969 research outputs found
Polylogarithmic Approximation for Generalized Minimum Manhattan Networks
Given a set of terminals, which are points in -dimensional Euclidean
space, the minimum Manhattan network problem (MMN) asks for a minimum-length
rectilinear network that connects each pair of terminals by a Manhattan path,
that is, a path consisting of axis-parallel segments whose total length equals
the pair's Manhattan distance. Even for , the problem is NP-hard, but
constant-factor approximations are known. For , the problem is
APX-hard; it is known to admit, for any \eps > 0, an
O(n^\eps)-approximation.
In the generalized minimum Manhattan network problem (GMMN), we are given a
set of terminal pairs, and the goal is to find a minimum-length
rectilinear network such that each pair in is connected by a Manhattan
path. GMMN is a generalization of both MMN and the well-known rectilinear
Steiner arborescence problem (RSA). So far, only special cases of GMMN have
been considered.
We present an -approximation algorithm for GMMN (and, hence,
MMN) in dimensions and an -approximation algorithm for 2D.
We show that an existing -approximation algorithm for RSA in 2D
generalizes easily to dimensions.Comment: 14 pages, 5 figures; added appendix and figure
Topological Stability of Kinetic -Centers
We study the -center problem in a kinetic setting: given a set of
continuously moving points in the plane, determine a set of (moving)
disks that cover at every time step, such that the disks are as small as
possible at any point in time. Whereas the optimal solution over time may
exhibit discontinuous changes, many practical applications require the solution
to be stable: the disks must move smoothly over time. Existing results on this
problem require the disks to move with a bounded speed, but this model is very
hard to work with. Hence, the results are limited and offer little theoretical
insight. Instead, we study the topological stability of -centers.
Topological stability was recently introduced and simply requires the solution
to change continuously, but may do so arbitrarily fast. We prove upper and
lower bounds on the ratio between the radii of an optimal but unstable solution
and the radii of a topologically stable solution---the topological stability
ratio---considering various metrics and various optimization criteria. For we provide tight bounds, and for small we can obtain nontrivial
lower and upper bounds. Finally, we provide an algorithm to compute the
topological stability ratio in polynomial time for constant
Covering Points by Disjoint Boxes with Outliers
For a set of n points in the plane, we consider the axis--aligned (p,k)-Box
Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together
contain n-k points. In this paper, we consider the boxes to be either squares
or rectangles, and we want to minimize the area of the largest box. For general
p we show that the problem is NP-hard for both squares and rectangles. For a
small, fixed number p, we give algorithms that find the solution in the
following running times:
For squares we have O(n+k log k) time for p=1, and O(n log n+k^p log^p k time
for p = 2,3. For rectangles we get O(n + k^3) for p = 1 and O(n log n+k^{2+p}
log^{p-1} k) time for p = 2,3.
In all cases, our algorithms use O(n) space.Comment: updated version: - changed problem from 'cover exactly n-k points' to
'cover at least n-k points' to avoid having non-feasible solutions. Results
are unchanged. - added Proof to Lemma 11, clarified some sections - corrected
typos and small errors - updated affiliations of two author
On the Fine-Grained Complexity of Small-Size Geometric Set Cover and Discrete k-Center for Small k
We study the time complexity of the discrete k-center problem and related (exact) geometric set cover problems when k or the size of the cover is small. We obtain a plethora of new results:
- We give the first subquadratic algorithm for rectilinear discrete 3-center in 2D, running in O?(n^{3/2}) time.
- We prove a lower bound of ?(n^{4/3-?}) for rectilinear discrete 3-center in 4D, for any constant ? > 0, under a standard hypothesis about triangle detection in sparse graphs.
- Given n points and n weighted axis-aligned unit squares in 2D, we give the first subquadratic algorithm for finding a minimum-weight cover of the points by 3 unit squares, running in O?(n^{8/5}) time. We also prove a lower bound of ?(n^{3/2-?}) for the same problem in 2D, under the well-known APSP Hypothesis. For arbitrary axis-aligned rectangles in 2D, our upper bound is O?(n^{7/4}).
- We prove a lower bound of ?(n^{2-?}) for Euclidean discrete 2-center in 13D, under the Hyperclique Hypothesis. This lower bound nearly matches the straightforward upper bound of O?(n^?), if the matrix multiplication exponent ? is equal to 2.
- We similarly prove an ?(n^{k-?}) lower bound for Euclidean discrete k-center in O(k) dimensions for any constant k ? 3, under the Hyperclique Hypothesis. This lower bound again nearly matches known upper bounds if ? = 2.
- We also prove an ?(n^{2-?}) lower bound for the problem of finding 2 boxes to cover the largest number of points, given n points and n boxes in 12D . This matches the straightforward near-quadratic upper bound
Cache-Oblivious Selection in Sorted X+Y Matrices
Let X[0..n-1] and Y[0..m-1] be two sorted arrays, and define the mxn matrix A
by A[j][i]=X[i]+Y[j]. Frederickson and Johnson gave an efficient algorithm for
selecting the k-th smallest element from A. We show how to make this algorithm
IO-efficient. Our cache-oblivious algorithm performs O((m+n)/B) IOs, where B is
the block size of memory transfers
Kinetic Euclidean 2-centers in the black-box model
We study the 2-center problem for moving points in the plane. Given a set P of n points, the Euclidean 2-center problem asks for two congruent disks of minimum size that together cover P. Our methods work in the black-box KDS model, where we receive the locations of the points at regular time steps and we know an upper bound d_max on the maximum displacement of any point within one time step.
We show how to maintain a (1 + e)-approximation of the Euclidean 2-center in amortized sub-linear time per time step, under certain assumptions on the distribution of the point set P. In many cases --namely when the distance between the centers of the disks is relatively large or relatively small-- the solution we maintain is actually optimal
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