Range Assignment Problem Research Articles

Bounded-Hop Communication Networks

We study the problem of assigning transmission ranges to radio stations in the plane such that any pair of stations can communicate within a bounded number of hops h and the cost of the network is minimized. We consider two settings of the problem: collinear station locations and arbitrary locations. For the case of collinear stations, we introduce the pioneer polynomial-time exact algorithm for any $$\alpha \ge 1$$ and constant h, and thus conclude that the 1D version of the problem, where h is a constant, is in $$\mathcal {P}$$ . Furthermore, we provide a 3 / 2-approximation algorithm for the case where h is an arbitrary number and $$\alpha =1$$ , improving the previously best known approximation ratio of 2. For the case of stations placed arbitrarily in the plane and $$\alpha =1$$ , we first present a $$(1.5+ \varepsilon )$$ -approximation algorithm for a case where a deviation of one hop ( $$h+1$$ hops in total) is acceptable. Then, we show a $$(3+\varepsilon )$$ -approximation algorithm that complies with the exact hop bound. To achieve that, we introduce the following two auxiliary problems, which are of independent interest. The first is the bounded-hop multi-sink range problem, for which we present a PTAS which can be applied to compute a $$(1+\varepsilon )$$ -approximation for the bounded diameter minimum spanning tree, for any $$\varepsilon >0$$ . The second auxiliary problem is the bounded-hop dual-sink pruning problem, for which we show a polynomial-time algorithm. To conclude, we consider the initial bounded-hop all-to-all range assignment problem and show that a combined application of the aforementioned problems yields the $$(3+\varepsilon )$$ -approximation ratio for this problem, which improves the previously best known approximation ratio of $$4(9^{h-2})/(\root h \of {2}-1)$$ .

Dual Decomposition for Computational Optimization of Minimum-Power Shared Broadcast Tree in Wireless Networks

We consider the problem of constructing a shared broadcast tree (SBT) in wireless networks, such that the total power required for supporting broadcast initiated by all source nodes is minimal. In the well-studied minimum-energy broadcast (MEB) problem, the optimal tree varies by source. In contrast, SBT is source-independent, thus substantially reducing the overhead for information storage and processing. The SBT problem also differs from the range assignment problem (RAP), because the power for message forwarding in SBT, although being source-independent, depends on from which tree neighbor the message is received. We approach SBT from a computational optimization standpoint, and present a dual decomposition method applied to an optimization model that embeds multiple directed trees into a shared tree. For the dual decomposition method, some of the constraints in the model are preferably formulated implicitly. The dual decomposition scheme is coupled with a fast local search algorithm. We report computational results demonstrating the effectiveness of the proposed approach. In average, the performance gap to global optimality is less than three percent.

Submodular Formulations for Range Assignment Problems

Abstract We devise two new integer programming models for range assignment problems arising in wireless network design. Building on an arbitrary set of feasible network topologies, e.g., all spanning trees, we explicitly model the power consumption at a given node as a weighted maximum over edge variables. We show that the standard ILP model is an extended formulation of the new models. We observe that the objective functions in the compact models are submodular. This enables us to derive complete polyhedral descriptions in the unconstrained case where all topologies are allowed. These results give rise to tight relaxations even in the constrained case. We can show experimentally that the compact formulations compare favorably to the standard approach.

Homogeneous 2-hop broadcast in 2D

In this paper, two variations of the minimum cost homogeneous range assignment problem for 2-hop broadcast from a given source are considered. A set S = { s 0 , s 1 , … , s n } of radio stations are pre-placed in R 2 , and a source station s 0 (say) is marked. In our first problem, the objective is to find a real number r such that 2-hop homogeneous broadcast from s 0 is possible with range r, and the total power consumption of the entire network is minimum. In the second problem, a real number r is given and the objective is to identify the smallest subset of S for which range r can be assigned to accomplish the 2-hop broadcast from s 0 , provided such an assignment is possible. The first problem is solved in O ( n 2.376 log n ) time and O ( n 2 ) space. For the second problem, a 2-factor approximation algorithm is proposed that runs in O ( n 2 ) time and O ( n ) space.

Approximate hierarchical facility location and applications to the bounded depth Steiner tree and range assignment problems

The paper concerns a new variant of the hierarchical facility location problem on metric powers ( HFL β [ h ] ), which is a multi-level uncapacitated facility location problem defined as follows. The input consists of a set F of locations that may open a facility, subsets D 1 , D 2 , … , D h − 1 of locations that may open an intermediate transmission station and a set D h of locations of clients. Each client in D h must be serviced by an open transmission station in D h − 1 and every open transmission station in D l must be serviced by an open transmission station on the next lower level, D l − 1 . An open transmission station on the first level, D 1 must be serviced by an open facility. The cost of assigning a station j on level l ⩾ 1 to a station i on level l − 1 is c i j . For i ∈ F , the cost of opening a facility at location i is f i ⩾ 0 . It is required to find a feasible assignment that minimizes the total cost. A constant ratio approximation algorithm is established for this problem. This algorithm is then used to develop constant ratio approximation algorithms for the bounded depth Steiner tree problem and the bounded hop strong-connectivity range assignment problem.

On the hardness of range assignment problems

AbstractWe investigate the computational hardness of the connectivity, the strong connectivity, and the broadcast type of range assignment problems in ℝ2 and ℝ3. We present new reductions for the connectivity problem, which are easily adapted to suit the other two problems. All reductions are considerably simpler than the technically quite involved ones used in earlier works on these problems. Using our constructions, we can for the first time prove NP‐hardness of these problems for all real distance‐power gradients α > 0 (resp. α > 1 for broadcast) in 2D, and prove APX‐hardness of all three problems in 3D for all α > 1. Our reductions yield improved lower bounds on the approximation ratios for all problems where APX‐hardness was known before already. In particular, we derive the overall first APX‐hardness proof for broadcast. This was an open problem posed in earlier work in this area, as was the question whether (strong) connectivity remains NP‐hard for α = 1. In addition, we give the first hardness results for so‐called well‐spread instances. On the positive side, we prove that two natural greedy algorithms are 2‐approximations for (strong) connectivity, and show that the factor 2 is tight in ℝ2 for α > 1. We also analyze the performance guarantee of the well‐known MST‐heuristic as a function of the input size. © 2008 Wiley Periodicals, Inc. NETWORKS, 2008

Weighted broadcast in linear radio networks

The weighted version of the broadcast range assignment problem in ad hoc wireless network is studied. Efficient algorithms are presented for the unbounded and bounded-hop broadcast problems for the linear radio network, where radio stations are placed on a straight line. For the unbounded case of the problem, the proposed algorithm runs in O ( n 2 ) time and using O ( n ) space, where n is the number of radio stations in the network. For the h-hop broadcast problem, the time and space complexities of our algorithm are O ( h n 2 log n ) and O ( h n ) , respectively. This improves time complexity of the existing results for the same two problems by a factor of n and n 2 log n , respectively [C. Ambuhl, A.E.F. Clementi, M.D. Ianni, G. Rossi, A. Monti, R. Silvestri, The range assignment problem in non-homogeneous static ad hoc networks, in: Proc. 18th Int. Parallel and Distributed Precessing Symposium, 2004].

IMPROVED ALGORITHM FOR MINIMUM COST RANGE ASSIGNMENT PROBLEM FOR LINEAR RADIO NETWORKS

In the unbounded version of the range assignment problem for all-to-all communication in 1D, a set of n radio-stations are placed arbitrarily on a line; the objective is to assign ranges to these radio-stations such that each of them can communicate with the others (using at most n - 1 hops) and the total power consumption is minimum. A simple incremental algorithm for this problem is proposed which produces optimum solution in O(n3) time and O(n2) space. This is an improvement in the running time by a factor of n over the best known existing algorithm for the same problem.

The minimum-area spanning tree problem

Motivated by optimization problems in sensor coverage, we formulate and study the Minimum-Area Spanning Tree ( mast) problem: Given a set P of n points in the plane, find a spanning tree of P of minimum “area”, where the area of a spanning tree T is the area of the union of the n − 1 disks whose diameters are the edges in T . We prove that the Euclidean minimum spanning tree of P is a constant-factor approximation for mast. We then apply this result to obtain constant-factor approximations for the Minimum-Area Range Assignment ( mara) problem, for the Minimum-Area Connected Disk Graph ( macdg) problem, and for the Minimum-Area Tour ( mat) problem. The first problem is a variant of the power assignment problem in radio networks, the second problem is a related natural problem, and the third problem is a variant of the traveling salesman problem.

Range assignment for energy efficient broadcasting in linear radio networks

Given a set S of n radio-stations located on a d-dimensional space, a source node s ( ∈ S ) and an integer h ( 1 ⩽ h ⩽ n - 1 ) , the h-hop broadcast range assignment problem deals with assigning ranges to the members in S so that s can communicate with all other members in S in at most h-hops, and the total power consumption is minimum. The problem is known to be NP-hard for d ⩾ 2 . We propose an O ( n 2 ) time algorithm for the one dimensional version ( d = 1 ) of the problem. This is an improvement over the existing result on this problem by a factor of h [A.E.F. Clementi et al. The minimum broadcast range assignment problem on linear multi-hop wireless networks, Theoret. Comput. Sci. 299 (2003) 751–761].

Assign ranges in general ad-hoc networks

In this paper we theoretically study the M INIMUM R ANGE A SSIGNMENT problem in static ad-hoc networks with arbitrary structure, where the transmission distances can violate triangle inequality. We consider two versions of the M INIMUM R ANGE A SSIGNMENT problem, where the communication graph has to fulfill either the h-strong connectivity condition (M IN-R ANGE( h -SC)) or the h-broadcast condition (M IN-R ANGE( h -B)). Both homogeneous and non-homogeneous cases are studied. By approximating arbitrary edge-weighted graphs by paths, we present probabilistic O ( log n ) -approximation algorithms for M IN-R ANGE( h -SC) and M IN-R ANGE( h -B), which improves the previous best ratios O ( log n log log n ) and O ( n 2 log n log log n ) , respectively [D. Ye, H. Zhang, The range assignment problem in static ad-hoc networks on metric spaces, Proceedings of the 11th Colloquium on Structural Information and Communication Complexity, Sirocco 2004, Lecture Notes in Computer Science, vol. 3104, pp. 291–302]. The result for M IN-R ANGE( h -B) matches the lower bound [G. Rossi, The range assignment problem in static ad-hoc wireless networks. Ph.D. Thesis, 2003] for the case that triangle inequality for transmission distance holds (which is a special case of our model). Furthermore, we show that if the network fulfils certain property and the distance power gradient α is sufficiently small, the approximation ratio is improved to O ( ( log log n ) α ) . Finally we discuss the applications of our algorithms in mobile ad-hoc networks.

On the approximability of the range assignment problem on radio networks in presence of selfish agents

We consider the range assignment problem in ad-hoc wireless networks in the context of selfish agents: A network manager aims to assigning transmission ranges to the stations in order to achieve strong connectivity of the network within a minimal overall power consumption. Station is not directly controlled by the manager and may refuse to transmit with a certain transmission range because it might be costly in terms of power consumption. We investigate the existence of payment schemes which induce the stations to follow the decisions of a network manager in computing a range assignment, that is, truthful mechanisms for the range assignment problem. We provide both positive and negative results on the existence of truthful VCG-based mechanisms for this N P-hard problem. We prove that (i) in general, every polynomial-time truthful VCG-based mechanism computes a solution of cost far-off the optimum, unless P=N P and (ii) there exists a polynomial-time truthful VCG-based mechanism achieving constant approximation for practically relevant, still N P-hard versions, i.e., the metric and the well-spread case.

The minimum broadcast range assignment problem on linear multi-hop wireless networks

Given a set N of radio stations located on an Euclidean space, a source station s and an integer h(1⩽h⩽|N|−1), the minimum bounded-hop broadcast range assignment problem consists in finding a range assignment for N of minimum total power consumption that allows broadcast operations from s to every station in N in at most h hops. The problem is known to be NP-hard on d-dimensional spaces for any d⩾2 (18th Annual Symp. on Theoretical Aspects of Computer Science (STACS’01), Lecture Notes in Computer Science, Vol. 1770, 2000, pp. 651–660.) and some efficient approximation algorithms have been given in Clementi et al. and Wann et al. (18th Annual Symp. on Theoretical Aspects of Computer Science (STACS’01), Lecture Notes in Computer Science, Vol. 1770, 2000, pp. 651–660, IEEE INFOCOM’01, 2001). In this paper, we address the case in which the stations are arbitrarily located along a line (i.e., the linear case). We provide the first polynomial-time algorithm that returns an optimal solution for any instance of the linear case. The algorithm works in O(h|N|2) time.

The Minimum Range Assignment Problem on Linear Radio Networks

Given a set S of radio stations located on a line and an integer h ≥ 1 , the MIN ASSIGNMENT problem consists in finding a range assignment of minimum power consumption provided that any pair of stations can communicate in at most h hops. Previous positive results for this problem are only known when h=|S|-1 or in the uniform chain case (i.e., when the stations are equally spaced). As for the first case, Kirousis et al. [7] provided a polynomial-time algorithm while, for the second case, they derive a polynomial-time approximation algorithm. This paper presents the first polynomial-time, approximation algorithm for the MIN ASSIGNMENT problem. The algorithm guarantees a 2-approximation ratio and runs in O(hn 3 ) time. We also prove that, for fixed h and for ``well spaced'' instances (a broad generalization of the uniform chain case), the problem admits a polynomial-time approximation scheme . This result significantly improves over the approximability result given by Kirousis {et al}. Both our approximation results are obtained via new algorithms that exactly solve two natural variants of the MIN ASSIGNMENT problem: the problem in which every station must reach a fixed one in at most h hops and the problem in which the goal is to select a subset of bases such that all the other stations must reach one base in at most h-1 hops. Finally, we show that for h=2 the MIN ASSIGNMENT problem can be exactly solved in O(n 3 ) time.