Abstract. With the proliferation of database views and curated databases,

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An -appro ximate quan tile summary of a sequence of N elements is a data structure that can answer quantile queries about the sequence to within a precision of N.We presen t a new online algorithm for computing -appro ximate quantile summaries of very large data sequences. The algorithm has a worst-case space requirement o f O 1 log N. This improves upon the previous best result ofO 1 log 2 N. Moreover, in con trast to earlier deterministic algorithms, our algorithm does not require a priori knowledge of the length of the input sequence.Finally, the actual space bounds obtained on experimental data are signi cantly better than the worst case guarantees of our algorithm as well as the observed space requirements of earlier algorithms.

Abstract. The multiple knapsack problem (MKP) is a natural and well-known generalization of the single knapsack problem and is defined as follows. We are given a set of n items and m bins (knapsacks) such that each item i has a profit p(i) and a size s(i), and each bin j has a capacity c(j). The goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins. MKP is a special case of the generalized assignment problem (GAP) where the profit and the size of an item can vary based on the specific bin that it is assigned to. GAP is APX-hard and a 2-approximation, for it is implicit in the work of Shmoys and Tardos [Math. Program. A, 62 (1993), pp. 461-474], and thus far, this was also the best known approximation for MKP. The main result of this paper is a polynomial time approximation scheme (PTAS) for MKP.Apart from its inherent theoretical interest as a common generalization of the well-studied knapsack and bin packing problems, it appears to be the strongest special case of GAP that is not APX-hard. We substantiate this by showing that slight generalizations of MKP are APX-hard. Thus our results help demarcate the boundary at which instances of GAP become APX-hard. An interesting aspect of our approach is a PTAS-preserving reduction from an arbitrary instance of MKP to an instance with O(log n) distinct sizes and profits.