We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed efficiently stored set. Our algorithms are for the unit-cost word-level RAM with multiplication and extend to give optimal dynamic algorithms. The lower bounds are proved in a much stronger communication game model, but they apply to the cell probe and RAM models and to both static and dynamic predecessor problems.

We address the problem of minimizing the communication involved in the exchange of similar documents. We consider two users, A and B, who hold documents x and y respectively. Neither of the users has any information about the other's document. They exchange messages so that B computes x; it may be required that A compute y as well. Our goal is to design communication protocols with the main objective of minimizing the total number of bits they exchange; other objectives are minimizing the number of rounds and the complexity of internal computations. An important notion which determines the efficiency of the protocols is how one measures the distance between x and y. We consider several metrics for measuring this distance, namely the Hamming metric, the Levenshtein metric (edit distance), and a new LZ metric, which is introduced in this paper. We show how to estimate the distance between x and y using a single message of logarithmic size. For each metric, we present the first communica...

We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed compactly stored set. Our algorithms are for the unit-cost word RAM with multiplication and are extended to give dynamic algorithms. The lower bounds are proved for a large class of problems, including both static and dynamic predecessor problems, in a much stronger communication game model, but they apply to the cell probe and RAM models.

We introduce a novel technique for converting static polynomial space search structures for ordered sets into fullydynamic linear space data structures. Based on this we present optimal bounds for dynamic integer searching, including finger search, and exponentially improved bounds for priority queues.

We consider dictionaries of size n over the finite universe U = and introduce a new technique for their implementation: error correcting codes. The use of such codes makes it possible to replace the use of strong forms of hashing, such as universal hashing, with much weaker forms, such as clustering. We use

In this paper we consider solutions to the dictionary problem on AC RAMs, i.e.

A randomized sorting algorithm is presented, doing as described in the title. 1 Introduction In this paper we consider sorting on a very simple RAM where the only word-operations are addition, shift, and bit-wise boolean operations. Besides these word-operations, we have direct and indirect addressing, jumps, and conditional statements. Such a RAM has been referred to as a Practical RAM [Mil96]. In this paper we show Theorem 1 On a Practical RAM, there is a randomized algorithm sorting n words in O(n log log n) time and linear space. The above algorithm only makes shifts by powers of two, and it only needs O(log n) random words. Our time bound matches that of the current fastest sorting algorithm by Andersson, Hagerup, Raman, and Nilsson [AHNR95]. Their algorithm has two variants: one is deterministic uses space 2 "w , where w is the word length and " is a positive constant. Thus the space is unbounded in terms of n. The other variant is randomized and uses linear space like ou...

. We show that on a RAM with addition, subtraction, bitwise Boolean operations and shifts, but no multiplication, there is a transdichotomous solution to the static dictionary problem using linear space and with query time p log n(log log n) 1+o(1) . On the way, we show that two w-bit words can be multiplied in time (log w) 1+o(1) and that time \Omega (log w) is necessary, and that \Theta(log log w) time is necessary and sufficient for identifying the least significant set bit of a word. 1 Introduction Consider a problem (like sorting or searching) whose instances consists of collections of members of the universe U = f0; 1g w of w-bit bit strings (or numbers between 0 and 2 w \Gamma 1). An increasingly popular theoretical model for studying such problems is the trans-dichotomous model of computation [13, 14, 1, 7, 8, 3, 2, 20, 18, 9, 4, 21, 6], where one assumes a random access machine where each register is capable of holding exactly one element of the universe, i.e. we...

. We consider solving the static dictionary problem with n keys from the universe f0; : : : ; m \Gamma 1g on a RAM with direct and indirect addressing, conditional jump, addition, bitwise Boolean operations, and arbitrary shifts (a Practical RAM). For any ffl ? 0, tries yield constant query time using space m ffl , provided that n = m o(1) . We show that this is essentially optimal: Any scheme with constant query time requires space m ffl for some ffl ? 0, even if n (log m) 2 . 1 Introduction The static dictionary problem is the following: Given a subset S of size n of the universe U = f0; : : : ; m \Gamma 1g, store it as a data structure OE S in the memory of a unit cost random access machine, using few memory registers, each containing O(logm) bits, so that membership queries "Is x 2 S?" can be answered efficiently for any value of x. The set S can be stored as a sorted table using n memory registers. Then queries can be answered using binary search in O(logn) time. Yao...

This chapter reviews randomization algorithms developed in the last few years to solve a wide range of geometric optimization problems. We rst review a number of general techniques, including randomized binary search, randomized linear-programming algorithms, and random sampling. Next, we describe several applications of these and other techniques, including facility location, proximity problems, statistical estimators, nearest neighbor searching, and Euclidean TSP.