A rounding algorithm for approximating minimum Manhattan networks

Chepoi, Victor; Nouioua, Karim; Vaxès, Yann

doi:10.1016/j.tcs.2007.10.013

Cited by 27 publications

(33 citation statements)

References 14 publications

(28 reference statements)

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“…Let also

F ″

be the set of all ordered pairs

({boldt}_{j'}, {boldt}_{l})

and

({boldt}_{l}, {boldt}_{j'})

such that there exists a staircase

{scriptS}_{i, j | i', j'}

with t l belonging to the set

T_{i, j}

of all terminals defining the corners of

{scriptS}_{i, j | i', j'}

. The proof of the following result closely follows the proof of Lemma 3.3 of. Lemma

F : = F' \cup F ″

is a generating set . Proof The set

F_{\emptyset}

of empty pairs is clearly a generating set.…”

Section: Strips Staircases and Generating Setsmentioning

confidence: 85%

“…In this section, we briefly recall the notions of strips and staircases defined and studied in ; we refer to this article for proofs and some missing details. An empty rectangle

R_{i, j}

is called a vertical strip if the x ‐coordinates of t i and t j are consecutive entries of the sorted list of all x ‐coordinates of the terminals.…”

Section: Strips Staircases and Generating Setsmentioning

confidence: 99%

“…Nouioua and Fuchs and Schulze presented two simple

O (n \log n)

‐time 3‐approximation algorithms. The first correct 2‐approximation algorithm (solving the first open question from ) was presented by Chepoi, Nouioua, and Vaxès and is based on a strip‐staircase decomposition and a rounding method applied to the linear program from . In his PhD thesis, Nouioua described an

O (n \log n)

‐time 2‐approximation algorithm based on the primal‐dual method.…”

Section: Introductionmentioning

confidence: 99%

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Bidirected minimum Manhattan network problem

et al. 2016

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International audienceIn the bidirected minimum Manhattan network problem, given a set T of n terminals in the plane, no two terminals on the same horizontal or vertical line, we need to construct a network N(T) of minimum total length with the property that the edges of N(T) belong to the axis-parallel grid defined by T and are oriented in a such a way that every ordered pair of terminals is connected in N(T) by a directed Manhattan path. In this article, we present a polynomial factor 2-approximation algorithm for the bidirected minimum Manhattan network problem

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“…Let also

F ″

be the set of all ordered pairs

({boldt}_{j'}, {boldt}_{l})

and

({boldt}_{l}, {boldt}_{j'})

such that there exists a staircase

{scriptS}_{i, j | i', j'}

with t l belonging to the set

T_{i, j}

of all terminals defining the corners of

{scriptS}_{i, j | i', j'}

. The proof of the following result closely follows the proof of Lemma 3.3 of. Lemma

F : = F' \cup F ″

is a generating set . Proof The set

F_{\emptyset}

of empty pairs is clearly a generating set.…”

Section: Strips Staircases and Generating Setsmentioning

confidence: 85%

“…In this section, we briefly recall the notions of strips and staircases defined and studied in ; we refer to this article for proofs and some missing details. An empty rectangle

R_{i, j}

is called a vertical strip if the x ‐coordinates of t i and t j are consecutive entries of the sorted list of all x ‐coordinates of the terminals.…”

Section: Strips Staircases and Generating Setsmentioning

confidence: 99%

“…Nouioua and Fuchs and Schulze presented two simple

O (n \log n)

O (n \log n)

‐time 2‐approximation algorithm based on the primal‐dual method.…”

Section: Introductionmentioning

confidence: 99%

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Bidirected minimum Manhattan network problem

et al. 2016

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“…Below, we construct an instance that requires a generating set of size Ω(n 2 ). The idea of using linear-size generating sets is exploited by several algorithms for 2D-MMN [5,11]. The following theorem shows that these approaches do not easily carry over to 3D.…”

Section: Quadratic Lower Bound For Generating Sets In 3dmentioning

confidence: 99%

“…This outperforms our grid-based algorithm when k ∈ o(n ε ). Whereas 2D-MMN admits a 2-approximation [5,10,15], it remains open whether O(1)or O(log n)-approximation algorithms exist for higher dimensions.…”

Section: Open Problemsmentioning

confidence: 99%

Approximating Minimum Manhattan Networks in Higher Dimensions

Das

Gansner

Kaufmann

et al. 2011

Algorithms – ESA 2011

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We study the minimum Manhattan network problem, which is defined as follows. Given a set of points called terminals in R d , find a minimum-length network such that each pair of terminals is connected by a set of axis-parallel line segments whose total length is equal to the pair's Manhattan (that is, L1-) distance. The problem is NP-hard 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 algorithm. For 3D, we also give a 4(k − 1)-approximation algorithm for the case that the terminals are contained in the union of k ≥ 2 parallel planes. s s t t (a) an M-network for T = {s, s , t, t } s s t t (b) a minimum Mnetwork for T s s t t (c) a minimum Mnetwork in 3D t s t s (d) M-paths missing each other Fig. 1: Examples of M-networks in 2D and 3D. network problem (MMN) consists of finding, for a given set T of terminals, a minimum-weight M-network. For examples, see Fig. 1. M-networks have important applications in several areas such as VLSI layout and computational biology. For example, Lam et al. [12] used them in gene alignment in order to reduce the size of the search space of the Viterbi algorithm for pair hidden Markov models. Previous workThe 2D-version of the problem, 2D-MMN, was introduced by Gudmundsson et al. [9]. They gave an 8-and a 4-approximation algorithm. Later, the approximation ratio was improved to 3 [3,8] and then to 2, which is currently the best possible. It was achieved in three different ways: via linear programming [5], using the primal-dual scheme [15] and with purely geometric arguments [10]. The last two algorithms run in O(n log n) time, given a set of n points in the plane. A ratio of 1.5 was claimed [16], but apparently the proof is incomplete [8]. Chin et al. [6] finally settled the complexity of 2D-MMN by proving it NP-hard.A little earlier, Muñoz et al.[14] considered 3D-MMN. They showed that the problem is NP-hard and that it is NP-hard to approximate beyond a factor of 1.00002. For the special case of 3D-MMN, where any cuboid spanned by two terminals contains other terminals or is a rectangle, they gave a 2α-approximation algorithm, where α denotes the best approximation ratio for 2D-MMN. They posed the design of approximation algorithms for general 3D-MMN as an open problem. Related problemsAs we observe in Section 2.3, MMN is a special case of the directed Steiner forest problem (DSF). More precisely, an instance of MMN can be decomposed into a constant number of DSF instances. The input of DSF is an edge-weighted directed graph G and a set of vertex pairs. The goal is to find a minimum-cost subgraph of G (not necessarily a forest) that connects all given vertex pairs. Recently, Feldman et al. [7] reported, for any ε > 0, an O(n 4/5+ε )-approximation algorithm for DSF, where n is the number of vertices of the given graph. This bound carries over to dD-MMN.An important special case of DSF is the directed Steiner tree problem (DST). Here, the inpu...

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