16 research outputs found
Deterministic and Probabilistic Binary Search in Graphs
We consider the following natural generalization of Binary Search: in a given
undirected, positively weighted graph, one vertex is a target. The algorithm's
task is to identify the target by adaptively querying vertices. In response to
querying a node , the algorithm learns either that is the target, or is
given an edge out of that lies on a shortest path from to the target.
We study this problem in a general noisy model in which each query
independently receives a correct answer with probability (a
known constant), and an (adversarial) incorrect one with probability .
Our main positive result is that when (i.e., all answers are
correct), queries are always sufficient. For general , we give an
(almost information-theoretically optimal) algorithm that uses, in expectation,
no more than queries, and identifies the target correctly with probability at
leas . Here, denotes the
entropy. The first bound is achieved by the algorithm that iteratively queries
a 1-median of the nodes not ruled out yet; the second bound by careful repeated
invocations of a multiplicative weights algorithm.
Even for , we show several hardness results for the problem of
determining whether a target can be found using queries. Our upper bound of
implies a quasipolynomial-time algorithm for undirected connected
graphs; we show that this is best-possible under the Strong Exponential Time
Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs
with non-uniform node querying costs, the problem is PSPACE-complete. For a
semi-adaptive version, in which one may query nodes each in rounds, we
show membership in in the polynomial hierarchy, and hardness
for
Finding the maximum and minimum
AbstractWe consider the problem of finding the maximum out of a list of n ordered items with binary comparisons where the pth fraction of the answers may be false. It is shown that the maximum can be determined iff p < 12 and that a successful strategy needs Θ(11−p)n questions. A few similar problems are also discussed, including the problem of finding the maximum and minimum simultaneously with lies and in the nuts and bolts model
Result-Sensitive Binary Search with Noisy Information
We describe new algorithms for the predecessor problem in the Noisy Comparison Model. In this problem, given a sorted list L of n (distinct) elements and a query q, we seek the predecessor of q in L: denoted by u, the largest element less than or equal to q. In the Noisy Comparison Model, the result of a comparison between two elements is non-deterministic. Moreover, multiple comparisons of the same pair of elements might have different results: each is generated independently, and is correct with probability p > 1/2. Given an overall error tolerance Q, the cost of an algorithm is measured by the total number of noisy comparisons; these must guarantee the predecessor is returned with probability at least 1 - Q. Feige et al. showed that predecessor queries can be answered by a modified binary search with Theta(log (n/Q)) noisy comparisons.
We design result-sensitive algorithms for answering predecessor queries. The query cost is related to the index, k, of the predecessor u in L. Our first algorithm answers predecessor queries with O(log ((log^{*(c)} n)/Q) + log (k/Q)) noisy comparisons, for an arbitrarily large constant c. The function log^{*(c)} n iterates c times the iterated-logarithm function, log^* n. Our second algorithm is a genuinely result-sensitive algorithm whose expected query cost is bounded by O(log (k/Q)), and is guaranteed to terminate after at most O(log((log n)/Q)) noisy comparisons.
Our results strictly improve the state-of-the-art bounds when k is in omega(1) intersected with o(n^epsilon), where epsilon > 0 is some constant. Moreover, we show that our result-sensitive algorithms immediately improve not only predecessor-query algorithms, but also binary-search-like algorithms for solving key applications
Robust and Adaptive Search
Binary search finds a given element in a sorted array with an optimal number of log n queries. However, binary search fails even when the array is only slightly disordered or access to its elements is subject to errors. We study the worst-case query complexity of search algorithms that are robust to imprecise queries and that adapt to perturbations of the order of the elements. We give (almost) tight results for various parameters that quantify query errors and that measure array disorder. In particular, we exhibit settings where query complexities of log n + ck, (1+epsilon) log n + ck, and sqrt(cnk)+o(nk) are best-possible for parameter value k, any epsilon > 0, and constant c
Noisy Sorting Without Searching: Data Oblivious Sorting with Comparison Errors
We provide and study several algorithms for sorting an array of n comparable distinct elements subject to probabilistic comparison errors. In this model, the comparison of two elements returns the wrong answer according to a fixed probability, p_e < 1/2, and otherwise returns the correct answer. The dislocation of an element is the distance between its position in a given (current or output) array and its position in a sorted array. There are various algorithms that can be utilized for sorting or near-sorting elements subject to probabilistic comparison errors, but these algorithms are not data oblivious because they all make heavy use of noisy binary searching. In this paper, we provide new methods for sorting with comparison errors that are data oblivious while avoiding the use of noisy binary search methods. In addition, we experimentally compare our algorithms and other sorting algorithms
Noisy Computing of the and Functions
We consider the problem of computing a function of variables using noisy
queries, where each query is incorrect with some fixed and known probability . Specifically, we consider the computation of the
function of bits (where queries correspond to noisy readings of the bits)
and the function of real numbers (where queries correspond
to noisy pairwise comparisons). We show that an expected number of queries of
is
both sufficient and necessary to compute both functions with a vanishing error
probability , where denotes the
Kullback-Leibler divergence between and
distributions. Compared to previous work, our results tighten the dependence on
in both the upper and lower bounds for the two functions