in which a planar
convex body is determined by its covariogram. Moreover, we prove that there is
no pair of non-congruent planar polyominoes consisting of less than 9 points
that have equal discrete covariogram.Comment: 15 pages, 7 figures, accepted for publication on Mathematik
On the reconstruction of planar lattice-convex sets from the covariogram
A finite subset of is said to be lattice-convex if is
the intersection of with a convex set. The covariogram of
is the function associating to each u \in
\integer^d the cardinality of . Daurat, G\'erard, and Nivat and
independently Gardner, Gronchi, and Zong raised the problem on the
reconstruction of lattice-convex sets from . We provide a partial
positive answer to this problem by showing that for and under mild extra
assumptions, determines up to translations and reflections. As a
complement to the theorem on reconstruction we also extend the known
counterexamples (i.e., planar lattice-convex sets which are not
reconstructible, up to translations and reflections) to an infinite family of
counterexamples.Comment: accepted in Discrete and Computational Geometr
String Reconstruction from Substring Compositions
Motivated by mass-spectrometry protein sequencing, we consider a
simply-stated problem of reconstructing a string from the multiset of its
substring compositions. We show that all strings of length 7, one less than a
prime, or one less than twice a prime, can be reconstructed uniquely up to
reversal. For all other lengths we show that reconstruction is not always
possible and provide sometimes-tight bounds on the largest number of strings
with given substring compositions. The lower bounds are derived by
combinatorial arguments and the upper bounds by algebraic considerations that
precisely characterize the set of strings with the same substring compositions
in terms of the factorization of bivariate polynomials. The problem can be
viewed as a combinatorial simplification of the turnpike problem, and its
solution may shed light on this long-standing problem as well. Using well known
results on transience of multi-dimensional random walks, we also provide a
reconstruction algorithm that reconstructs random strings over alphabets of
size in optimal near-quadratic time
Set Unification
The unification problem in algebras capable of describing sets has been
tackled, directly or indirectly, by many researchers and it finds important
applications in various research areas--e.g., deductive databases, theorem
proving, static analysis, rapid software prototyping. The various solutions
proposed are spread across a large literature. In this paper we provide a
uniform presentation of unification of sets, formalizing it at the level of set
theory. We address the problem of deciding existence of solutions at an
abstract level. This provides also the ability to classify different types of
set unification problems. Unification algorithms are uniformly proposed to
solve the unification problem in each of such classes.
The algorithms presented are partly drawn from the literature--and properly
revisited and analyzed--and partly novel proposals. In particular, we present a
new goal-driven algorithm for general ACI1 unification and a new simpler
algorithm for general (Ab)(Cl) unification.Comment: 58 pages, 9 figures, 1 table. To appear in Theory and Practice of
Logic Programming (TPLP
Structures from Distances in Two and Three Dimensions using Stochastic Proximity Embedding
The point placement problem is to determine the locations of a set of distinct points uniquely (up to translation and reflection) by making the fewest possible pairwise distance queries of an adversary. Deterministic and randomized algorithms are available if distances are known exactly. In this thesis, we discuss a 1-round algorithm for approximate point placement in the plane in an adversarial model. The distance query graph presented to the adversary is chordal. The remaining distances are uniquely determined using the Stochastic Proximity Embedding (SPE) method due to Agrafiotis, and the layout of the points is also generated from SPE. We have also computed the distances uniquely using a distance matrix completion algorithm for chordal graphs, based on a result by Bakonyi and Johnson. The layout of the points is determined using the traditional Young- Householder approach. We compared the layout of both the method and discussed briefly inside. The modified version of SPE is proposed to overcome the highest translation embedding that the method faces when dealing with higher learning rates. We also discuss the computation of molecular structures in three-dimensional space, with only a subset of the pairwise atomic distances available. The subset of distances is obtained using the Philips model for creating artificial backbone chain of molecular structures. We have proposed the Degree of Freedom Approach to solve this problem and carried out our implementation using SPE and the Distance matrix completion Approac
Experiments with Point Placement Algorithms and Recognition of Line Rigid Graphs
The point placement problem is to determine the position of n distinct points on a line, up to translation and reflection by fewest possible pairwise adversarial distance queries. This masters thesis focusses on two aspects of point placement problem. In one part we focusses on an experimental study of a number of deterministic point placement algorithms and an incremental randomized algorithm, with the goal of obtaining a greater insight into the behavior of these algorithms, particularly of the randomize algorithm. The pairwise distance queries in the point placement problem creates a type of graph, called point placement graph. A point placement graph G is dened as line rigid graph if and only if the vertices of G has unique placement on a line. The other part of this thesis focusses on recognizing line rigid graph of certain class based on structural property of an arbitrarily given graph. Layer graph drawing and rectangular drawing are used as key idea in recognizing line rigid graphs
Algorithmic Aspects of Some Problems in Computational Biology
Given a sequence of pairs of numbers ( a i , l i ), i = 1, 2, ..., n , with l i \u3e 0, and another pair of numbers L and U , the length-constrained maximum density segment problem is to find a subsequence [ a i , a j ]whose density is the maximum under the constraint L ≤ [Special characters omitted.] Sjs=i l s ≤ U . It has application to DNA sequence analysis in Computational Biology, particularly in the determination of the percentage of CG contents in a DNA sequence. A linear time geometric algorithm is presented that is more powerful than the existing linear time algorithms. The method is extended to solve the k length-constrained maximum density segments problem. Previously, there was no known algorithm with non-trivial time complexity for this problem. We present a linear time algorithm to solve the length-constrained maximum sum segment problem. It is extended to solve the k length-constrained maximum sum segments problem in O(n+k) time. The algorithms are extended to solve the problem of finding all the length-constrained segments satisfying user specified sum or density lower bound in O(n+h) time, where h is the size of the output. The point placement problem is to determine the positions of a linear set of points uniquely up to translation and reflection from the fewest possible distance queries between pairs of points. The motivation comes from a problem known as the restriction site mapping. We present 2-round algorithms with queries 10 n /7+ O (1), 4 n /3+ O (1) and 9 n /7+ O (1) respectively. The lower bound for 2 rounds is improved from 17 n /16 to 9 n /8. We also present a modification of a geometric method called MSPocket for detection of ligand binding sites on protein surfaces. Experimentation using 48 benchmark dataset of bound protein structures shows that the success rate of our method is slightly better than that of MSPocket. (Abstract shortened by UMI.