scholarly journals A Unified Model and Algorithms for Temporal Map Labeling

Algorithmica ◽  
2020 ◽  
Vol 82 (10) ◽  
pp. 2709-2736
Author(s):  
Andreas Gemsa ◽  
Benjamin Niedermann ◽  
Martin Nöllenburg
Keyword(s):  
Approximation Algorithms ◽  
Traffic Control ◽  
Constant Factor ◽  
Maximization Problem ◽  
Fixed Position ◽  
Time Intervals ◽  
Map Labeling ◽  
Polynomial Time Algorithms ◽  
Dynamic Map ◽  
Flexible Model

Abstract We consider map labeling for the case that a map undergoes a sequence of operations such as rotation, zoom and translation over a specified time span. We unify and generalize several previous models for dynamic map labeling into one versatile and flexible model. In contrast to previous research, we completely abstract from the particular operations and express the labeling problem as a set of time intervals representing the labels’ presences, activities and conflicts. One of the model’s strength is manifested in its simplicity and broad range of applications. In particular, it supports label selection both for map features with fixed position as well as for moving entities (e.g., for tracking vehicles in logistics or air traffic control). We study the active range maximization problem in this model. We prove that the problem is -complete and [1]-hard, and present constant-factor approximation algorithms. In the restricted, yet practically relevant case that no more than k labels can be active at any time, we give polynomial-time algorithms as well as constant-factor approximation algorithms.

EFFICIENT APPROXIMATION ALGORITHMS FOR TWO-LABEL POINT LABELING

2001 ◽  
Vol 11 (04) ◽  
pp. 455-464 ◽  
Author(s):  
BINHAI ZHU ◽  
C. K. POON
Keyword(s):  
Approximation Algorithms ◽  
Mild Condition ◽  
Distinct Point ◽  
Constant Factor ◽  
Approximation Schemes ◽  
Uniform Size ◽  
Map Labeling ◽  
Label Point ◽  
Efficient Approximation

In this paper we propose and study two practical variations of the map labeling problem: Given a set S of n distinct (point) sites in the plane, label each site with: (1) a pair of non-intersecting squares of maximum possible size, (2) a pair of non-intersecting circles of maximum possible size (all the squares and circles are topologically open and are of uniform size). Almost nothing has been done before in this aspect, i.e., multi-label map labeling. We obtain constant-factor approximation algorithms for these problems. We also study bicriteria approximation schemes for these problems under a mild condition.

scholarly journals Approximation algorithms on consistent dynamic map labeling

2016 ◽  
Vol 640 ◽  
pp. 84-93 ◽  
Author(s):  
Chung-Shou Liao ◽  
Chih-Wei Liang ◽  
Sheung Hung Poon
Keyword(s):  
Approximation Algorithms ◽  
Map Labeling ◽  
Dynamic Map

Two-stage submodular maximization problem beyond nonnegative and monotone

2021 ◽  
pp. 1-16
Author(s):  
Zhicheng Liu ◽  
Hong Chang ◽  
Ran Ma ◽  
Donglei Du ◽  
Xiaoyan Zhang
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Objective Function ◽  
Modular Function ◽  
Submodular Function ◽  
Maximization Problem ◽  
Cardinality Constraint ◽  
Two Stage ◽  
Time Efficiency ◽  
Submodular Maximization

Abstract We consider a two-stage submodular maximization problem subject to a cardinality constraint and k matroid constraints, where the objective function is the expected difference of a nonnegative monotone submodular function and a nonnegative monotone modular function. We give two bi-factor approximation algorithms for this problem. The first is a deterministic $\left( {{1 \over {k + 1}}\left( {1 - {1 \over {{e^{k + 1}}}}} \right),1} \right)$ -approximation algorithm, and the second is a randomized $\left( {{1 \over {k + 1}}\left( {1 - {1 \over {{e^{k + 1}}}}} \right) - \varepsilon ,1} \right)$ -approximation algorithm with improved time efficiency.

APPROXIMATION ALGORITHMS FOR A VARIANT OF DISCRETE PIERCING SET PROBLEM FOR UNIT DISKS

2013 ◽  
Vol 23 (06) ◽  
pp. 461-477 ◽  
Author(s):  
MINATI DE ◽  
GAUTAM K. DAS ◽  
PAZ CARMI ◽  
SUBHAS C. NANDY
Keyword(s):  
Approximation Algorithms ◽  
Simple Algorithm ◽  
Constant Factor ◽  
Performance Ratio ◽  
Approximation Result ◽  
Worst Case ◽  
Approximation Factor ◽  
Minimum Number ◽  
Unit Disks ◽  
Set Of Points

In this paper, we consider constant factor approximation algorithms for a variant of the discrete piercing set problem for unit disks. Here a set of points P is given; the objective is to choose minimum number of points in P to pierce the unit disks centered at all the points in P. We first propose a very simple algorithm that produces 12-approximation result in O(n log n) time. Next, we improve the approximation factor to 4 and then to 3. The worst case running time of these algorithms are O(n8 log n) and O(n15 log n) respectively. Apart from the space required for storing the input, the extra work-space requirement for each of these algorithms is O(1). Finally, we propose a PTAS for the same problem. Given a positive integer k, it can produce a solution with performance ratio [Formula: see text] in nO(k) time.

scholarly journals Optimizing active ranges for consistent dynamic map labeling

2010 ◽  
Vol 43 (3) ◽  
pp. 312-328 ◽  
Author(s):  
Ken Been ◽  
Martin Nöllenburg ◽  
Sheung-Hung Poon ◽  
Alexander Wolff
Keyword(s):  
Map Labeling ◽  
Dynamic Map

Unique Coverage with Rectangular Regions

2018 ◽  
Vol 28 (04) ◽  
pp. 341-363
Author(s):  
Rom Aschner ◽  
Paz Carmi ◽  
Yael Stein
Keyword(s):  
Wireless Networks ◽  
Polynomial Time ◽  
Optimal Solution ◽  
Directional Antenna ◽  
Constant Factor ◽  
Polynomial Time Algorithms ◽  
Half Strip ◽  
Coverage Problems

We study unique coverage problems with rectangle and half-strip regions, motivated by wireless networks in the context of coverage using directional antennae without interference. Given a set [Formula: see text] of points (clients) and a set [Formula: see text] of directional antennae in the plane, the goal is to assign a direction to each directional antenna in [Formula: see text], such that the number of clients in [Formula: see text] that are uniquely covered by the directional antennae is maximized. A client is covered uniquely if it is covered by exactly one antenna. We consider two types of rectangular regions representing half-strip directional antennae: unbounded half-strips and half-strips bounded by a range [Formula: see text] (i.e., [Formula: see text]-sided rectangular regions and rectangular regions). The directional antennae can be directed up or down. We present two polynomial time algorithms: an optimal solution for the problem with the [Formula: see text]-sided rectangular regions, and a constant factor approximation for the rectangular regions.

SHORTEST DESCENDING PATHS: TOWARDS AN EXACT ALGORITHM

2011 ◽  
Vol 21 (04) ◽  
pp. 431-466 ◽  
Author(s):  
MUSTAQ AHMED ◽  
ANNA LUBIW
Keyword(s):  
Approximation Algorithms ◽  
Polynomial Time ◽  
Shortest Path ◽  
Exact Algorithm ◽  
Full Characterization ◽  
Polynomial Time Algorithms ◽  
Straight Line ◽  
Bend Angles ◽  
Special Classes

A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. Although a shortest path on a terrain unfolds to a straight line, a shortest descending path (SDP) does not. We give a full characterization of the bend angles of an SDP, showing that they follow a generalized form of Snell's law of refraction of light. The complexity of finding SDPs is open—only approximation algorithms are known. We reduce the SDP problem to the problem of finding an SDP through a given sequence of faces. We give polynomial time algorithms for SDPs on two special classes of terrains, but argue that the general case will be difficult.

CUTTING OUT POLYGONS WITH LINES AND RAYS

2006 ◽  
Vol 16 (02n03) ◽  
pp. 227-248 ◽  
Author(s):  
OVIDIU DAESCU ◽  
JUN LUO
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Convex Polygon ◽  
Linear Time ◽  
Constant Factor ◽  
Open Problems ◽  
Cutting Out ◽  
Cutting Sequence

We present approximation algorithms for cutting out a polygon P with n vertices from another convex polygon Q with m vertices by line cuts and ray cuts. For line cuts we require both P and Q are convex while for ray cuts we require Q is convex and P is ray cuttable. Our results answer a number of open problems and are either the first solutions or significantly improve over previously known solutions. For the line cutting version, we prove a key property that leads to a simple, constant factor approximation algorithm. For the ray cutting version, we prove it is possible to compute in almost linear time a cutting sequence that is an O( log 2 n)-factor approximation of an optimal cutting sequence. No algorithms were previously known for the ray cutting version.

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