1,463 research outputs found
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
Fat Polygonal Partitions with Applications to Visualization and Embeddings
Let be a rooted and weighted tree, where the weight of any node
is equal to the sum of the weights of its children. The popular Treemap
algorithm visualizes such a tree as a hierarchical partition of a square into
rectangles, where the area of the rectangle corresponding to any node in
is equal to the weight of that node. The aspect ratio of the
rectangles in such a rectangular partition necessarily depends on the weights
and can become arbitrarily high.
We introduce a new hierarchical partition scheme, called a polygonal
partition, which uses convex polygons rather than just rectangles. We present
two methods for constructing polygonal partitions, both having guarantees on
the worst-case aspect ratio of the constructed polygons; in particular, both
methods guarantee a bound on the aspect ratio that is independent of the
weights of the nodes.
We also consider rectangular partitions with slack, where the areas of the
rectangles may differ slightly from the weights of the corresponding nodes. We
show that this makes it possible to obtain partitions with constant aspect
ratio. This result generalizes to hyper-rectangular partitions in
. We use these partitions with slack for embedding ultrametrics
into -dimensional Euclidean space: we give a -approximation algorithm for embedding -point ultrametrics
into with minimum distortion, where denotes the spread
of the metric, i.e., the ratio between the largest and the smallest distance
between two points. The previously best-known approximation ratio for this
problem was polynomial in . This is the first algorithm for embedding a
non-trivial family of weighted-graph metrics into a space of constant dimension
that achieves polylogarithmic approximation ratio.Comment: 26 page
The Homogeneous Broadcast Problem in Narrow and Wide Strips
Let be a set of nodes in a wireless network, where each node is modeled
as a point in the plane, and let be a given source node. Each node
can transmit information to all other nodes within unit distance, provided
is activated. The (homogeneous) broadcast problem is to activate a minimum
number of nodes such that in the resulting directed communication graph, the
source can reach any other node. We study the complexity of the regular and
the hop-bounded version of the problem (in the latter, must be able to
reach every node within a specified number of hops), with the restriction that
all points lie inside a strip of width . We almost completely characterize
the complexity of both the regular and the hop-bounded versions as a function
of the strip width .Comment: 50 pages, WADS 2017 submissio
Fine-Grained Complexity Analysis of Two Classic TSP Variants
We analyze two classic variants of the Traveling Salesman Problem using the
toolkit of fine-grained complexity. Our first set of results is motivated by
the Bitonic TSP problem: given a set of points in the plane, compute a
shortest tour consisting of two monotone chains. It is a classic
dynamic-programming exercise to solve this problem in time. While the
near-quadratic dependency of similar dynamic programs for Longest Common
Subsequence and Discrete Frechet Distance has recently been proven to be
essentially optimal under the Strong Exponential Time Hypothesis, we show that
bitonic tours can be found in subquadratic time. More precisely, we present an
algorithm that solves bitonic TSP in time and its bottleneck
version in time. Our second set of results concerns the popular
-OPT heuristic for TSP in the graph setting. More precisely, we study the
-OPT decision problem, which asks whether a given tour can be improved by a
-OPT move that replaces edges in the tour by new edges. A simple
algorithm solves -OPT in time for fixed . For 2-OPT, this is
easily seen to be optimal. For we prove that an algorithm with a runtime
of the form exists if and only if All-Pairs
Shortest Paths in weighted digraphs has such an algorithm. The results for
may suggest that the actual time complexity of -OPT is
. We show that this is not the case, by presenting an algorithm
that finds the best -move in time for
fixed . This implies that 4-OPT can be solved in time,
matching the best-known algorithm for 3-OPT. Finally, we show how to beat the
quadratic barrier for in two important settings, namely for points in the
plane and when we want to solve 2-OPT repeatedly.Comment: Extended abstract appears in the Proceedings of the 43rd
International Colloquium on Automata, Languages, and Programming (ICALP 2016
The Dominating Set Problem in Geometric Intersection Graphs
We study the parameterized complexity of dominating sets in geometric intersection graphs. In one dimension, we investigate intersection graphs induced by translates of a fixed pattern Q that consists of a finite number of intervals and a finite number of isolated points. We prove that Dominating Set on such intersection graphs is polynomially solvable whenever Q contains at least one interval, and whenever Q contains no intervals and for any two point pairs in Q the distance ratio is rational. The remaining case where Q contains no intervals but does contain an irrational distance ratio is shown to be NP-complete and contained in FPT (when parameterized by the solution size). In two and higher dimensions, we prove that Dominating Set is contained in W[1] for intersection graphs of semi-algebraic sets with constant description complexity. This generalizes known results from the literature. Finally, we establish W[1]-hardness for a large class of intersection graphs
On One-Round Discrete Voronoi Games
Let V be a multiset of n points in R^d, which we call voters, and let k >=slant 1 and l >=slant 1 be two given constants. We consider the following game, where two players P and Q compete over the voters in V: First, player P selects a set P of k points in R^d, and then player Q selects a set Q of l points in R^d. Player P wins a voter v in V iff dist(v,P) <=slant dist(v,Q), where dist(v,P) := min_{p in P} dist(v,p) and dist(v,Q) is defined similarly. Player P wins the game if he wins at least half the voters. The algorithmic problem we study is the following: given V, k, and l, how efficiently can we decide if player P has a winning strategy, that is, if P can select his k points such that he wins the game no matter where Q places her points.
Banik et al. devised a singly-exponential algorithm for the game in R^1, for the case k=l. We improve their result by presenting the first polynomial-time algorithm for the game in R^1. Our algorithm can handle arbitrary values of k and l. We also show that if d >= 2, deciding if player P has a winning strategy is Sigma_2^P-hard when k and l are part of the input. Finally, we prove that for any dimension d, the problem is contained in the complexity class exists for all R, and we give an algorithm that works in polynomial time for fixed k and l
Covering many points with a small-area box
Let be a set of points in the plane. We show how to find, for a given
integer , the smallest-area axis-parallel rectangle that covers points
of in time. We also consider the problem of,
given a value , covering as many points of as possible with an
axis-parallel rectangle of area at most . For this problem we give a
probabilistic -approximation that works in near-linear time:
In time we find an
axis-parallel rectangle of area at most that, with high probability,
covers at least points, where
is the maximum possible number of points that could be
covered
Unions of Fat Convex Polytopes Have Short Skeletons
The skeleton of a polyhedral set is the union of its edges and vertices. Let be a set of fat, convex polytopes in three dimensions with n vertices in total, and let f max be the maximum complexity of any face of a polytope in . We prove that the total length of the skeleton of the union of the polytopes in is at most O(a(n)·log* n·logf max) times the sum of the skeleton lengths of the individual polytopes
Computing the visibility map of fat objects
AbstractWe give an output-sensitive algorithm for computing the visibility map of a set of n constant-complexity convex fat polyhedra or curved objects in 3-space. Our algorithm runs in O((n+k) polylog n) time, where k is the combinatorial complexity of the visibility map. This is the first algorithm for computing the visibility map of fat objects that does not require a depth order on the objects and is faster than the best known algorithm for general objects. It is also the first output-sensitive algorithm for curved objects that does not require a depth order
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