Simple Proofs of Occupancy Tail Bounds

We give short proofs of some occupancy tail bounds using themethod of bounded differences in expected form and the notion ofnegative association

Talagrand’s Inequality in Hereditary Settings

We develop a nicely packaged form of Talagrand's inequality thatcan be applied to prove concentration of measure for functions defined by hereditary properties. We illustrate the framework with several applications from combinatorics and algorithms. We also give an extension of the inequality valid in spaces satisfying a certain negative dependence property and give some applications

Transforming Comparison Model Lower Bounds to the PRAM

This note provides general transformations of lower bounds in Valiant'sparallel comparison decision tree model to lower bounds in the priorityconcurrent-read concurrent-write parallel-random-access-machine model.The proofs rely on standard Ramsey-theoretic arguments that simplifythe structure of the computation by restricting the input domain. Thetransformation of comparison model lower bounds, which are usually easierto obtain, to the parallel-random-access-machine, unifies some knownlower bounds and gives new lower bounds for several problems

On p-Separability

We introduce the notion of p-separability in analogy with the recursion-theoretic notion of recursive separability. The existence of p-inseparable sets in NP is related to structural properties of complexity classes. Sparseness is related to p-separability and structural conditions for the existence of sparse p-inseparable sets NP are given. Using the notion we obtain sets hard for the $\sum_{2}^{0}$ and $\prod_{2}^{0}$ levels of the Kleene Arithmetic Hierarchy. Some independence results are shown to follow

Simple Proofs of Occupancy Tail Bounds

We give short proofs of some occupancy tail bounds using themethod of bounded differences in expected form and the notion ofnegative association

Algorithmic Investigations in P-Adic Fields

This thesis is concerned with algorithmic investigations in p-adically closed fields, of which Hensel's field of p-adic numbers is prototypical. The well known analogies between the field of real numbers and the field of p-adic numbers are supplemented from a computational standpoint. We resolve the complexity of the Decision Problem for Fields in the p-adic case. We work in the new $n$th power formalism for p-adic fields where the correspondence to the reals is most transparent. First we give an alternating exponential time algorithm for deciding linear sentences in the theory of p-adically closed fields. This also translates into a deterministic algorithm running in exponential space or double exponential time. A deterministic quantifier-elimination procedure for the linear fragment running in double exponential time and space is also presented. Next we employ a quantitative version of a Cell Decomposition Lemma due to Denef to give an alternating exponential time decision procedure for the full theory. As usual this also yields a deterministic decision procedure running in double exponential time or in exponential space, and a quantifier elimination procedure running in double exponential time and space. These complexity bounds are demonstrated to be essentially optimal by proving matching lower bounds on the respective problems. We give a simple algorithm to determine all roots among the p-adic integers of a given polynomial equation. This algorithm is a purely symbolic (as opposed to numerical) p-adic version of the classical Newton and Horner iteration methods and has a natural parallel implementation. We also give algorithms for some problems in valued fields and in p-adic semi-algebraic geometry. Finally we give some additional elementary evidence to support the thesis that certain cosets of $n$th powers are the proper p-adic analogues to signs in the real case. This is done by showing that these coset representatives display similar behavior with respect to functions and their derivatives, as do the signs in the real case

Inclusion-Exclusion(3) Implies Inclusion-Exclusion(n)

We consider a natural generalisation of the familiar inclusion-exclusion formula for sets in the setting of ranked lattices. We show that the generalised inclusion-exclusion formula holds in a lattice if and only if the lattice is distributive and the rank function is modular. As a consequence it turns out (perhaps surprisingly) that the inclusion-exclusion formula for three elements implies the inclusion-exclusion formula for an arbitrary number of elements

Transforming Comparison Model Lower Bounds to the PRAM

This note provides general transformations of lower bounds in Valiant's parallel comparison decision tree model to lower bounds in the priority concurrent-read concurrent-write parallel-random-access-machine model. The proofs rely on standard Ramsey--theoretic arguments that simplify the structure of the computation by restricting the input domain. The transformation of comparison model lower bounds, which are usually easier to obtain, to the parallel-random-access-machine, unifies some known lower bounds and gives new lower bounds for several problems

Transforming Comparison Model

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