Timothy Y. Chow

Fast optical layer mesh protection using pre-cross-connected trails

Conventional optical networks are based on SONET rings, but since rings are known to use bandwidth inefficiently, there has been much research into shared mesh protection, which promises significant bandwidth savings. Unfortunately, most shared mesh protection schemes cannot guarantee that failed traffic will be restored within the 50-ms timeframe that SONET standards specify. A notable exception is the p-cycle scheme of Grover and Stamatelakis. We argue, however, that p-cycles have certain limitations, e.g., there is no easy way to adapt p-cycles to a path-based protection scheme, and p-cycles seem more suited to static traffic than to dynamic traffic. In this paper we show that the key to fast restoration times is not a ring-like topology per se, but rather the ability to pre-cross-connect protection paths. This leads to the concept of a pre-cross-connected trail or PXT, which is a structure that is more flexible than rings and that adapts readily to both path-based and link-based schemes and to both static and dynamic traffic. The PXT protection scheme achieves fast restoration speeds, and our simulations, which have been carefully chosen using ideas from experimental design theory, show that the bandwidth efficiency of the PXT protection scheme is comparable to that of conventional shared mesh protection schemes.

Forbidden subsequences and Chebyshev polynomials

Abstract In (West, Discrete Math. 157 (1996) 363–374) it was shown using transfer matrices that the number | S n (123; 3214)| of permutations avoiding the patterns 123 and 3214 is the Fibonacci number F 2 n (as are also | S n (213; 1234)| and | S n (213; 4123)|). We now find the transfer matrix for | S n (123; r , r − 1,…,2, 1, r − 1)|, | S n (213; 1,2,…, r , r + 1)|, and | S n (213; r + 1, 1, 2,…, r )|, determine its characteristic polynomial in terms of the Chebyshev polynomials, and go on to determine the generating function as a quotient of modified Chebyshev polynomials. This leads to an asymptotic result for each r which collapses to the exact results 2 n when r = 2 and F 2 n when r = 3 and to the Catalan number c n as r → ∞. We observe that our generating function also enumerates certain lattice paths, plane trees, and directed animals, giving hope that these areas of combinatorics can be applied to enumerating permutations with excluded subsequences.

The ring grooming problem

The problem of minimizing the number of bidirectional SONET rings required to support a given traffic demand has been studied by several researchers. Here we study the related ring-grooming problem of minimizing the number of add/drop locations instead of the number of rings; in a number of situations this is a better approximation to the true equipment cost. Our main result is a new lower bound for the case of uniform traffic. This allows us to prove that a certain simple algorithm for uniform traffic is, in fact, a constant-factor approximation algorithm, and it also demonstrates that known lower bounds for the general problem—in particular, the linear programming relaxation—are not within a constant factor of the optimum. We also show that our results for uniform traffic extend readily to the more practically important case of quasi-uniform traffic. Finally, we show that if the number of nodes on the ring is fixed, then ring grooming is solvable in polynomial time; however, whether ring grooming is fixed-parameter tractable is still an open question.

What is a Closed-Form Number?

If a student asks for an antiderivative of exp(x^2), there is a standard reply: the answer is not an elementary function. But if a student asks for a closed-form expression for the real root of x = cos(x), there is no standard reply. We propose a definition of a closed-form expression for a number (as opposed to a *function*) that we hope will become standard. With our definition, the question of whether the root of x = cos(x) has a closed form is, perhaps surprisingly, still open. We show that Schanuels conjecture in transcendental number theory resolves questions like this, and we also sketch some connections with Tarskis problem of the decidability of the first-order theory of the reals with exponentiation. Many (hopefully accessible) open problems are described.

Descents, Quasi-Symmetric Functions, Robinson-Schensted for Posets, and the Chromatic Symmetric Function

We investigate an apparent hodgepodge of topics: a Robinson-Schensted algorithm for (3 + 1)-free posets, Chung and Grahams G-descent expansion of the chromatic polynomial, a quasi-symmetric expansion of the path-cycle symmetric function, and an expansion of Stanleys chromatic symmetric function XG in terms of a new symmetric function basis. We show how the theory of P-partitions (in particular, Stanleys quasi-symmetric function expansion of the chromatic symmetric function XG) unifies them all, subsuming two old results and implying two new ones. Perhaps our most interesting result relates to the still-open problem of finding a Robinson-Schensted algorithm for (3 + 1)-free posets. (Magid has announced a solution but it appears to be incorrect.) We show that such an algorithm ought to “respect descents”, and that the best partial algorithm so far—due to Sundquist, Wagner, and West—respects descents if it avoids a certain induced subposet.

Additive Partitions and Continued Fractions

A set S of positive integers is avoidable if there exists a partition of the positive integers into two disjoint sets such that no two distinct integers from the same set sum to an element of S. Much previous work has focused on proving the avoidability of very special sets of integers. We vastly broaden the class of avoidable sets by establishing a previously unnoticed connection with the elementary theory of continued fractions.