[Paper Review] Efficiently decoding strings from their shingles

Aryeh Kontorovich, Ari Trachtenberg|arXiv (Cornell University)|Apr 15, 2012

Algorithms and Data Compression18 references3 citations

TL;DR

This paper presents a streaming, linear-time algorithm that determines whether a string can be uniquely reconstructed from its bigrams (2-character shingles), achieving O(|Σ|) space and Θ(n) time complexity. The method leverages a novel graph-theoretic insight to enable efficient, online decoding, and extends to a rateless protocol for string reconciliation with minimal communication overhead.

ABSTRACT

Determining whether an unordered collection of overlapping substrings (called shingles) can be uniquely decoded into a consistent string is a problem that lies within the foundation of a broad assortment of disciplines ranging from networking and information theory through cryptography and even genetic engineering and linguistics. We present three perspectives on this problem: a graph theoretic framework due to Pevzner, an automata theoretic approach from our previous work, and a new insight that yields a time-optimal streaming algorithm for determining whether a string of $n$ characters over the alphabet $Σ$ can be uniquely decoded from its two-character shingles. Our algorithm achieves an overall time complexity $Θ(n)$ and space complexity $O(|Σ|)$. As an application, we demonstrate how this algorithm can be extended to larger shingles for efficient string reconciliation.

Motivation & Objective

• To develop an efficient, online algorithm for testing whether a string is uniquely decodable from its bigrams.
• To reduce the time and space complexity of existing algorithms for unique string decoding.
• To extend the algorithm to a practical, rateless protocol for string reconciliation in distributed systems.
• To provide theoretical bounds on the number of shingle merges and communication complexity under random string models.

Proposed method

• The algorithm uses a graph-theoretic framework based on de Bruijn graphs, where each bigram corresponds to a directed edge.
• It constructs a modified de Bruijn graph from the shingles and checks for unique decodability via Eulerian cycle existence.
• The method operates in a streaming fashion, processing characters sequentially with constant pre-processing time.
• A key innovation is the use of a linear-time algorithm to detect whether the shingle collection admits a unique Eulerian path.
• The approach is extended to the α-edits problem by using a two-phase protocol: set reconciliation followed by shingle merge identification.
• A rateless encoding scheme combines reconciliation data and merge indices, enabling one-way, adaptive communication for arbitrary edit distances.

Experimental results

Research questions

• RQ1 Can a string be uniquely decoded from its bigrams in linear time and constant space per alphabet character?
• RQ2 What is the minimal communication cost for reconciling two strings differing in α edits using shingle-based methods?
• RQ3 How can the number of shingle merge operations be minimized in practice, especially under random string assumptions?
• RQ4 Can the algorithm be generalized to test for multiple possible decodings beyond uniqueness?
• RQ5 What is the optimal shingle size l that minimizes communication while ensuring high probability of unique decodability?

Key findings

• The proposed algorithm achieves Θ(n) time complexity and O(|Σ|) space complexity, significantly improving upon the prior O(n|Σ|³) time and Θ(|Σ|³) space bound.
• For random strings with i.i.d. bits, shingle size l = O(log n) ensures high probability of unique decodability, avoiding costly merge operations.
• Communication complexity of the reconciliation protocol is O(α log² n) for large n when l is chosen optimally, under random string assumptions.
• The algorithm enables a one-way, rateless streaming protocol for string reconciliation, adaptable to arbitrary edit distances.
• The number of shingle merges is minimized when l is chosen such that each node in the de Bruijn graph has a single outgoing edge, which occurs when l ≤ n + 1 + W(−ln(p)p⁻ⁿ)/ln p.
• The method provides a practical and efficient solution for applications in DNA sequencing, cryptography, and network reconciliation.