Gautam K. Das

ON THE DISCRETE UNIT DISK COVER PROBLEM

Given a set of n points and a set of m unit disks on a 2-dimensional plane, the discrete unit disk cover (DUDC) problem is (i) to check whether each point in is covered by at least one disk in or not and (ii) if so, then find a minimum cardinality subset such that the unit disks in cover all the points in . The discrete unit disk cover problem is a geometric version of the general set cover problem which is NP-hard. The general set cover problem is not approximable within , for some constant c, but the DUDC problem was shown to admit a constant factor approximation. In this paper, we provide an algorithm with constant approximation factor 18. The running time of the proposed algorithm is . The previous best known tractable solution for the same problem was a 22-factor approximation algorithm with running time .

On the discrete unit disk cover problem

Given a set P of n points and a set D of m unit disks on a 2-dimensional plane, the discrete unit disk cover (DUDC) problem is (i) to check whether each point in P is covered by at least one disk in D or not and (ii) if so, then find a minimum cardinality subset D* ⊆ D such that unit disks in D* cover all the points in P. The discrete unit disk cover problem is a geometric version of the general set cover problem which is NP-hard [14]. The general set cover problem is not approximable within c log |P|, for some constant c, but the DUDC problem was shown to admit a constant factor approximation. In this paper, we provide an algorithm with constant approximation factor 18. The running time of the proposed algorithm is O(n log n+m log m+mn). The previous best known tractable solution for the same problem was a 22-factor approximation algorithm with running time O(m2n4).

Unit Disk Cover Problem in 2D

In this paper we consider the discrete unit disk cover problem and the rectangular region cover problem as follows.

Approximation algorithms for maximum independent set of a unit disk graph

Abstract We propose a 2-approximation algorithm for the maximum independent set problem for a unit disk graph. The time and space complexities are O ( n 3 ) and O ( n 2 ) , respectively. For a penny graph, our proposed 2-approximation algorithm works in O ( n log ⁡ n ) time using O ( n ) space. We also propose a polynomial-time approximation scheme (PTAS) for the maximum independent set problem for a unit disk graph. Given an integer k > 1 , it produces a solution of size 1 ( 1 + 1 k ) 2 | OPT | in O ( k 4 n σ k log ⁡ k + n log ⁡ n ) time and O ( n + k log ⁡ k ) space, where OPT is the subset of disks in an optimal solution and σ k ≤ 7 k 3 + 2 . For a penny graph, the proposed PTAS produces a solution of size 1 ( 1 + 1 k ) | OPT | in O ( 2 2 σ k n k + n log ⁡ n ) time using O ( 2 σ k + n ) space.

VARIATIONS OF BASE-STATION PLACEMENT PROBLEM ON THE BOUNDARY OF A CONVEX REGION

Due to the recent growth in the demand of mobile communication services in several typical environments, the development of efficient s ystems for providing specialized services has become an important issue in mobile communication research. An important sub-problem in this area is the base-station placement problem, where the objective is to identify the location for placing the basestations. Mobile terminals communicate with their respective nearest base station, and the base stations communicate with each other over scarce wireless channels in a multi-hop fashion by receiving and transmitting radio signals. Each base station emits signal periodically and all the mobile terminals within its range can identify it as its nearest base station after receiving such radio signal. Here the problem is to position the base stations such that each point in the entire area can communicate with at least one base-station, and total power required for all the base-stations in the network is minimized. A different variation of this problem arises when some portions of the target region is not suitable for placing the base-stations, but the communication inside those regions need to be provided. For example, we may consider the large water bodies or the stiff mountains. In such cases, we need some specialized algorithms for efficiently placing the base-stations on the boundary of the f orbidden zone to provide services inside that region.

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

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(n8log n) and O(n15log 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 in nO(k) time.

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(n2) 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].

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.

An efficient heuristic algorithm for 2D h-hops range assignment problem

Given a set S of n radio-stations on a 2D plane and an integer h, the range assignment problem is to assign ranges to the members in S such that each member of S can communicate with all other members in S using at most h hops, and the sum of powers required for all the members in S is minimized. The general 2D h-hop range assignment problem is known to be NP-hard (A.E.F. Clementi et al, Proc. Symp. on Theor. Aspects of Comp. Sci. (STACS-00), pp. 651-660, 2000). We first consider some simplified variations of the problem and propose an efficient polynomial time algorithm for obtaining optimal solution. In the homogeneous version, where the range assigned to each radio-station is same (/spl rho/), we can obtain the minimum value of /spl rho/ in O(n/sup 3/logn) time in the worst case. In addition, if we consider the unbounded version of the homogeneous range assignment problem (i.e. h=n-1), then the optimal value of /spl rho/ can be obtained in O(n/sup 2/logn) time. Finally, we propose an efficient heuristic algorithm for the general h-hop range assignment problem in 2D, where the range of the radio stations may not be equal. Experimental results demonstrate that our heuristic algorithm runs fast and produces near-optimal solutions on randomly generated instances.

Distributed construction of connected dominating set in unit disk graphs

Let G=(V,E) be a unit disk graph corresponding to a given set P of n points in R2. We propose a distributed approximation algorithm to find a minimum connected dominating set of G. The maintenance of the connected dominating set constructed is fully localized. Our algorithm produces a connected dominating set of size (104opt+52), where opt is the size of a minimum connected dominating set. The time and message complexities of our algorithm are O() and O(n) respectively, where and n are the maximum node degree and number of nodes in G respectively. Our distributed approximation algorithm outperforms existing algorithms in terms of its time and message complexities.We also propose a scheduling scheme that obtains O() conflict-free time slots to deal with interference. We proposed a distributed algorithm to find a connected dominating set for unit disk graphs.Our algorithm produces a connected dominating set of size (104opt+52), where opt is the size of an optimum solution.Running time and message complexities of the algorithm are O() and O(n) respectively.Scheduling scheme to deal interference of the network (graph).