332 research outputs found
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
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
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
,
where is a set of possible data to be stored,
is a set of possible queries (for natural problems, we
have ) and is
the answer to question about data .
A solution is given by a
representation and a query algorithm
so that . The time of the query algorithm
is the number of bits it reads in .
In this paper, we consider the case of {em succinct} representations
where for some {em redundancy} .
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 , 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 lower bounds were known on
in the general model for explicit functions, even for very small
redundancies.
By restricting our attention to {em systematic} or {em index}
structures satisfying for some
map (where 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
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
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