205,037 research outputs found
Covering Paths and Trees for Planar Grids
Given a set of points in the plane, a covering path is a polygonal path that
visits all the points. In this paper we consider covering paths of the vertices
of an n x m grid. We show that the minimal number of segments of such a path is
except when we allow crossings and , in which case the
minimal number of segments of such a path is , i.e., in this case
we can save one segment. In fact we show that these are true even if we
consider covering trees instead of paths.
These results extend previous works on axis-aligned covering paths of n x m
grids and complement the recent study of covering paths for points in general
position, in which case the problem becomes significantly harder and is still
open
Fast Generation of Random Spanning Trees and the Effective Resistance Metric
We present a new algorithm for generating a uniformly random spanning tree in
an undirected graph. Our algorithm samples such a tree in expected
time. This improves over the best previously known bound
of -- that follows from the work of
Kelner and M\k{a}dry [FOCS'09] and of Colbourn et al. [J. Algorithms'96] --
whenever the input graph is sufficiently sparse.
At a high level, our result stems from carefully exploiting the interplay of
random spanning trees, random walks, and the notion of effective resistance, as
well as from devising a way to algorithmically relate these concepts to the
combinatorial structure of the graph. This involves, in particular,
establishing a new connection between the effective resistance metric and the
cut structure of the underlying graph
Mobile vs. point guards
We study the problem of guarding orthogonal art galleries with horizontal
mobile guards (alternatively, vertical) and point guards, using "rectangular
vision". We prove a sharp bound on the minimum number of point guards required
to cover the gallery in terms of the minimum number of vertical mobile guards
and the minimum number of horizontal mobile guards required to cover the
gallery. Furthermore, we show that the latter two numbers can be calculated in
linear time.Comment: This version covers a previously missing case in both Phase 2 &
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