25 research outputs found
Improved Algorithms for Integer Complexity
The integer complexity of a positive integer is defined as the
minimum number of 1's needed to represent , using additions, multiplications
and parentheses. We present two simple and faster algorithms for computing the
integer complexity:
1) A near-optimal -time algorithm for
computing the integer complexity of all , improving the previous
one [Cordwell et al., 2017].
2) The first sublinear-time algorithm for computing the integer complexity of
a single , with running time . The previous algorithms for
computing a single require computing all
Simple and Faster Algorithms for Knapsack
In this paper, we obtain a number of new simple pseudo-polynomial time
algorithms on the well-known knapsack problem, focusing on the running time
dependency on the number of items , the maximum item weight
, and the maximum item profit . Our results
include:
- An -time
randomized algorithm for 0-1 knapsack, improving the previous
[Bringmann and Cassis, ESA'23] for the small case.
- An -time
randomized algorithm for bounded knapsack, improving the previous
[Polak, Rohwedder and
Wegrzyck, ICALP'21]
More Dynamic Data Structures for Geometric Set Cover with Sublinear Update Time
We study geometric set cover problems in dynamic settings, allowing insertions and deletions of points and objects. We present the first dynamic data structure that can maintain an O(1)-approximation in sublinear update time for set cover for axis-aligned squares in 2D . More precisely, we obtain randomized update time O(n^{2/3+?}) for an arbitrarily small constant ? > 0. Previously, a dynamic geometric set cover data structure with sublinear update time was known only for unit squares by Agarwal, Chang, Suri, Xiao, and Xue [SoCG 2020]. If only an approximate size of the solution is needed, then we can also obtain sublinear amortized update time for disks in 2D and halfspaces in 3D . As a byproduct, our techniques for dynamic set cover also yield an optimal randomized O(nlog n)-time algorithm for static set cover for 2D disks and 3D halfspaces, improving our earlier O(nlog n(log log n)^{O(1)}) result [SoCG 2020]
The Complexity of Distributed Edge Coloring with Small Palettes
The complexity of distributed edge coloring depends heavily on the palette
size as a function of the maximum degree . In this paper we explore the
complexity of edge coloring in the LOCAL model in different palette size
regimes.
1. We simplify the \emph{round elimination} technique of Brandt et al. and
prove that -edge coloring requires
time w.h.p. and time deterministically, even on trees.
The simplified technique is based on two ideas: the notion of an irregular
running time and some general observations that transform weak lower bounds
into stronger ones.
2. We give a randomized edge coloring algorithm that can use palette sizes as
small as , which is a natural barrier for
randomized approaches. The running time of the algorithm is at most
, where is the complexity of a
permissive version of the constructive Lovasz local lemma.
3. We develop a new distributed Lovasz local lemma algorithm for
tree-structured dependency graphs, which leads to a -edge
coloring algorithm for trees running in time. This algorithm
arises from two new results: a deterministic -time LLL algorithm for
tree-structured instances, and a randomized -time graph
shattering method for breaking the dependency graph into independent -size LLL instances.
4. A natural approach to computing -edge colorings (Vizing's
theorem) is to extend partial colorings by iteratively re-coloring parts of the
graph. We prove that this approach may be viable, but in the worst case
requires recoloring subgraphs of diameter . This stands
in contrast to distributed algorithms for Brooks' theorem, which exploit the
existence of -length augmenting paths
Further Results on Colored Range Searching
We present a number of new results about range searching for colored (or
"categorical") data:
1. For a set of colored points in three dimensions, we describe
randomized data structures with space that can
report the distinct colors in any query orthogonal range (axis-aligned box) in
expected time, where is the number of
distinct colors in the range, assuming that coordinates are in
. Previous data structures require query time. Our result also implies improvements in higher constant
dimensions.
2. Our data structures can be adapted to halfspace ranges in three dimensions
(or circular ranges in two dimensions), achieving expected query
time. Previous data structures require query time.
3. For a set of colored points in two dimensions, we describe a data
structure with space that can answer colored
"type-2" range counting queries: report the number of occurrences of every
distinct color in a query orthogonal range. The query time is , where is the number of distinct colors in
the range. Naively performing uncolored range counting queries would
require time.
Our data structures are designed using a variety of techniques, including
colored variants of randomized incremental construction (which may be of
independent interest), colored variants of shallow cuttings, and bit-packing
tricks.Comment: full version of a SoCG'20 pape
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
The Energy Complexity of Broadcast
Energy is often the most constrained resource in networks of battery-powered
devices, and as devices become smaller, they spend a larger fraction of their
energy on communication (transceiver usage) not computation. As an imperfect
proxy for true energy usage, we define energy complexity to be the number of
time slots a device transmits/listens; idle time and computation are free.
In this paper we investigate the energy complexity of fundamental
communication primitives such as broadcast in multi-hop radio networks. We
consider models with collision detection (CD) and without (No-CD), as well as
both randomized and deterministic algorithms. Some take-away messages from this
work include:
1. The energy complexity of broadcast in a multi-hop network is intimately
connected to the time complexity of leader election in a single-hop (clique)
network. Many existing lower bounds on time complexity immediately transfer to
energy complexity. For example, in the CD and No-CD models, we need
and energy, respectively.
2. The energy lower bounds above can almost be achieved, given sufficient
() time. In the CD and No-CD models we can solve broadcast using
energy and energy,
respectively.
3. The complexity measures of Energy and Time are in conflict, and it is an
open problem whether both can be minimized simultaneously. We give a tradeoff
showing it is possible to be nearly optimal in both measures simultaneously.
For any constant , broadcast can be solved in
time with
energy, where is the diameter of the network
Faster Approximation Algorithms for Geometric Set Cover
We improve the running times of -approximation algorithms for the set
cover problem in geometric settings, specifically, covering points by disks in
the plane, or covering points by halfspaces in three dimensions. In the
unweighted case, Agarwal and Pan [SoCG 2014] gave a randomized -time, -approximation algorithm, by using variants of the
multiplicative weight update (MWU) method combined with geometric data
structures. We simplify the data structure requirement in one of their methods
and obtain a deterministic -time algorithm. With
further new ideas, we obtain a still faster randomized -time algorithm.
For the weighted problem, we also give a randomized -time, -approximation algorithm, by simple modifications to the MWU
method and the quasi-uniform sampling technique