[Paper Review] String Reconstruction from Substring Compositions

Jayadev Acharya, Hirakendu Das+3 more|arXiv (Cornell University)|Mar 10, 2014

Algorithms and Data Compression19 references6 citations

TL;DR

This paper studies the problem of reconstructing a binary string from the multiset of its substring compositions (Parikh vectors), showing that all strings of length 7, one less than a prime, or one less than twice a prime can be uniquely reconstructed up to reversal. Using algebraic techniques based on bivariate polynomial factorization, the authors characterize equicomposable strings and establish tight bounds on the number of confusable strings, resolving a combinatorial simplification of the turnpike problem.

ABSTRACT

Motivated by mass-spectrometry protein sequencing, we consider a simply-stated problem of reconstructing a string from the multiset of its substring compositions. We show that all strings of length 7, one less than a prime, or one less than twice a prime, can be reconstructed uniquely up to reversal. For all other lengths we show that reconstruction is not always possible and provide sometimes-tight bounds on the largest number of strings with given substring compositions. The lower bounds are derived by combinatorial arguments and the upper bounds by algebraic considerations that precisely characterize the set of strings with the same substring compositions in terms of the factorization of bivariate polynomials. The problem can be viewed as a combinatorial simplification of the turnpike problem, and its solution may shed light on this long-standing problem as well. Using well known results on transience of multi-dimensional random walks, we also provide a reconstruction algorithm that reconstructs random strings over alphabets of size $\ge4$ in optimal near-quadratic time.

Motivation & Objective

• To investigate the conditions under which a string can be uniquely reconstructed from the multiset of its substring compositions.
• To characterize when two strings are equicomposable (i.e., have the same multiset of substring compositions).
• To determine the maximum number of strings that can share the same substring composition multiset, especially for lengths not covered by prior constructions.
• To develop efficient algorithms for reconstructing random strings using combinatorial and probabilistic methods.
• To explore connections between this problem and the long-standing turnpike problem in combinatorics.

Proposed method

• Represent strings as bivariate generating polynomials to model substring compositions algebraically.
• Use polynomial factorization to characterize equicomposable strings, showing they correspond to factorizations of the same polynomial.
• Apply cyclotomic polynomials and algebraic number theory to derive upper bounds on the number of confusable strings.
• Use combinatorial constructions involving interleaving strings to generate lower bounds on the number of equicomposable strings.
• Leverage results on transience of multi-dimensional random walks to design a near-optimal reconstruction algorithm for random strings over alphabets of size ≥4.
• Reduce the string reconstruction problem to a turnpike-like problem with O(n²) total length, enabling polynomial-time solutions.

Experimental results

Research questions

• RQ1 For which string lengths is unique reconstruction from substring compositions always possible, up to reversal?
• RQ2 What is the maximum number of distinct strings that can share the same multiset of substring compositions?
• RQ3 Can all confusable strings be fully characterized using algebraic methods such as polynomial factorization?
• RQ4 How does the structure of the generating polynomial relate to the combinatorial properties of the string?
• RQ5 Can efficient reconstruction algorithms be designed that exploit the algebraic structure of the problem?

Key findings

• All strings of length 7, one less than a prime, or one less than twice a prime can be uniquely reconstructed up to reversal.
• The maximum number of equicomposable strings is tightly bounded using cyclotomic polynomials, with the lower bound being tight when the length is one less than a prime power or twice a prime power.
• A construction using interleaved strings yields (n+1)log₃2 mutually equicomposable n-bit strings.
• For random strings over alphabets of size ≥4, a backtracking algorithm reconstructs the correct string (up to reversal) in near-quadratic time with high probability.
• For alphabet size 3, the expected length of the solution set is bounded by O(log n), and the algorithm runs in time O(n^(0.6/δ)) with probability ≥1−δ.
• The problem is a combinatorial simplification of the turnpike problem, and its solution provides structural insights into the original problem.