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Tight Multi-Pass Streaming Lower Bounds for Fundamental Problems


Core Concepts
Any k-pass streaming algorithm that solves the coin problem or distinguishes the needle problem distributions requires Ω(log n/k) bits of memory.
Abstract
The paper introduces a new notion of multi-pass information complexity (MIC) and uses it to prove tight lower bounds for fundamental streaming problems that require multiple passes over the input. For the coin problem, where the goal is to compute the majority of a stream of n i.i.d. uniform bits, the paper shows that any k-pass streaming algorithm requires Ω(log n/k) bits of memory to solve this problem with high probability. This significantly extends the previous Ω(log n) lower bound for single-pass algorithms. For the needle problem, where the goal is to distinguish between a stream of n i.i.d. uniform samples and a stream where each item independently equals a randomly chosen "needle" with probability p, the paper shows a tight multi-pass lower bound of kps^2n = Ω(1), where s is the space used by the algorithm. This resolves an open question and improves upon the previous Ω(1/p^2n log n) lower bound. The paper also presents applications of these multi-pass lower bounds to problems like approximate counting in strict turnstile streams, multi-ℓp-estimation, ℓ2-point query, ℓ2-heavy hitters, and sparse recovery in compressed sensing.
Stats
Any k-pass streaming algorithm that computes the majority of a stream of n i.i.d. uniform bits with probability at least 0.999 requires Ω(log n/k) bits of memory. Any k-pass streaming algorithm that distinguishes the uniform and needle distributions with high probability, where p is the needle probability, n is the stream length, and s is the space, satisfies kps^2n = Ω(1), provided the domain size t = Ω(n^2).
Quotes
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Deeper Inquiries

How can the multi-pass information complexity measure be generalized or adapted to other streaming problems beyond the coin and needle problems

The multi-pass information complexity measure introduced in the context can be generalized and adapted to other streaming problems beyond the coin and needle problems by considering the underlying distribution of the input data stream and the memory states of the algorithm. This measure captures the residual independence in the streaming data even after multiple passes, allowing for a unified approach to obtaining lower bounds for various streaming problems. To adapt this measure to other streaming problems, one would need to define the relevant memory states for the algorithm, the input distribution, and the information shared between the memory states and the input data at each time step. By carefully considering the information complexity framework and how it accounts for dependencies and interactions in the streaming process, one can apply similar principles to analyze and derive lower bounds for different streaming problems. For example, in problems related to frequency estimation, heavy hitters, or point queries in data streams, one could define the appropriate memory states and information shared between the algorithm's memory and the input data. By extending the multi-pass information complexity measure to these scenarios, one can analyze the memory requirements and information processing capabilities needed to solve these problems accurately in a streaming setting.

Can the techniques used to prove the lower bounds be applied to obtain tight multi-pass lower bounds for other fundamental streaming problems

The techniques used to prove the lower bounds for the coin and needle problems can indeed be applied to obtain tight multi-pass lower bounds for other fundamental streaming problems. The key lies in understanding the information complexity framework, the trade-offs between memory usage and information processing, and the simulation techniques used to establish lower bounds for multi-pass algorithms. By applying similar simulation techniques, round elimination strategies, and information complexity analysis to different streaming problems, one can uncover the inherent complexities and memory requirements for solving these problems accurately in a streaming environment. The ability to embed hard instances for lower-pass algorithms within the input distribution obtained by conditioning on the output of higher-pass algorithms allows for a systematic approach to deriving tight lower bounds for a variety of streaming problems. Furthermore, by generalizing the concepts and methodologies used in proving lower bounds for the coin and needle problems, researchers can extend these techniques to address a wide range of streaming problems, providing valuable insights into the memory complexity and computational challenges of streaming algorithms in diverse applications.

What are the implications of the tight multi-pass lower bounds for the practical design of streaming algorithms in real-world applications

The tight multi-pass lower bounds obtained for fundamental streaming problems have significant implications for the practical design of streaming algorithms in real-world applications. These lower bounds provide insights into the minimum memory requirements and computational complexity needed to solve these problems accurately in a streaming setting, guiding the development of efficient and effective streaming algorithms. By understanding the information complexity and memory constraints highlighted by the lower bounds, algorithm designers can optimize their approaches to ensure that streaming algorithms are capable of handling large-scale data streams with limited memory resources. The lower bounds also serve as benchmarks for evaluating the performance of streaming algorithms, helping researchers and practitioners assess the efficiency and scalability of their solutions. In practical terms, the tight multi-pass lower bounds can inform the development of streaming algorithms for various applications such as data analytics, real-time processing, and machine learning. By incorporating the insights gained from these lower bounds, algorithm designers can enhance the robustness, accuracy, and efficiency of streaming algorithms in real-world scenarios, ultimately improving the quality and reliability of data processing in streaming environments.
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