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SelfApproaching Graphs
"... Abstract. In this paper we introduce selfapproaching graph drawings. A straightline drawing of a graph is selfapproaching if, for any origin vertex s and any destination vertex t, there is an stpath in the graph such that, for any point q on the path, as a point p moves continuously along the pa ..."
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Abstract. In this paper we introduce selfapproaching graph drawings. A straightline drawing of a graph is selfapproaching if, for any origin vertex s and any destination vertex t, there is an stpath in the graph such that, for any point q on the path, as a point p moves continuously along the path from the origin to q, the Euclidean distance from p to q is always decreasing. This is a more stringent condition than a greedy drawing (where only the distance between vertices on the path and the destination vertex must decrease), and guarantees that the drawing is a 5.33spanner. We study three topics: (1) recognizing selfapproaching drawings; (2) constructing selfapproaching drawings of a given graph; (3) constructing a selfapproaching Steiner network connecting a given set of points. Weshowthat: (1)thereare efficient algorithms totest ifapolygonal path is selfapproaching inR 2 and R 3, butit is NPhardtotest ifagiven graph drawing in R 3 has a selfapproaching uvpath; (2) we can characterize the trees that have selfapproaching drawings; (3) for any given set of terminal points in the plane, we can find a linear sized network that has a selfapproaching path between any ordered pair of terminals.
Approximating Minimum Manhattan Networks in Higher Dimensions
"... We consider the minimum Manhattan network problem, which is defined as follows. Given a set of points called terminals in R d, find a minimumlength network such that each pair of terminals is connected by a set of axisparallel line segments whose total length is equal to the pair’s Manhattan (tha ..."
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We consider the minimum Manhattan network problem, which is defined as follows. Given a set of points called terminals in R d, find a minimumlength network such that each pair of terminals is connected by a set of axisparallel line segments whose total length is equal to the pair’s Manhattan (that is, L1) distance. The problem is NPhard in 2D and there is no PTAS for 3D (unless P = N P). Approximation algorithms are known for 2D, but not for 3D. We present, for any fixed dimension d and any ε> 0, an O(n ε)approximation. For 3D, we also give a 4(k − 1)approximation for the case that the terminals are contained in the union of k ≥ 2 parallel planes.
A.: Approximating the generalized minimum Manhattan network problem. Arxiv report
, 2012
"... Abstract. We consider the generalized minimum Manhattan network problem (GMMN). The input to this problem is a set R of n pairs of terminals, which are points in R 2 . The goal is to find a minimumlength rectilinear network that connects every pair in R by a Manhattan path, that is, a path of axis ..."
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Abstract. We consider the generalized minimum Manhattan network problem (GMMN). The input to this problem is a set R of n pairs of terminals, which are points in R 2 . The goal is to find a minimumlength rectilinear network that connects every pair in R by a Manhattan path, that is, a path of axisparallel line segments whose total length equals the pair's Manhattan distance. This problem is a natural generalization of the extensively studied minimum Manhattan network problem (MMN) in which R consists of all possible pairs of terminals. Another important special case is the wellknown rectilinear Steiner arborescence problem (RSA). As a generalization of these problems, GMMN is NPhard. No approximation algorithms are known for general GMMN. We obtain an O(log n)approximation algorithm for GMMN. Our solution is based on a stabbing technique, a novel way of attacking Manhattan network problems. Some parts of our algorithm generalize to higher dimensions, yielding a simple O(log d+1 n)approximation algorithm for the problem in arbitrary fixed dimension d. As a corollary, we obtain an exponential improvement upon the previously best O(n ε )ratio for MMN in d dimensions [ESA'11]. En route, we show that an existing O(log n)approximation algorithm for 2DRSA generalizes to higher dimensions.
LowPower Gated Bus Synthesis for 3D IC via Rectilinear Shortestpath Steiner Graph
, 2012
"... In this paper, we propose a new approach for gated bus synthesis [16] with minimum wire capacitance per transaction in threedimensional (3D) ICs. The 3D IC technology connects different device layers with throughsilicon vias (TSV), which need to be considered differently from metal wire due to rel ..."
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In this paper, we propose a new approach for gated bus synthesis [16] with minimum wire capacitance per transaction in threedimensional (3D) ICs. The 3D IC technology connects different device layers with throughsilicon vias (TSV), which need to be considered differently from metal wire due to reliability issues and a larger footprint. Practically, the number of TSVs is bounded between layers; thus, we first devise dynamic programming and local search techniques to determine the optimal TSV locations. We then employ two approximation algorithms to generate a rectilinear shortestpath Steiner graph in each device layer. One algorithm extends the wellknown greedy heuristic for the Rectilinear Steiner Arborescence problem and handles large cases with high efficiency. The other algorithm utilizes a linear programming relaxation and rounding technique which costs more time and generates a nearlyoptimal Steiner graph. Experimental results show that our algorithms can construct shortestpath Steiner graphs with 22 % less total wire length than the previous method of Wang et al. [16].
The Generalized Minimum Manhattan Network Problem (GMMN) – ScaleDiversity Aware Approximation and a PrimalDual Algorithm
"... In the ddimensional Generalized Minimum Manhattan Network (dGMMN) problem one is interested in finding a minimum cost rectilinear network N connecting a given set R of n pairs of points in Rd such that each pair is connected in N via a shortest Manhattan path. The problem is known to be NPcomp ..."
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In the ddimensional Generalized Minimum Manhattan Network (dGMMN) problem one is interested in finding a minimum cost rectilinear network N connecting a given set R of n pairs of points in Rd such that each pair is connected in N via a shortest Manhattan path. The problem is known to be NPcomplete and does not admit a FPTAS, the best known upper bound is an O(logd+1 n)approximation for d> 2 and an O(log n)approximation for d = 2 by Das et al. [3]. In this paper we provide some more insight into the problem and develop two new algorithms, a ‘scalediversity aware ’ algorithm with an O(D) approximation guarantee for d = 2. Here D is a measure for the different ‘scales ’ that appear in the input, D ∈ O(log n) but potentially much smaller depending on the problem instance. Moreover, this implies that a potential proof of O(1)inapproximability for 2GMMN requires gadgets of many different scales in the construction. The other algorithm is based on a primaldual scheme solving a more general path covering problem. On 2GMMN it performs pretty well in practice with good a posteriori, instancebased approximation guarantees. Furthermore, it can be extended naturally to deal with obstacle avoiding requirements.
A Simulated Annealing approach for solving Minimum Manhattan Network Problem
"... In this paper we address the Minimum Manhattan Network (MMN) problem. It is an important geometric problem with vast applications. As it is an NPcomplete discrete combinatorial optimization problem we employ a simple metaheuristic namely Simulated Annealing. We have also developed benchmark datas ..."
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In this paper we address the Minimum Manhattan Network (MMN) problem. It is an important geometric problem with vast applications. As it is an NPcomplete discrete combinatorial optimization problem we employ a simple metaheuristic namely Simulated Annealing. We have also developed benchmark datasets and tested our algorithm with the dataset.
Polylogarithmic Approximation for Generalized Minimum Manhattan Networks
"... Abstract. Given a set of n terminals, which are points in ddimensional Euclidean space, the minimum Manhattan network problem (MMN) asks for a minimumlength rectilinear network that connects each pair of terminals by a Manhattan path, that is, a path consisting of axisparallel segments whose tota ..."
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Abstract. Given a set of n terminals, which are points in ddimensional Euclidean space, the minimum Manhattan network problem (MMN) asks for a minimumlength rectilinear network that connects each pair of terminals by a Manhattan path, that is, a path consisting of axisparallel segments whose total length equals the pair’s Manhattan distance. Even for d = 2, the problem is NPhard, but constantfactor approximations are known. For d ≥ 3, the problem is APXhard; it is known to admit, for any ε> 0, an O(nε)approximation. In the generalized minimum Manhattan network problem (GMMN), we are given a set R of n terminal pairs, and the goal is to find a minimumlength rectilinear network such that each pair in R is connected by a Manhattan path. GMMN is a generalization of both MMN and the wellknown rectilinear Steiner arborescence problem (RSA). So far, only special cases of GMMN have been considered. We present an O(logd+1 n)approximation algorithm for GMMN (and, hence, MMN) in d ≥ 2 dimensions and anO(logn)approximation algorithm for 2D. We show that an existingO(logn)approximation algorithm for RSA in 2D generalizes easily to d> 2 dimensions. 1