Reconstructing Numbers from Pairwise Function Values

Chen, Shiteng; Huang, Zhiyi; Kannan, S.

doi:10.1007/978-3-642-10631-6_16

Cited by 5 publications

(3 citation statements)

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“…Several variation of the problem have been recently considered, e.g. [14,15] and references therein. A related problem of characterizing strings that have the same set of compositions, instead of multisets, is considered in [16].…”

Section: Relation To Other Workmentioning

confidence: 99%

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String Reconstruction from Substring Compositions

Acharya¹,

Das

Milenković³

et al. 2015

SIAM J. Discrete Math.

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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 ≥ 4 in optimal near-quadratic time. * UCSD.

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Section: Relation To Other Workmentioning

confidence: 99%

“…String reconstruction from substring multisets is a combinatorial simplification of the turnpike problem [15,17]. To see that, convert every string-reconstruction instance to a turnpike problem whose solution implies the original string.…”

Section: Relation To Other Workmentioning

confidence: 99%

String Reconstruction from Substring Compositions

Acharya¹,

Das

Milenković³

et al. 2015

SIAM J. Discrete Math.

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“…When one knows the whole (unlabeled) set of pairwise products, various tractable methods have been proposed to perfectly recover all |ai|'s (e.g. [27]). Here, we employ a simple sorting algorithm as presented in Algorithm 1.…”

Section: Step 1: Recovering Unlabeled Sets Of Correlation Coefficient...mentioning

confidence: 99%

An algorithm for exact super-resolution and phase retrieval

Chen

Eldar

Goldsmith

2014

2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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We explore a fundamental problem of super-resolving a signal of interest from a few measurements of its low-pass magnitudes. We propose a 2-stage tractable algorithm that, in the absence of noise, admits perfect super-resolution of an r-sparse signal from 2r 2 − 2r + 2 low-pass magnitude measurements. The spike locations of the signal can assume any value over a continuous disk, without increasing the required sample size. The proposed algorithm first employs a conventional super-resolution algorithm (e.g. the matrix pencil approach) to recover unlabeled sets of signal correlation coefficients, and then applies a simple sorting algorithm to disentangle and retrieve the true parameters in a deterministic manner. Our approach can be adapted to multi-dimensional spike models and random Fourier sampling by replacing its first step with other harmonic retrieval algorithms.

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