No abstract

We study what we call functional monitoring problems. We have k players each tracking their inputs, say player i tracking a multiset Ai(t) up until time t, and communicating with a central coordinator. The coordinator's task is to monitor a given function f computed over the union of the inputs ∪iAi(t), continuously at all times t. The goal is to minimize the number of bits communicated between the players and the coordinator. A simple example is when f is the sum, and the coordinator is required to alert when the sum of a distributed set of values exceeds a given threshold τ . Of interest is the approximate version where the coordinator outputs 1 if f ≥ τ and 0 if f ≤ (1 − )τ . This defines the (k, f, τ, ) distributed, functional monitoring problem.Functional monitoring problems are fundamental in distributed systems, in particular sensor networks, where we must minimize communication; they also connect to problems in communication complexity, communication theory, and signal processing. Yet few formal bounds are known for functional monitoring.We give upper and lower bounds for the (k, f, τ, ) problem for some of the basic f 's. In particular, we study frequency moments (F0, F1, F2). For F0 and F1, we obtain continuously monitoring algorithms with costs almost the same as their one-shot computation algorithms. However, for F2 the monitoring problem seems much harder. We give a carefully constructed multi-round algorithm that uses "sketch summaries" at multiple levels of detail and solves the (k, F2, τ, ) problem with communicationÕ(k 2 / + ( √ k/ ) 3 ). Since frequency moment estimation is central to other problems, our results have immediate applications to histograms, wavelet computations, and others. Our algorithmic techniques are likely to be useful for other functional monitoring problems as well.

Nearest neighbor (NN) search in high dimensional space is an important problem in many applications. Ideally, a practical solution (i) should be implementable in a relational database, and (ii) its query cost should grow sub-linearly with the dataset size, regardless of the data and query distributions. Despite the bulk of NN literature, no solution fulfills both requirements, except locality sensitive hashing (LSH). The existing LSH implementations are either rigorous or adhoc. Rigorous-LSH ensures good quality of query results, but requires expensive space and query cost. Although adhoc-LSH is more efficient, it abandons quality control, i.e., the neighbor it outputs can be arbitrarily bad. As a result, currently no method is able to ensure both quality and efficiency simultaneously in practice.Motivated by this, we propose a new access method called the locality sensitive B-tree (LSB-tree) that enables fast highdimensional NN search with excellent quality. The combination of several LSB-trees leads to a structure called the LSB-forest that ensures the same result quality as rigorous-LSH, but reduces its space and query cost dramatically. The LSB-forest also outperforms adhoc-LSH, even though the latter has no quality guarantee. Besides its appealing theoretical properties, the LSB-tree itself also serves as an effective index that consumes linear space, and supports efficient updates. Our extensive experiments confirm that the LSB-tree is faster than (i) the state of the art of exact NN search by two orders of magnitude, and (ii) the best (linear-space) method of approximate retrieval by an order of magnitude, and at the same time, returns neighbors with much better quality.