111 research outputs found
Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus, and Minor-Free Graphs
A (1 + eps)-approximate distance oracle for a graph is a data structure that
supports approximate point-to-point shortest-path-distance queries. The most
relevant measures for a distance-oracle construction are: space, query time,
and preprocessing time. There are strong distance-oracle constructions known
for planar graphs (Thorup, JACM'04) and, subsequently, minor-excluded graphs
(Abraham and Gavoille, PODC'06). However, these require Omega(eps^{-1} n lg n)
space for n-node graphs. We argue that a very low space requirement is
essential. Since modern computer architectures involve hierarchical memory
(caches, primary memory, secondary memory), a high memory requirement in effect
may greatly increase the actual running time. Moreover, we would like data
structures that can be deployed on small mobile devices, such as handhelds,
which have relatively small primary memory. In this paper, for planar graphs,
bounded-genus graphs, and minor-excluded graphs we give distance-oracle
constructions that require only O(n) space. The big O hides only a fixed
constant, independent of \epsilon and independent of genus or size of an
excluded minor. The preprocessing times for our distance oracle are also faster
than those for the previously known constructions. For planar graphs, the
preprocessing time is O(n lg^2 n). However, our constructions have slower query
times. For planar graphs, the query time is O(eps^{-2} lg^2 n). For our
linear-space results, we can in fact ensure, for any delta > 0, that the space
required is only 1 + delta times the space required just to represent the graph
itself
RMT 555 - PERSEKITARAN PERUNDANGAN APRIL-MAY 06.
Abstract. We present a linear time algorithm for computing an implicit linear space representation of a minimum cycle basis (MCB) in weighted partial 2-trees, i.e., graphs of treewidth two. The implicit representation can be made explicit in a running time that is proportional to the size of the MCB. For planar graphs, Borradaile, Sankowski, and Wulff-Nilsen [Min st-cut Oracle for Planar Graphs with Near-Linear Preprocessing Time, FOCS 2010] showed how to compute an implicit O(n log n) space representation of an MCB in O(n log 5 n) time. For the special case of partial 2-trees, our algorithm improves this result to linear time and space. Such an improvement was achieved previously only for outerplanar graphs [Liu and Lu: Minimum Cycle Bases of Weighted Outerplanar Graphs, IPL 110:970–974, 2010]
Capacitated Vehicle Routing with Non-Uniform Speeds
The capacitated vehicle routing problem (CVRP) involves distributing
(identical) items from a depot to a set of demand locations, using a single
capacitated vehicle. We study a generalization of this problem to the setting
of multiple vehicles having non-uniform speeds (that we call Heterogenous
CVRP), and present a constant-factor approximation algorithm.
The technical heart of our result lies in achieving a constant approximation
to the following TSP variant (called Heterogenous TSP). Given a metric denoting
distances between vertices, a depot r containing k vehicles with possibly
different speeds, the goal is to find a tour for each vehicle (starting and
ending at r), so that every vertex is covered in some tour and the maximum
completion time is minimized. This problem is precisely Heterogenous CVRP when
vehicles are uncapacitated.
The presence of non-uniform speeds introduces difficulties for employing
standard tour-splitting techniques. In order to get a better understanding of
this technique in our context, we appeal to ideas from the 2-approximation for
scheduling in parallel machine of Lenstra et al.. This motivates the
introduction of a new approximate MST construction called Level-Prim, which is
related to Light Approximate Shortest-path Trees. The last component of our
algorithm involves partitioning the Level-Prim tree and matching the resulting
parts to vehicles. This decomposition is more subtle than usual since now we
need to enforce correlation between the size of the parts and their distances
to the depot
Matroid and Knapsack Center Problems
In the classic -center problem, we are given a metric graph, and the
objective is to open nodes as centers such that the maximum distance from
any vertex to its closest center is minimized. In this paper, we consider two
important generalizations of -center, the matroid center problem and the
knapsack center problem. Both problems are motivated by recent content
distribution network applications. Our contributions can be summarized as
follows:
1. We consider the matroid center problem in which the centers are required
to form an independent set of a given matroid. We show this problem is NP-hard
even on a line. We present a 3-approximation algorithm for the problem on
general metrics. We also consider the outlier version of the problem where a
given number of vertices can be excluded as the outliers from the solution. We
present a 7-approximation for the outlier version.
2. We consider the (multi-)knapsack center problem in which the centers are
required to satisfy one (or more) knapsack constraint(s). It is known that the
knapsack center problem with a single knapsack constraint admits a
3-approximation. However, when there are at least two knapsack constraints, we
show this problem is not approximable at all. To complement the hardness
result, we present a polynomial time algorithm that gives a 3-approximate
solution such that one knapsack constraint is satisfied and the others may be
violated by at most a factor of . We also obtain a 3-approximation
for the outlier version that may violate the knapsack constraint by
.Comment: A preliminary version of this paper is accepted to IPCO 201
Selection from read-only memory with limited workspace
Given an unordered array of elements drawn from a totally ordered set and
an integer in the range from to , in the classic selection problem
the task is to find the -th smallest element in the array. We study the
complexity of this problem in the space-restricted random-access model: The
input array is stored on read-only memory, and the algorithm has access to a
limited amount of workspace. We prove that the linear-time prune-and-search
algorithm---presented in most textbooks on algorithms---can be modified to use
bits instead of words of extra space. Prior to our
work, the best known algorithm by Frederickson could perform the task with
bits of extra space in time. Our result separates
the space-restricted random-access model and the multi-pass streaming model,
since we can surpass the lower bound known for the latter
model. We also generalize our algorithm for the case when the size of the
workspace is bits, where . The running time
of our generalized algorithm is ,
slightly improving over the
bound of Frederickson's algorithm. To obtain the improvements mentioned above,
we developed a new data structure, called the wavelet stack, that we use for
repeated pruning. We expect the wavelet stack to be a useful tool in other
applications as well.Comment: 16 pages, 1 figure, Preliminary version appeared in COCOON-201
Locked and Unlocked Chains of Planar Shapes
We extend linkage unfolding results from the well-studied case of polygonal
linkages to the more general case of linkages of polygons. More precisely, we
consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are
hinged together sequentially at rotatable joints. Our goal is to characterize
the families of planar shapes that admit locked chains, where some
configurations cannot be reached by continuous reconfiguration without
self-intersection, and which families of planar shapes guarantee universal
foldability, where every chain is guaranteed to have a connected configuration
space. Previously, only obtuse triangles were known to admit locked shapes, and
only line segments were known to guarantee universal foldability. We show that
a surprisingly general family of planar shapes, called slender adornments,
guarantees universal foldability: roughly, the distance from each edge along
the path along the boundary of the slender adornment to each hinge should be
monotone. In contrast, we show that isosceles triangles with any desired apex
angle less than 90 degrees admit locked chains, which is precisely the
threshold beyond which the inward-normal property no longer holds.Comment: 23 pages, 25 figures, Latex; full journal version with all proof
details. (Fixed crash-induced bugs in the abstract.
Improved Approximation Algorithms for Box Contact Representations ⋆
Abstract. We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is called CONTACT REPRESENTATION OF WORD NETWORKS (CROWN) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. CROWN is known to be NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, MAX-CROWN, in which realizing each desired adjacency yields a certain profit. We present the first O(1)-approximation algorithm for the general case, when the input is a complete weighted graph, and for the bipartite case. Since the subgraph of realized adjacencies is necessarily planar, we also consider several planar graph classes (namely stars, trees, outerplanar, and planar graphs), improving upon the known results. For some graph classes, we also describe improvements in the unweighted case, where each adjacency yields the same profit. Finally, we show that the problem is APX-hard on bipartite graphs of bounded maximum degree.
On the Treewidth of Dynamic Graphs
Dynamic graph theory is a novel, growing area that deals with graphs that
change over time and is of great utility in modelling modern wireless, mobile
and dynamic environments. As a graph evolves, possibly arbitrarily, it is
challenging to identify the graph properties that can be preserved over time
and understand their respective computability.
In this paper we are concerned with the treewidth of dynamic graphs. We focus
on metatheorems, which allow the generation of a series of results based on
general properties of classes of structures. In graph theory two major
metatheorems on treewidth provide complexity classifications by employing
structural graph measures and finite model theory. Courcelle's Theorem gives a
general tractability result for problems expressible in monadic second order
logic on graphs of bounded treewidth, and Frick & Grohe demonstrate a similar
result for first order logic and graphs of bounded local treewidth.
We extend these theorems by showing that dynamic graphs of bounded (local)
treewidth where the length of time over which the graph evolves and is observed
is finite and bounded can be modelled in such a way that the (local) treewidth
of the underlying graph is maintained. We show the application of these results
to problems in dynamic graph theory and dynamic extensions to static problems.
In addition we demonstrate that certain widely used dynamic graph classes
naturally have bounded local treewidth
Affective Man-Machine Interface: Unveiling human emotions through biosignals
As is known for centuries, humans exhibit an electrical profile. This profile is altered through various psychological and physiological processes, which can be measured through biosignals; e.g., electromyography (EMG) and electrodermal activity (EDA). These biosignals can reveal our emotions and, as such, can serve as an advanced man-machine interface (MMI) for empathic consumer products. However, such a MMI requires the correct classification of biosignals to emotion classes. This chapter starts with an introduction on biosignals for emotion detection. Next, a state-of-the-art review is presented on automatic emotion classification. Moreover, guidelines are presented for affective MMI. Subsequently, a research is presented that explores the use of EDA and three facial EMG signals to determine neutral, positive, negative, and mixed emotions, using recordings of 21 people. A range of techniques is tested, which resulted in a generic framework for automated emotion classification with up to 61.31% correct classification of the four emotion classes, without the need of personal profiles. Among various other directives for future research, the results emphasize the need for parallel processing of multiple biosignals
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