Reconstruction of Integers from Pairwise Distances

Given a set of integers, one can easily construct the set of their pairwise distances. We consider the inverse problem: given a set of pairwise distances, find the integer set which realizes the pairwise distance set. This problem arises in a lot of fields in engineering and applied physics, and has confounded researchers for over 60 years. It is one of the few fundamental problems that are neither known to be NP-hard nor solvable by polynomial-time algorithms. Whether unique recovery is possible also remains an open question. In many practical applications where this problem occurs, the integer set is naturally sparse (i.e., the integers are sufficiently spaced), a property which has not been explored. In this work, we exploit the sparse nature of the integer set and develop a polynomial-time algorithm which provably recovers the set of integers (up to linear shift and reversal) from the set of their pairwise distances with arbitrarily high probability if the sparsity is O(n^{1/2-\eps}). Numerical simulations verify the effectiveness of the proposed algorithm.Comment: 14 pages, 4 figures, submitted to ICASSP 201

Molecular solutions for double and partial digest problems in polynomial time

A fundamental problem in computational biology is the construction of physical maps of chromosomes from the hybridization experiments between unique probes and clones of chromosome fragments. Double and partial digest problems are two intractable problems used to construct physical maps of DNA molecules in bioinformatics. Several approaches, including exponential algorithms and heuristic algorithms, have been proposed to tackle these problems. In this paper we present two polynomial time molecular algorithms for both problems. For this reason, a molecular model similar to Adleman and Lipton model is presented. The presented operations are simple and performed in polynomial time. Our algorithms are computationally simulated

Sparse Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms

The problem of signal recovery from its Fourier transform magnitude is of paramount importance in various fields of engineering and has been around for over 100 years. Due to the absence of phase information, some form of additional information is required in order to be able to uniquely identify the signal of interest. In this work, we focus our attention on discrete-time sparse signals (of length n). We first show that, if the DFT dimension is greater than or equal to 2n, then almost all signals with aperiodic support can be uniquely identified by their Fourier transform magnitude (up to time-shift, conjugate-flip and global phase). Then, we develop an efficient Two-stage Sparse Phase Retrieval algorithm (TSPR), which involves: (i) identifying the support, i.e., the locations of the non-zero components, of the signal using a combinatorial algorithm (ii) identifying the signal values in the support using a convex algorithm. We show that TSPR can provably recover most O(n^(1/2-ϵ)-sparse signals (up to a timeshift, conjugate-flip and global phase). We also show that, for most O(n^(1/4-ϵ)-sparse signals, the recovery is robust in the presence of measurement noise. These recovery guarantees are asymptotic in nature. Numerical experiments complement our theoretical analysis and verify the effectiveness of TSPR

The Restriction Mapping Problem Revisited

In computational molecular biology, the aim of restriction mapping is to locate the restriction sites of a given enzyme on a DNA molecule. Double digest and partial digest are two well-studied techniques for restriction mapping. While double digest is NP-complete, there is no known polynomial algorithm for partial digest. Another disadvantage of the above techniques is that there can be multiple solutions for reconstruction. In thi