On-Line Reevaluation of Functions

Given a finite set S and a function f : S^n -> S^m, we consider the problem of making a data structure which maintains a value of x in S^n and allows us to efficiently change an arbitrary coordinate of x and efficiently evaluate f_i(x) for arbitrary i. We both examine the problem for specific choices of f and relate the possibility of an efficient solution to general properties of f: expressibility as a formula, space complexity and time complexity

On Data Structures and Asymmetric Communication Complexity

In this paper we consider two party communication complexity when the input sizes of the two players differ significantly, the ``asymmetric'' case. Most of previous work on communication complexity only considers the total number of bits sent, but we study tradeoffs between the number of bits the first player sends and the number of bits the second sends. These types of questions are closely related to the complexity of static data structure problems in the cell probe model. We derive two generally applicable methods of proving lower bounds, and obtain several applications. These applications include new lower bounds for data structures in the cell probe model. Of particular interest is our ``round elimination'' lemma, which is interesting also for the usual symmetric communication case. This lemma generalizes and abstracts in a very clean form the ``round reduction'' techniques used in many previous lower bound proofs

Circuit Depth Relative to a Random Oracle

The study of separation of complexity classes with respect to random oracles was initiated by Bennett and Gill and continued by many other authors. Wilson defined relativized circuit depth and constructed various oracles A for which   P^A ¬ NC^A NC^A_k ¬ NC^A_k+varepsilon, AC^A_k ¬ AC^A_k+varepsilon, AC^A_k ¬ subset= AC^A_k+1-varepsilon, and NC^A_k not subset= AC^A_ k-varepsilon,for all positive rational k and varepsilon, thus separating those classes for which no trivial argument shows inclusion. In this note we show that as a consequence of a single lemma, these separations (or improvements of them) hold with respect to a random oracle A

The Cell Probe Complexity of Succinct Data Structures

In the cell probe model with word size 1 (the bit probe model), a static data structure problem is given by a map $f: {0,1}^n imes {0,1}^m ightarrow {0,1}$, where ${0,1}^n$ is a set of possible data to be stored, ${0,1}^m$ is a set of possible queries (for natural problems, we have $m ll n$) and $f(x,y)$ is the answer to question $y$ about data $x$. A solution is given by a representation $phi: {0,1}^n ightarrow {0,1}^s$ and a query algorithm $q$ so that $q(phi(x), y) = f(x,y)$. The time $t$ of the query algorithm is the number of bits it reads in $phi(x)$. In this paper, we consider the case of {em succinct} representations where $s = n + r$ for some {em redundancy} $r ll n$. For a boolean version of the problem of polynomial evaluation with preprocessing of coefficients, we show a lower bound on the redundancy-query time tradeoff of the form [ (r+1) t geq Omega(n/log n).] In particular, for very small redundancies $r$, we get an almost optimal lower bound stating that the query algorithm has to inspect almost the entire data structure (up to a logarithmic factor). We show similar lower bounds for problems satisfying a certain combinatorial property of a coding theoretic flavor. Previously, no $omega(m)$ lower bounds were known on $t$ in the general model for explicit functions, even for very small redundancies. By restricting our attention to {em systematic} or {em index} structures $phi$ satisfying $phi(x) = x cdot phi^*(x)$ for some map $phi^*$ (where $cdot$ denotes concatenation) we show similar lower bounds on the redundancy-query time tradeoff for the natural data structuring problems of Prefix Sum and Substring Search

Truthful approximations to range voting

We consider the fundamental mechanism design problem of approximate social welfare maximization under general cardinal preferences on a finite number of alternatives and without money. The well-known range voting scheme can be thought of as a non-truthful mechanism for exact social welfare maximization in this setting. With m being the number of alternatives, we exhibit a randomized truthful-in-expectation ordinal mechanism implementing an outcome whose expected social welfare is at least an Omega(m^{-3/4}) fraction of the social welfare of the socially optimal alternative. On the other hand, we show that for sufficiently many agents and any truthful-in-expectation ordinal mechanism, there is a valuation profile where the mechanism achieves at most an O(m^{-{2/3}) fraction of the optimal social welfare in expectation. We get tighter bounds for the natural special case of m = 3, and in that case furthermore obtain separation results concerning the approximation ratios achievable by natural restricted classes of truthful-in-expectation mechanisms. In particular, we show that for m = 3 and a sufficiently large number of agents, the best mechanism that is ordinal as well as mixed-unilateral has an approximation ratio between 0.610 and 0.611, the best ordinal mechanism has an approximation ratio between 0.616 and 0.641, while the best mixed-unilateral mechanism has an approximation ratio bigger than 0.660. In particular, the best mixed-unilateral non-ordinal (i.e., cardinal) mechanism strictly outperforms all ordinal ones, even the non-mixed-unilateral ordinal ones

Derandomizing Arthur-Merlin Games using Hitting Sets

We prove that AM (and hence Graph Nonisomorphism) is in NPif for some epsilon > 0, some language in NE intersection coNE requires nondeterministiccircuits of size 2^(epsilon n). This improves recent results of Arvindand K¨obler and of Klivans and Van Melkebeek who proved the sameconclusion, but under stronger hardness assumptions, namely, eitherthe existence of a language in NE intersection coNE which cannot be approximatedby nondeterministic circuits of size less than 2^(epsilon n) or the existenceof a language in NE intersection coNE which requires oracle circuits of size 2^(epsilon n)with oracle gates for SAT (satisfiability).The previous results on derandomizing AM were based on pseudorandomgenerators. In contrast, our approach is based on a strengtheningof Andreev, Clementi and Rolim's hitting set approach to derandomization.As a spin-off, we show that this approach is strong enoughto give an easy (if the existence of explicit dispersers can be assumedknown) proof of the following implication: For some epsilon > 0, if there isa language in E which requires nondeterministic circuits of size 2^(epsilon n),then P=BPP. This differs from Impagliazzo and Wigderson's theorem"only" by replacing deterministic circuits with nondeterministicones