April Rasala

Route oscillations in I-BGP with route reflection

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.

Strictly non-blocking WDM cross-connects for heterogeneous networks

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. *Supported by a Lucent Technologies GRPW fellowship. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed fbr profit or commercial advantage and that copies bear this notice and the lull citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. STOC 2000 Portland Oregon USA Copyright ACM 2000 1-58113r 184-4/00/5...

Wide-Sense Nonblocking WDM Cross-Connects

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Strictly Nonblocking WDM Cross-connects

We consider the problem of minimizing the number of wavelength interchangers in the design of wide-sense nonblocking cross-connects for wavelength division multiplexed (WDM) optical networks. The problem is modeled as a graph theoretic problem that we call dynamic edge coloring. In dynamic edge coloring the nodes of a graph are fixed but edges appear and disappear, and must be colored at the time of appearance without assigning the same color to adjacent edges.For wide-sense nonblocking WDM cross-connects with k input and k output fibers, it is straightforward to show that 2k-1 wavelength interchangers are always sufficient. We show that there is a constant c > 0 such that if there are at least ck2 wavelengths then 2k-1 wavelength interchangers are also necessary. This improves previous exponential bounds. When there are only 2 or 3 wavelengths available, we show that far fewer than 2k-1 wavelength interchangers are needed. However we also prove that for any ? > 0 and k > 1/2?, if the number of wavelengths is at least 1/?2 then 2(1-?)k wavelength interchangers are needed.

Approximating the smallest grammar: Kolmogorov complexity in natural models

Using (WDM) technology, an optical network can route multiple signals simultaneously along a single optical fiber by encoding each signal on its own wavelength. If the network contains places where multiple fibers connect together and signals are allowed to be moved from any of the incoming fibers to any of the outgoing fibers, then the network is said to contain cross-connects. More precisely, a

Existence theorems, lower bounds and algorithms for scheduling to meet two objectives

k_1\,\times\,k_2

Strictly non-blocking WDM cross-connects

WDM cross-connect has k1 input fibers and k2 output fibers. Each of the k1 input fibers supports the same n1 input wavelengths and each of the k2 output fibers supports the same n2 output wavelengths. Since a signal on input wavelength

Improved bicriteria existence theorems for scheduling

\lambda

Existence Theorems for Scheduling to Meet Two Objectives

can be routed from its input fiber to an output fiber such that it arrives on the output fiber using wavelength

On list update and work function algorithms

\gamma