Manhattan Networks Research Articles

Approximating Minimum Manhattan Networks in Higher Dimensions

We study the minimum Manhattan network problem, which is defined as follows. Given a set of points called terminals in ${\mathbb {R}}^{d}$ , 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, L 1-) distance. The problem is NP-hard in 2D and there is no PTAS for 3D (unless ${\mathcal{P}} = \mathcal{NP}$ ). Approximation algorithms are known for 2D, but not for 3D. We present, for any fixed dimension d and any ?>0, an O(n ? )-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.

A rounding algorithm for approximating minimum Manhattan networks

For a set T of n points (terminals) in the plane, a Manhattan network on T is a network N ( T ) = ( V , E ) with the property that its edges are horizontal or vertical segments connecting points in V ⊇ T and for every pair of terminals, the network N ( T ) contains a shortest l 1 -path between them. A minimum Manhattan network on T is a Manhattan network of minimum possible length. The problem of finding minimum Manhattan networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan [J. Gudmundsson, C. Levcopoulos, G. Narasimhan, Approximating a minimum Manhattan network, Nordic Journal of Computing 8 (2001) 219–232. Proc. APPROX’99, 1999, pp. 28–37] and its complexity status is unknown. Several approximation algorithms (with factors 8, 4, and 3) have been proposed; recently Kato, Imai, and Asano [R. Kato, K. Imai, T. Asano, An improved algorithm for the minimum Manhattan network problem, ISAAC’02, in: LNCS, vol. 2518, 2002, pp. 344–356] have given a factor 2-approximation algorithm, however their correctness proof is incomplete. In this paper, we propose a rounding 2-approximation algorithm based on an LP-formulation of the minimum Manhattan network problem.

A Spectral Study of the Manhattan Networks

The multidimensional Manhattan networks are a family of digraphs with many appealing properties, such as vertex symmetry (in fact they are Cayley digraphs), easy routing, Hamiltonicity, and modular structure. From the known structural properties of these digraphs, we fully determine their spectra, which always contain the spectra of hypercubes. In particular, in the standard (two-dimensional) case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity.

The Multidimensional Manhattan Network

The n-dimensional Manhattan network Mn—a special case of n-regular digraph—is formally defined and some of its structural properties are studied. In particular, it is shown that Mn is a Cayley digraph, which can be seen as a subgroup of the n-dim version of the wallpaper group pgg. These results induce a useful new presentation of Mn, which can be applied to design a (shortest-path) local routing algorithm and to study some other metric properties. Also it is shown that the n-dim Manhattan networks are Hamiltonian and, in the standard case (that is, dimension two), they can be decomposed in two arc-disjoint Hamiltonian cycles. Finally, some results on the connectivity and distance-related parameters of Mn, such as the distribution of the node distances and the diameter are presented.

The minimum Manhattan network problem: Approximations and exact solutions

Given a set of points in the plane and a constant t ⩾ 1 , a Euclidean t- spanner is a network in which, for any pair of points, the ratio of the network distance and the Euclidean distance of the two points is at most t. Such networks have applications in transportation or communication network design and have been studied extensively. In this paper we study 1-spanners under the Manhattan (or L 1 -) metric. Such networks are called Manhattan networks. A Manhattan network for a set of points is a set of axis-parallel line segments whose union contains an x- and y-monotone path for each pair of points. It is not known whether it is NP-hard to compute minimum Manhattan networks (MMN), i.e., Manhattan networks of minimum total length. In this paper we present an approximation algorithm for this problem. Given a set P of n points, our algorithm computes in O ( n log n ) time and linear space a Manhattan network for P whose length is at most 3 times the length of an MMN of P. We also establish a mixed-integer programming formulation for the MMN problem. With its help we extensively investigate the performance of our factor-3 approximation algorithm on random point sets.

Picking alignments from (Steiner) trees.

The application of Needleman-Wunsch alignment techniques to biological sequences is complicated by two serious problems when the sequences are long: the running time, which scales as the product of the lengths of sequences, and the difficulty in obtaining suitable parameters that produce meaningful alignments. The running time problem is often corrected by reducing the search space, using techniques such as banding, or chaining of high-scoring pairs. The parameter problem is more difficult to fix, partly because the probabilistic model, which Needleman-Wunsch is equivalent to, does not capture a key feature of biological sequence alignments, namely the alternation of conserved blocks and seemingly unrelated nonconserved segments. We present a solution to the problem of designing efficient search spaces for pair hidden Markov models that align biological sequences by taking advantage of their associated features. Our approach leads to an optimization problem, for which we obtain a 2-approximation algorithm, and that is based on the construction of Manhattan networks, which are close relatives of Steiner trees. We describe the underlying theory and show how our methods can be applied to alignment of DNA sequences in practice, successfully reducing the Viterbi algorithm search space of alignment PHMMs by three orders of magnitude.

An optical switch architecture for Manhattan networks

An electronically controlled optical packet deflection switch that is based on space and time switching at intermediate nodes is described. The switch uses optical delay lines to store and switch packets in the optical domain so that optical bandwidth can be achieved across the network. The optical switch is controlled by an associated electronic mechanism that provides the necessary versatility and processing power. This approach combines the advantages of the optics and the electronics. The use and performance of the switch are demonstrated in the context of the Manhattan deflection network. >

Deterministic access protocols for packet radio networks

AbstractIn this paper, multiple‐phase deterministic protocols for packet radio networks are introduced and analysed. Two modes of information transfer are considered, namely (a) broadcasting and (b) point‐to‐point transmission. We explore systematic ways of designing multiple‐phase protocols and apply them on Manhattan networks. The proposed protocols are studied primarily from the point of view of throughput efficiency. Delay analysis is also presented.