Optimal Quantile Approximation in Streams (1603.05346v2)

Abstract: This paper resolves one of the longest standing basic problems in the streaming computational model. Namely, optimal construction of quantile sketches. An $\varepsilon$ approximate quantile sketch receives a stream of items $x_1,\ldots,x_n$ and allows one to approximate the rank of any query up to additive error $\varepsilon n$ with probability at least $1-\delta$. The rank of a query $x$ is the number of stream items such that $x_i \le x$. The minimal sketch size required for this task is trivially at least $1/\varepsilon$. Felber and Ostrovsky obtain a $O((1/\varepsilon)\log(1/\varepsilon))$ space sketch for a fixed $\delta$. To date, no better upper or lower bounds were known even for randomly permuted streams or for approximating a specific quantile, e.g.,\ the median. This paper obtains an $O((1/\varepsilon)\log \log (1/\delta))$ space sketch and a matching lower bound. This resolves the open problem and proves a qualitative gap between randomized and deterministic quantile sketching. One of our contributions is a novel representation and modification of the widely used merge-and-reduce construction. This subtle modification allows for an analysis which is both tight and extremely simple. Similar techniques should be useful for improving other sketching objectives and geometric coreset constructions.

• The paper establishes both an improved upper bound and a matching lower bound by achieving O((1/ε)log log(1/δ)) space for randomized quantile sketches.
• It introduces a refined merge-and-reduce algorithm that efficiently reduces memory usage in processing data streams.
• The results highlight the distinct advantages of randomized over deterministic methods, motivating further research in streaming algorithms.

Optimal Quantile Approximation in Streams

The paper "Optimal Quantile Approximation in Streams" addresses the computational challenge of efficiently constructing quantile sketches in the streaming model. A quantile sketch is a data structure that estimates the rank of an element within a stream with an additive error of $\varepsilon n$, where $n$ is the number of observed elements. This paper makes a significant contribution by establishing both upper and lower bounds for randomized quantile sketching, and introducing a novel merge-and-reduce approach that optimizes space complexity.

Key Contributions

The paper successfully resolves the problem by deriving an $O((1/\varepsilon)\log \log (1/\delta))$ space sketch for estimating quantiles with a failure probability $\delta$, and proving a matching lower bound. This advances the previous best result from Felber and Ostrovsky, who achieved an $O((1/\varepsilon)\log(1/\varepsilon))$ space complexity. The result demonstrates a qualitative gap between the capacities of deterministic and randomized algorithms for this problem class.

A pivotal feature of the proposed methodology is a refined variant of the merge-and-reduce algorithm. This modification allows for a tightly bound analysis that remains straightforward, setting the groundwork for potential improvements in other sketching functions and geometric coreset constructions.

Implications and Extensions

The implications of achieving this optimized sketch size are substantial in both theoretical and practical contexts. Practically, the enhanced space efficiency allows the processing of data streams with reduced memory usage, which is essential for applications in systems with constrained resources such as sensors and mobile computing environments.

Theoretically, the result stimulates further exploration into the advantages of randomized over deterministic approaches for similar streaming problems. It highlights areas where randomness offers a clear space advantage, which could influence the design of future algorithms in data stream management and beyond.

Future Developments

The research opens several avenues for future investigation. One potential direction is exploring the transferability of the merge-and-reduce technique modifications to other summary statistics in data streams, like heavy hitters or frequency moments. Moreover, further research could focus on improving other streaming algorithms or functional approximations beyond quantiles, applying the lessons learned from the space versus error trade-offs elucidated here.

Additionally, this paper paves the way for exploring enhanced data structures that balance mergeability with optimal space complexity in randomized settings. The interplay between mergeability and space efficiency remains a fertile ground for innovation, particularly in distributed systems where data is naturally partitioned across nodes.

In summary, "Optimal Quantile Approximation in Streams" stands as a thorough exploration of the quantile sketching problem, providing both a practical solution with optimal space efficiency and a theoretical expansion of understanding how randomness can be leveraged in streaming algorithms. Its findings have the potential to influence a wide array of applications and future research endeavors in the vast landscape of data stream processing.