611 research outputs found
Algorithmic and Combinatorial Results on Fence Patrolling, Polygon Cutting and Geometric Spanners
The purpose of this dissertation is to study problems that lie at the intersection of geometry and computer science. We have studied and obtained several results from three different areas, namely–geometric spanners, polygon cutting, and fence patrolling. Specifically, we have designed and analyzed algorithms along with various combinatorial results in these three areas. For geometric spanners, we have obtained combinatorial results regarding lower bounds on worst case dilation of plane spanners. We also have studied low degree plane lattice spanners, both square and hexagonal, of low dilation. Next, for polygon cutting, we have designed and analyzed algorithms for cutting out polygon collections drawn on a piece of planar material
using the three geometric models of saw, namely, line, ray and segment cuts. For fence patrolling, we have designed several strategies for robots patrolling both open and closed fences
Fault-tolerant additive weighted geometric spanners
Let S be a set of n points and let w be a function that assigns non-negative
weights to points in S. The additive weighted distance d_w(p, q) between two
points p,q belonging to S is defined as w(p) + d(p, q) + w(q) if p \ne q and it
is zero if p = q. Here, d(p, q) denotes the (geodesic) Euclidean distance
between p and q. A graph G(S, E) is called a t-spanner for the additive
weighted set S of points if for any two points p and q in S the distance
between p and q in graph G is at most t.d_w(p, q) for a real number t > 1.
Here, d_w(p,q) is the additive weighted distance between p and q. For some
integer k \geq 1, a t-spanner G for the set S is a (k, t)-vertex fault-tolerant
additive weighted spanner, denoted with (k, t)-VFTAWS, if for any set S'
\subset S with cardinality at most k, the graph G \ S' is a t-spanner for the
points in S \ S'. For any given real number \epsilon > 0, we obtain the
following results:
- When the points in S belong to Euclidean space R^d, an algorithm to compute
a (k,(2 + \epsilon))-VFTAWS with O(kn) edges for the metric space (S, d_w).
Here, for any two points p, q \in S, d(p, q) is the Euclidean distance between
p and q in R^d.
- When the points in S belong to a simple polygon P, for the metric space (S,
d_w), one algorithm to compute a geodesic (k, (2 + \epsilon))-VFTAWS with
O(\frac{k n}{\epsilon^{2}}\lg{n}) edges and another algorithm to compute a
geodesic (k, (\sqrt{10} + \epsilon))-VFTAWS with O(kn(\lg{n})^2) edges. Here,
for any two points p, q \in S, d(p, q) is the geodesic Euclidean distance along
the shortest path between p and q in P.
- When the points in lie on a terrain T, an algorithm to compute a
geodesic (k, (2 + \epsilon))-VFTAWS with O(\frac{k n}{\epsilon^{2}}\lg{n})
edges.Comment: a few update
Lower bounds on the dilation of plane spanners
(I) We exhibit a set of 23 points in the plane that has dilation at least
, improving the previously best lower bound of for the
worst-case dilation of plane spanners.
(II) For every integer , there exists an -element point set
such that the degree 3 dilation of denoted by in the domain of plane geometric spanners. In the
same domain, we show that for every integer , there exists a an
-element point set such that the degree 4 dilation of denoted by
The
previous best lower bound of holds for any degree.
(III) For every integer , there exists an -element point set
such that the stretch factor of the greedy triangulation of is at least
.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2
table
Sparse geometric graphs with small dilation
Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we
show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum
degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For
any k, we also construct planar n-point sets for which any geometric graph with
n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if
the points of S are required to be in convex position
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