We give a deterministic polynomial-time method for finding a set cover in a set system (X, ~') of dual VC-dimension d such that the size of our cover is at most a factor of O(d log(dc)) from the optimal size, c. For constant VCdimensional set systems, which are common in computational geometry, our method gives an O(logc) approximation factor. This improves the previous O(logl XI) bound of the greedy method and challenges recent complexity-theoretic lower bounds for set covers (which do not make any assumptions about the VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those arising in threedimensional polytope approximation and two-dimensional disk covering, we can quickly find O(c)-sized covers. IntroductionA set system (X, ~q~) is a set X along with a collection ~q' of subsets of X, which are sometimes called ranges [25]. Such entities have also been called hypergraphs and range spaces in the computational geometry literature (e.g., see [5], [10]-[16], [20], [24], [25], [34]-[36], and [38]-[41]), and they can be used to model a number of interesting computational geometry problems.

Suppose a client, Alice, has outsourced her data to an external storage provider, Bob, because he has capacity for her massive data set, of size n, whereas her private storage is much smaller-say, of size O(n 1/r ), for some constant r > 1. Alice trusts Bob to maintain her data, but she would like to keep its contents private. She can encrypt her data, of course, but she also wishes to keep her access patterns hidden from Bob as well. We describe schemes for the oblivious RAM simulation problem with a small logarithmic or polylogarithmic amortized increase in access times, with a very high probability of success, while keeping the external storage to be of size O(n). To achieve this, our algorithmic contributions include a parallel MapReduce cuckoo-hashing algorithm and an external-memory dataoblivious sorting algorithm.

We present the software architecture and implementation of an efficient data structure for dynamically maintaining an authenticated dictionary. The building blocks of the data structure are skip lists and one-way commutative hash functions. We also present the results of a preliminary experiment on the performance of the data structure. Applications of our work include certificate revocation in public key infrastructure and the publication of data collections on the Internet.

We present a version of the Bloom filter data structure that supports not only the insertion, deletion, and lookup of key-value pairs, but also allows a complete listing of the pairs it contains with high probability, as long the number of keyvalue pairs is below a designed threshold. Our structure allows the number of keyvalue pairs to greatly exceed this threshold during normal operation. Exceeding the threshold simply temporarily prevents content listing and reduces the probability of a successful lookup. If entries are later deleted to return the structure below the threshold, everything again functions appropriately. We also show that simple variations of our structure are robust to certain standard errors, such as the deletion of a key without a corresponding insertion or the insertion of two distinct values for a key. The properties of our structure make it suitable for several applications, including database and networking applications that we highlight.

We give a deterministic polynomial time method for finding a set cover in a set system (X, 7?) of VC-dimension d such that the size of our cover is at most a factor of 0 (d log (de)) from the optimal size, c. For constant VC-dimension set systems, which are common in computational geometry, our method gives an O (log c) approximation factor. This improves the previous @(log IX 1) bound of the greedy method and beats recent complexitytheoretic lower bounds for set covers (which don't make any assumptions about VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those that arise in 3-d polytope approximation and 2-d disc covering, we can quickly find O(c) -sized covers.