Conference Paper

# Approximation Schemes for Broadcasting in Heterogenous Networks.

DOI: 10.1007/b99805 Conference: Approximation, Randomization, and Combinatorial Optimization, Algorithms and Techniques, 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2004, and 8th International Workshop on Randomization and Computation, RANDOM 2004, Cambridge, MA, USA, August 22-24, 2004, Proceedings
Source: DBLP

ABSTRACT

In the Minimum Common String Partition problem (MCSP) we are given two strings on input, and we wish to partition them into the same collection of substrings, minimimizing the number of the substrings in the partition. Even a special case, denoted 2-MCSP, where each letter occurs at most twice in each input string, is NP-hard. We study a greedy algorithm for MCSP that at each step extracts a longest common substring from the given strings. We show that the approximation ratio of this algorithm is between Ω(n 0·43 ) and O(n 0·69 ). In case of 2-MCSP, we show that the approximation ratio is equal to 3. For 4-MCSP, we give a lower bound of Ω(logn).

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ABSTRACT: We consider the problem of deterministic broadcasting in radio networks when the nodes have limited knowledge about the topology of the network. We show that for every deterministic broadcasting protocol there exists a network, of radius 2, for which the protocol takes at least $\Omega(\sqrt{n}) rounds for completing the broadcast. Our argument can be extended to prove a lower bound of Omega(\sqrt{nD}) rounds for broadcasting in radio networks of radius D. This resolves one of the open problems posed in [29], where in the authors proved a lower bound of$\Omega(n^{1/4}) rounds for broadcasting in constant diameter networks. We prove the new lower $\Omega(\sqrt{n})$ bound for a special family of radius 2 networks. Each network of this family consists of O(\sqrt{n}) components which are connected to each other via only the source node. At the heart of the proof is a novel simulation argument, which essentially says that any arbitrarily complicated strategy of the source node can be simulated by the nodes of the networks, if the source node just transmits partial topological knowledge about some component instead of arbitrary complicated messages. To the best of our knowledge this type of simulation argument is novel and may be useful in further improving the lower bound or may find use in other applications. Keywords: radio networks, deterministic broadcast, lower bound, advice string, simulation, selective families, limited topological knowledge.
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