Crossing sequence

Abstract: In this paper we consider the time and the crossing sequence complexities of one-tape off-line Turing machines. We show that the running time of each nondeterministic machine accepting a nonregular language must grow at least as n\log n, in the case all accepting computations are considered (accept measure). We also prove that the maximal length of the crossing sequences used in accepting computations must grow at least as \log n. On the other hand, it is known that if the time is measured considering, for each accepted string, only the faster accepting computation (weak measure), then there exist nonregular languages accepted in linear time. We prove that under this measure, each accepting computation should exhibit a crossing sequence of length at least \log\log n. We also present efficient implementations of algorithms accepting some unary nonregular languages.

Abstract: Let ck = crk (G) denote the minimum number of edge crossings when a graph G is drawn on an orientable surface of genus k. The (orientable) crossing sequence co, c1,c2…encodes the trade-off between adding handles and decreasing crossings. We focus on sequences of the type co > c1 > c2 = 0; equivalently, we study the planar and toroidal crossing number of doubly-toroidal graphs. For every e > 0 we construct graphs whose orientable crossing sequence satisfies c1-co > 5-6-e. In other words, we construct graphs where the addition of one handle can save roughly 1-6th of the crossings, but the addition of a second handle can save five times more crossings. We similarly define the non-orientable crossing sequence ~0,~1,~2, … for drawings on non-orientable surfaces. We show that for every ~0 >~1 > 0 there exists a graph with non-orientable crossing sequence ~0, ~1, 0. We conjecture that every strictly-decreasing sequence of non-negative integers can be both an orientable crossing sequence and a non-orientable crossing sequence (with different graphs). © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 230243, 2001

Abstract: We present four results on the complexity of VLSI computations: a) We further justify the Boolean circuit model [Vu, Sa, LS] by showing that it is able to model multi-directional VLSI devices (eg pass transistors, pre-charged bus drivers) b) We prove a general cutting theorem for compact regions in R^{d} (d\geq2) that allows us to drop the convexity assumption in lower bound proofs based on the crossing sequence argument c) We exhibit an \Omega(n^{1/3}) asymptotically tight lower bound on the area of strongly where-oblivious chips for transitive functions d) We prove a lower bound on the switching energy needed for computing transitive functions