[Paper Review] Communication and Streaming Complexity of Approximate Pattern Matching

Djamal Belazzougui|arXiv (Cornell University)|Nov 30, 2015

Algorithms and Data Compression11 references12 citations

TL;DR

This paper presents the first efficient deterministic one-way protocol for document exchange under edit distance, achieving O(n·polylog(n)) encoding/decoding time and O(k² + k log²n) message size. It uses deterministic string sampling and periodicity properties to enable robust reconstruction of a binary string from a compact, error-resilient signature, enabling a new error-correcting code for adversarial edit errors with near-linear time complexity and redundancy O(k³ + k² log²n).

ABSTRACT

Suppose that we have two parties that possess each a binary string. Suppose that the length of the first string (document) is $n$ and that the two strings (documents) have edit distance (minimal number of deletes, inserts and substitutions needed to transform one string into the other) at most $k$. The problem we want to solve is to devise an efficient protocol in which the first party sends a single message that allows the second party to guess the first party's string. In this paper we show an efficient deterministic protocol for this problem. The protocol runs in time $O(n\cdot \mathtt{polylog}(n))$ and has message size $O(k^2+k\log^2n)$ bits. To the best of our knowledge, ours is the first efficient deterministic protocol for this problem, if efficiency is measured in both the message size and the running time. As an immediate application of our new protocol, we show a new error correcting code that is efficient even for large numbers of (adversarial) edit errors.

Motivation & Objective

• To solve the open problem of designing an efficient deterministic one-way protocol for document exchange under edit distance.
• To achieve both low message size and polynomial-time encoding/decoding for arbitrary k.
• To construct a scalable error-correcting code resilient to adversarial edit errors with near-linear time complexity.
• To overcome limitations of prior randomized schemes and prior deterministic codes that are either inefficient or restricted to small k.

Proposed method

• Uses deterministic string sampling via Vishkin's lemma to create compact, unique signatures for substrings.
• Applies periodicity lemmas (Fine-Wilf and related) to identify and eliminate spurious matches in string matching.
• Employs a hierarchical decomposition of the document into blocks, recursively computing signatures based on period length and deterministic samples.
• Uses Reed-Solomon encoding on block signatures to enable error resilience in the message.
• Protects the final message with a (2k+1)-repetition code to tolerate up to k edit errors in the transmitted signature.
• Combines all components into a single-round, one-way protocol where the sender transmits a compact signature, and the receiver reconstructs the original string using periodicity and signature matching.

Experimental results

Research questions

• RQ1 Can a deterministic, one-way document exchange protocol be designed with both low message size and polynomial-time complexity for arbitrary k?
• RQ2 Is it possible to achieve message size close to the information-theoretic lower bound Θ(k log(n/k)) while maintaining polynomial-time decoding?
• RQ3 Can such a protocol be used to construct an efficient error-correcting code for adversarial edit errors?
• RQ4 How can deterministic string signatures replace randomized ones (e.g., Rabin-Karp) in this setting without sacrificing efficiency or correctness?
• RQ5 What is the minimal redundancy required for a deterministic code to correct k adversarial edit errors in time polynomial in n?

Key findings

• The protocol achieves message size O(k² + k log²n) bits, which is the first such deterministic bound with both efficiency and correctness.
• The encoding and decoding time is O(n·polylog(n)), making it efficient for large inputs and arbitrary k.
• The protocol enables a new error-correcting code with redundancy O(k³ + k² log²n) bits, scalable up to k = O(n^{1/3}).
• The scheme is robust against adversarial edit errors, with the receiver able to reconstruct the original string even if the message is corrupted by up to k edits.
• The use of deterministic sampling and periodicity lemmas ensures correctness without reliance on randomness, unlike prior randomized protocols.
• The protocol outperforms previous schemes in message size and time complexity for deterministic approaches, especially for larger k.