1,876 research outputs found
Conditional Lower Bounds for Dynamic Geometric Measure Problems
We give new polynomial lower bounds for a number of dynamic measure problems
in computational geometry. These lower bounds hold in the Word-RAM model,
conditioned on the hardness of either 3SUM, APSP, or the Online Matrix-Vector
Multiplication problem [Henzinger et al., STOC 2015]. In particular we get
lower bounds in the incremental and fully-dynamic settings for counting maximal
or extremal points in R^3, different variants of Klee's Measure Problem,
problems related to finding the largest empty disk in a set of points, and
querying the size of the i'th convex layer in a planar set of points. We also
answer a question of Chan et al. [SODA 2022] by giving a conditional lower
bound for dynamic approximate square set cover. While many conditional lower
bounds for dynamic data structures have been proven since the seminal work of
Patrascu [STOC 2010], few of them relate to computational geometry problems.
This is the first paper focusing on this topic. Most problems we consider can
be solved in O(n log n) time in the static case and their dynamic versions have
only been approached from the perspective of improving known upper bounds. One
exception to this is Klee's measure problem in R^2, for which Chan [CGTA 2010]
gave an unconditional lower bound on the worst-case update
time. By a similar approach, we show that such a lower bound also holds for an
important special case of Klee's measure problem in R^3 known as the
Hypervolume Indicator problem, even for amortized runtime in the incremental
setting.Comment: Improved presentation, improved the reduction for the Hypervolume
Indicator problem and added a reduction for dynamic approximate square set
cove
Defective Coloring on Classes of Perfect Graphs
In Defective Coloring we are given a graph and two integers ,
and are asked if we can -color so that the maximum
degree induced by any color class is at most . We show that this
natural generalization of Coloring is much harder on several basic graph
classes. In particular, we show that it is NP-hard on split graphs, even when
one of the two parameters , is set to the smallest possible
fixed value that does not trivialize the problem ( or ). Together with a simple treewidth-based DP algorithm this completely
determines the complexity of the problem also on chordal graphs. We then
consider the case of cographs and show that, somewhat surprisingly, Defective
Coloring turns out to be one of the few natural problems which are NP-hard on
this class. We complement this negative result by showing that Defective
Coloring is in P for cographs if either or is fixed; that
it is in P for trivially perfect graphs; and that it admits a sub-exponential
time algorithm for cographs when both and are unbounded
Simple and Robust Dynamic Two-Dimensional Convex Hull
The convex hull of a data set is the smallest convex set that contains
.
In this work, we present a new data structure for convex hull, that allows
for efficient dynamic updates. In a dynamic convex hull implementation, the
following traits are desirable: (1) algorithms for efficiently answering
queries as to whether a specified point is inside or outside the hull, (2)
adhering to geometric robustness, and (3) algorithmic simplicity.Furthermore, a
specific but well-motivated type of two-dimensional data is rank-based data.
Here, the input is a set of real-valued numbers where for any number its rank is its index in 's sorted order. Each value in can be mapped
to a point to obtain a two-dimensional point set. In this work,
we give an efficient, geometrically robust, dynamic convex hull algorithm, that
facilitates queries to whether a point is internal. Furthermore, our
construction can be used to efficiently update the convex hull of rank-ordered
data, when the real-valued point set is subject to insertions and deletions.
Our improved solution is based on an algorithmic simplification of the
classical convex hull data structure by Overmars and van Leeuwen~[STOC'80],
combined with new algorithmic insights. Our theoretical guarantees on the
update time match those of Overmars and van Leeuwen, namely ,
while we allow a wider range of functionalities (including rank-based data).
Our algorithmic simplification includes simplifying an 11-case check down to a
3-case check that can be written in 20 lines of easily readable C-code. We
extend our solution to provide a trade-off between theoretical guarantees and
the practical performance of our algorithm. We test and compare our solutions
extensively on inputs that were generated randomly or adversarially, including
benchmarking datasets from the literature.Comment: Accepted for ALENEX2
Maintaining the Union of Unit Discs Under Insertions with Near-Optimal Overhead
We present efficient data structures for problems on unit discs and arcs of their boundary in the plane. (i) We give an output-sensitive algorithm for the dynamic maintenance of the union of n unit discs under insertions in O(k log^2 n) update time and O(n) space, where k is the combinatorial complexity of the structural change in the union due to the insertion of the new disc. (ii) As part of the solution of (i) we devise a fully dynamic data structure for the maintenance of lower envelopes of pseudo-lines, which we believe is of independent interest. The structure has O(log^2 n) update time and O(log n) vertical ray shooting query time. To achieve this performance, we devise a new algorithm for finding the intersection between two lower envelopes of pseudo-lines in O(log n) time, using tentative binary search; the lower envelopes are special in that at x=-infty any pseudo-line contributing to the first envelope lies below every pseudo-line contributing to the second envelope. (iii) We also present a dynamic range searching structure for a set of circular arcs of unit radius (not necessarily on the boundary of the union of the corresponding discs), where the ranges are unit discs, with O(n log n) preprocessing time, O(n^{1/2+epsilon} + l) query time and O(log^2 n) amortized update time, where l is the size of the output and for any epsilon>0. The structure requires O(n) storage space
Dynamic Coloring of Unit Interval Graphs with Limited Recourse Budget
In this paper we study the problem of coloring a unit interval graph which changes dynamically. In our model the unit intervals are added or removed one at the time, and have to be colored immediately, so that no two overlapping intervals share the same color. After each update only a limited number of intervals are allowed to be recolored. The limit on the number of recolorings per update is called the recourse budget. In this paper we show, that if the graph remains k-colorable at all times, the updates consist of insertions only, and the final instance consists of n intervals, then we can achieve an amortized recourse budget of while maintaining a proper coloring with k colors. This is an exponential improvement over the result in [Bartłomiej Bosek et al., 2020] in terms of both k and n. We complement this result by showing the lower bound of on the amortized recourse budget in the fully dynamic setting. Our incremental algorithm can be efficiently implemented.
As an additional application of our techniques we include a new combinatorial result on coloring unit circular arc graphs. Let L be the maximum number of arcs intersecting in one point for some set of unit circular arcs . We show that if there is a set of non-intersecting unit arcs of size such that does not contain L+1 arcs intersecting in one point, then it is possible to color with L colors. This complements the work on circular arc coloring [Belkale and Chandran, 2009; Tucker, 1975; Valencia-Pabon, 2003], which specifies sufficient conditions needed to color with L+1 colors or more
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