New Unconditional Hardness Results for Dynamic and Online Problems

There has been a resurgence of interest in lower bounds whose truth rests on the conjectured hardness of well known computational problems. These conditional lower bounds have become important and popular due to the painfully slow progress on proving strong unconditional lower bounds. Nevertheless, the long term goal is to replace these conditional bounds with unconditional ones. In this paper we make progress in this direction by studying the cell probe complexity of two conjectured to be hard problems of particular importance: matrix-vector multiplication and a version of dynamic set disjointness known as Patrascu's Multiphase Problem. We give improved unconditional lower bounds for these problems as well as introducing new proof techniques of independent interest. These include a technique capable of proving strong threshold lower bounds of the following form: If we insist on having a very fast query time, then the update time has to be slow enough to compute a lookup table with the answer to every possible query. This is the first time a lower bound of this type has been proven

Upper and lower bounds for dynamic data structures on strings

We consider a range of simply stated dynamic data structure problems on strings. An update changes one symbol in the input and a query asks us to compute some function of the pattern of length $m$ and a substring of a longer text. We give both conditional and unconditional lower bounds for variants of exact matching with wildcards, inner product, and Hamming distance computation via a sequence of reductions. As an example, we show that there does not exist an $O(m^{1/2-\varepsilon})$ time algorithm for a large range of these problems unless the online Boolean matrix-vector multiplication conjecture is false. We also provide nearly matching upper bounds for most of the problems we consider.Comment: Accepted at STACS'1

The Unique Path of A. Leon Higginbotham, Jr. - A Voice for Equal Justice through Law

Symposium Honoring Judge A. Leon Higginbotham, Jr

Green's J-order and the rank of tropical matrices

We study Green's J-order and J-equivalence for the semigroup of all n-by-n matrices over the tropical semiring. We give an exact characterisation of the J-order, in terms of morphisms between tropical convex sets. We establish connections between the J-order, isometries of tropical convex sets, and various notions of rank for tropical matrices. We also study the relationship between the relations J and D; Izhakian and Margolis have observed that $D \neq J$ for the semigroup of all 3-by-3 matrices over the tropical semiring with $-\infty$, but in contrast, we show that $D = J$ for all full matrix semigroups over the finitary tropical semiring.Comment: 21 pages, exposition improve

On Second-Quantized Open Superstring Theory

The SO(32) theory, in the limit where it is an open superstring theory, is completely specified in the light-cone gauge as a second-quantized string theory in terms of a ``matrix string'' model. The theory is defined by the neighbourhood of a 1+1 dimensional fixed point theory, characterized by an Abelian gauge theory with type IB Green-Schwarz form. Non-orientability and SO(32) gauge symmetry arise naturally, and the theory effectively constructs an orientifold projection of the (weakly coupled) matrix type IIB theory (also discussed herein). The fixed point theory is a conformal field theory with boundary, defining the free string theory. Interactions involving the interior of open and closed strings are governed by a twist operator in the bulk, while string end-points are created and destroyed by a boundary twist operator.Comment: 20 pages,in harvmac.tex `b' mode; epsf.tex for 12 figure

Differential Calculus on qq-Deformed Light-Cone

We propose the ``short'' version of q-deformed differential calculus on the light-cone using twistor representation. The commutation relations between coordinates and momenta are obtained. The quasi-classical limit introduced gives an exact shape of the off-shell shifting.Comment: 11 pages, Standard LaTeX 2.0

On chains in HH-closed topological pospaces

We study chains in an $H$-closed topological partially ordered space. We give sufficient conditions for a maximal chain $L$ in an $H$-closed topological partially ordered space such that $L$ contains a maximal (minimal) element. Also we give sufficient conditions for a linearly ordered topological partially ordered space to be $H$-closed. We prove that any $H$-closed topological semilattice contains a zero. We show that a linearly ordered $H$-closed topological semilattice is an $H$-closed topological pospace and show that in the general case this is not true. We construct an example an $H$-closed topological pospace with a non-$H$-closed maximal chain and give sufficient conditions that a maximal chain of an $H$-closed topological pospace is an $H$-closed topological pospace.Comment: We have rewritten and substantially expanded the manuscrip

Multiple Realisations of N=1 Vacua in Six Dimensions

A while ago, examples of N=1 vacua in D=6 were constructed as orientifolds of Type IIB string theory compactified on the K3 surface. Among the interesting features of those models was the presence of D5-branes behaving like small instantons, and the appearance of extra tensor multiplets. These are both non-perturbative phenomena from the point of view of heterotic string theory. Although the orientifold models are a natural setting in which to study these non-perturbative Heterotic string phenomena, it is interesting and instructive to explore how such vacua are realised in Heterotic string theory, M-theory and F-theory, and consider the relations between them. In particular, we consider models of M-theory compactified on K3 x S^1/Z_2 with fivebranes present on the interval. There is a family of such models which yields the same spectra as a subfamily of the orientifold models. By further compactifying on T^2 to four dimensions we relate them to Heterotic string spectra. We then use Heterotic/Type IIA duality to deduce the existence of Calabi-Yau 3-folds which should yield the original six dimensional orientifold spectra if we use them to compactify F-theory. Finally, we show in detail how to take a limit of such an F-theory compactification which returns us to the Type IIB orientifold models.Comment: Uses harvmac.tex and epsf.tex, 22 pages (harvmac `b'), 1 figur