Ladislav Stacho

On minimizing the sum ofensor movements for barrier coverage of a line segment

A set of sensors establishes barrier coverage of a given line segment if every point of the segment is within the sensing range of a sensor. Given a line segment I, n mobile sensors in arbitrary initial positions on the line (not necessarily inside I) and the sensing ranges of the sensors, we are interested in finding final positions of sensors which establish a barrier coverage of I so that the sum of the distances traveled by all sensors from initial to final positions is minimized. It is shown that the problem is NP complete even to approximate up to constant factor when the sensors may have different sensing ranges. When the sensors have an identical sensing range we give several efficient algorithms to calculate the final destinations so that the sensors either establish a barrier coverage or maximize the coverage of the segment if complete coverage is not feasible while at the same time the sum of the distances traveled by all sensors is minimized. Some open problems are also mentioned.

Spanning Trees with Bounded Number of Branch Vertices

We introduce the following combinatorial optimization problem: Given a connected graph G, find a spanning tree T of G with the smallest number of branchv ertices (vertices of degree 3 or more in T). The problem is motivated by new technologies in the realm of optical networks. We investigate algorithmic and combinatorial aspects of the problem.

Spanning spiders and light-splitting switches

Motivated by a problem in the design of optical networks, we ask when a graph has a spanning spider (subdivision of a star), or, more generally, a spanning tree with a bounded number of branch vertices. We investigate the existence of these spanning subgraphs in analogy to classical studies of Hamiltonicity.

Prediction of 492 human protein kinase substrate specificities

BackgroundComplex intracellular signaling networks monitor diverse environmental inputs to evoke appropriate and coordinated effector responses. Defective signal transduction underlies many pathologies, including cancer, diabetes, autoimmunity and about 400 other human diseases. Therefore, there is high impetus to define the composition and architecture of cellular communications networks in humans. The major components of intracellular signaling networks are protein kinases and protein phosphatases, which catalyze the reversible phosphorylation of proteins. Here, we have focused on identification of kinase-substrate interactions through prediction of the phosphorylation site specificity from knowledge of the primary amino acid sequence of the catalytic domain of each kinase.ResultsThe presented method predicts 488 different kinase catalytic domain substrate specificity matrices in 478 typical and 4 atypical human kinases that rely on both positive and negative determinants for scoring individual phosphosites for their suitability as kinase substrates. This represents a marked advancement over existing methods such as those used in NetPhorest (179 kinases in 76 groups) and NetworKIN (123 kinases), which consider only positive determinants for kinase substrate prediction. Comparison of our predicted matrices with experimentally-derived matrices from about 9,000 known kinase-phosphosite substrate pairs revealed a high degree of concordance with the established preferences of about 150 well studied protein kinases. Furthermore for many of the better known kinases, the predicted optimal phosphosite sequences were more accurate than the consensus phosphosite sequences inferred by simple alignment of the phosphosites of known kinase substrates.ConclusionsApplication of this improved kinase substrate prediction algorithm to the primary structures of over 23, 000 proteins encoded by the human genome has permitted the identification of about 650, 000 putative phosphosites, which are posted on the open source PhosphoNET website (http://www.phosphonet.ca).

Two lower bounds for self-assemblies at temperature 1.

Using the Tile Assembly Model proposed by Rothemund and Winfree, we give two lower bounds on the minimum number of tile types needed to uniquely assemble a shape at temperature 1 under a natural assumption that there are no binding domain mismatches (any two adjacent tiles either form a bond or else both touching sides of the tiles are without glues). Rothemund and Winfree showed that uniquely assembling a full N x N square (a square where there is a bond between any two adjacent tiles) at temperature 1 requires N(2) distinct tile types, and conjectured that the minimum number of tile types needed to self-assemble an N x N square (not a full square) is 2N - 1. Our lower bounds imply that a tile system that uniquely assembles an N x N square without binding domains mismatches, requires at least 2N - 1 tile types.

Local construction of planar spanners in unit disk graphs with irregular transmission ranges

We give an algorithm for constructing a connected spanning subgraphs(panner) of a wireless network modelled as a unit disk graph with nodes of irregular transmission ranges, whereby for some parameter 0 < r ≤ 1 the transmission range of a node includes the entire disk around the node of radius at least r and it does not include any node at distance more than one. The construction of a spanner is distributed and local in the sense that nodes use only information at their vicinity, moreover for a given integer k ≥ 2 each node needs only consider all the nodes at distance at most k hops from it. The resulting spanner has maximum degree at most 3 +

Strong connectivity in sensor networks with given number of directional antennae of bounded angle

\frac{6}{\pi r}

New upper bounds for the chromatic number of a graph

+

Bisection width of transposition graphs

\frac{r+1}{r^{2}}

The Odd-Distance Plane Graph

, when 0 < r <1 (and at most five, when r = 1). Furthermore it is shown that the spanner is planar provided that the distance between any two nodes is at least