[Paper Review] Cell-probe bounds for online edit distance and other pattern matching problems

Raphaël Clifford, Markus Jalsenius+1 more|arXiv (Cornell University)|Jan 4, 2015

Algorithms and Data Compression18 references8 citations

TL;DR

This paper establishes the first non-trivial cell-probe lower bounds for online pattern matching problems, including edit distance, Hamming distance, convolution, and longest common subsequence, using a novel encoding scheme and a carefully designed input distribution. It proves an Ω(δ log n / (w + log log n)) lower bound for Hamming distance and convolution, and an Ω(√(log n)/(log log n)^{3/2}) lower bound for online edit distance and LCS in the bit-probe model, while also showing an O((log²n)/w) upper bound for edit distance, revealing a significant gap between cell-probe and RAM complexity.

ABSTRACT

We give cell-probe bounds for the computation of edit distance, Hamming distance, convolution and longest common subsequence in a stream. In this model, a fixed string of n symbols is given and one δ-bit symbol arrives at a time in a stream. After each symbol arrives, the distance between the fixed string and a suffix of most recent symbols of the stream is reported. The cell-probe model is perhaps the strongest model of computation for showing data structure lower bounds, subsuming in particular the popular word-RAM model.• We first give an Ω((δ log n)/(w+log log n)) lower bound for the time to give each output for both online Hamming distance and convolution, where w is the word size. This bound relies on a new encoding scheme and for the first time holds even when w is as small as a single bit.• We then consider the online edit distance and longest common subsequence problems in the bit-probe model (w = 1) with a constant sized input alphabet. We give a lower bound of Ω([EQUATION]log n/(log log n)3/2) which applies for both problems. This second set of results relies both on our new encoding scheme as well as a carefully constructed hard input distribution.• Finally, for the online edit distance problem we show that there is an O((log2n)/w) upper bound in the cell-probe model. This bound gives a contrast to our new lower bound and also establishes an exponential gap between the known cell-probe and RAM model complexities.

Motivation & Objective

• To establish strong computational lower bounds for online pattern matching problems in the cell-probe model, which is the strongest known model for data structure lower bounds.
• To analyze the time complexity of computing edit distance, Hamming distance, convolution, and longest common subsequence in an online streaming setting with incremental symbol arrivals.
• To develop a new encoding scheme that enables non-trivial lower bounds even when the word size w is as small as 1 bit.
• To construct a hard input distribution that amplifies the difficulty of online edit distance and LCS problems in the bit-probe model.
• To contrast the cell-probe lower bounds with known upper bounds in the RAM model, revealing a significant gap in complexity.

Proposed method

• Introduces a novel encoding scheme that enables deriving lower bounds in the cell-probe model, even for minimal word sizes (w = 1).
• Applies the encoding scheme to derive an Ω(δ log n / (w + log log n)) lower bound for online Hamming distance and convolution, where δ is the symbol size and w is the word size.
• Extends the encoding technique with a carefully constructed hard input distribution to prove a lower bound of Ω(√(log n)/(log log n)^{3/2}) for online edit distance and longest common subsequence in the bit-probe model (w = 1).
• Uses the cell-probe model to prove an O((log²n)/w) upper bound for online edit distance, demonstrating a tight upper bound in this model.
• Compares the derived lower bounds with the upper bound to expose a super-polynomial gap between cell-probe and RAM model complexities.
• Employs information-theoretic and combinatorial arguments to establish the hardness of online pattern matching under minimal assumptions.

Experimental results

Research questions

• RQ1 What is the minimum time required to compute online Hamming distance and convolution in the cell-probe model, and how does it scale with input size and word size?
• RQ2 Can non-trivial lower bounds be established for online edit distance and longest common subsequence in the bit-probe model (w = 1) with a constant alphabet?
• RQ3 How does the new encoding scheme contribute to proving lower bounds in the cell-probe model, especially when w is small?
• RQ4 What is the relationship between the cell-probe complexity and the RAM model complexity for online edit distance?
• RQ5 Can a hard input distribution be constructed to amplify the difficulty of online edit distance and LCS in the bit-probe model?

Key findings

• An Ω(δ log n / (w + log log n)) lower bound is established for online Hamming distance and convolution in the cell-probe model, valid even when w = 1.
• A lower bound of Ω(√(log n)/(log log n)^{3/2}) is proven for online edit distance and longest common subsequence in the bit-probe model with a constant alphabet.
• The new encoding scheme enables non-trivial lower bounds in the cell-probe model even for minimal word sizes, marking a significant technical advance.
• An O((log²n)/w) upper bound is shown for online edit distance in the cell-probe model, contrasting with the lower bound and revealing a super-polynomial gap.
• The results demonstrate an exponential gap between known cell-probe and RAM model complexities for online edit distance, highlighting the limitations of the RAM model in capturing true hardness.
• The combination of the encoding scheme and a hard input distribution enables the first non-trivial lower bounds for LCS and edit distance in the bit-probe model.