We study the route oscillation problem [16, 19] in the Internal Border Gateway Protocol (I-BGP)[18] when route reflection is used. We propose a formal model of I-BGP and use it to show that even deciding whether an I-BGP configuration with route reflection can converge is an NP-Complete problem. We then propose a modification to I-BGP and show that route reflection cannot cause the modified protocol to diverge. Moreover, we show that the modified protocol converges to the same stable routing configuration regardless of the order in which messages are sent or received.
A wavelength division multiplexed (WDM) network is a network of optical fibers and components that supports multiple wavelength channels along each fiber. A heterogeneous WDM network is one that is composed of sub-networks each supporting possibly different sets of wavelength channels. Thus the number of wavelength channels, nl, on the input fibers to a cross-connect connecting one such sub-network to another will possibly differ from the number of wavelength channels, n2, on its output fibers. In order to balance the number of channels coming into and going out of such a cross-connect it is possible then that the number of input fibers, kl, will differ from the number of output fibers, k2. We study the design of such cross-connects. In particular we consider strictly non-blocking WDM cross-connects where a cross-connect C is strictly non-blocking if it is always capable of routing any unused wavelength channel on any fiber entering C onto any unused wavelength channel on any fiber leaving C regardless of any other routes already in use within C. Any such WDM cross-connect will need some number of wavelength interchangers, expensive optical components that translate incoming wavelength channels onto different outgoing wavelength channels. We provide a simple method for constructing a kl × k2 strictly non-blocking WDM crossconnect having at most min(kl -t-k2 -1, nlkl) wavelength interchangers for any kl < k2. A proof is then given showing that these constructions are optimal in that any such WDM cross-connect for a heterogeneous network must have at least min(kl + k2 --1, nl kz) wavelength interchangers.
We study the route oscillation problem [16,19] in the Internal Border Gateway Protocol (I-BGP) [18] when route reflection is used. We propose a formal model of I-BGP and use it to show that even deciding whether an I-BGP configuration with route reflection can converge is an NP-Complete problem. We then propose a modification to I-BGP and show that route reflection cannot cause the modified protocol to diverge. Moreover, we show that the modified protocol converges to the same stable routing configuration regardless of the order in which messages are sent or received.
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We consider the problem of finding the smallest context-free grammar that generates exactly one given string of length n. The size of this grammar is of theoretical interest as an efficiently computable variant of Kolmogorov complexity. The problem is of practical importance in areas such as data compression and pattern extraction.The smallest grammar is known to be hard to approximate to within a constant factor, and an o(log n/log log n) approximation would require progress on a long-standing algebraic problem [10]. Previously, the best proved approximation ratio was O(n 1/2 ) for the Bisection algorithm [8]. Our main result is an exponential improvement of this ratio; we give an O(log(n/g * )) approximation algorithm, where g * is the size of the smallest grammar.We then consider other computable variants of Kolomogorov complexity. In particular we give an O(log 2 n) approximation for the smallest non-deterministic finite automaton with advice that produces a given string. We also apply our techniques to "advice-grammars" and "edit-grammars", two other natural models of string complexity. *
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