3 research outputs found
Approximating MIS over equilateral -VPG graphs
We present an approximation algorithm for the maximum independent set (MIS)
problem over the class of equilateral -VPG graphs. These are intersection
graphs of -shaped planar objects % (and their rotations by multiples of
) with both arms of each object being equal. We obtain a -approximate algorithm running in time for this problem,
where is the ratio and and denote
respectively the maximum and minimum length of any arm in the input equilateral
-representation of the graph. In particular, we obtain -factor
approximation of MIS for -VPG -graphs for which the ratio is bounded
by a constant. % formed by unit length -shapes. In fact, algorithm can be
generalized to an time and a -approximate MIS algorithm over arbitrary -VPG graphs. Here,
and denote respectively the analogues of when restricted to only
horizontal and vertical arms of members of the input. This is an improvement
over the previously best -approximate algorithm \cite{FoxP} (for
some fixed ), unless the ratio is exponentially large in .
In particular, -approximation of MIS is achieved for graphs with
Computing Longest Increasing Subsequence Over Sequential Data Streams
In this paper, we propose a data structure, a quadruple neighbor list
(QN-list, for short), to support real time queries of all longest increasing
subsequence (LIS) and LIS with constraints over sequential data streams. The
QN-List built by our algorithm requires space, where is the time
window size. The running time for building the initial QN-List takes time. Applying the QN-List, insertion of the new item takes
time and deletion of the first item takes time. To the best of our
knowledge, this is the first work to support both LIS enumeration and LIS with
constraints computation by using a single uniform data structure for real time
sequential data streams. Our method outperforms the state-of-the-art methods in
both time and space cost, not only theoretically, but also empirically.Comment: 20 pages (12+8
Complete Edge-Colored Permutation Graphs
We introduce the concept of complete edge-colored permutation graphs as
complete graphs that are the edge-disjoint union of "classical" permutation
graphs. We show that a graph is a complete edge-colored permutation
graph if and only if each monochromatic subgraph of is a "classical"
permutation graph and does not contain a triangle with~ different
colors. Using the modular decomposition as a framework we demonstrate that
complete edge-colored permutation graphs are characterized in terms of their
strong prime modules, which induce also complete edge-colored permutation
graphs. This leads to an -time recognition algorithm. We
show, moreover, that complete edge-colored permutation graphs form a superclass
of so-called symbolic ultrametrics and that the coloring of such graphs is
always a Gallai coloring