Getting Real with Spectraclick to expand

In “Inferring Peptide from Full Spectrum”, we considered an idealized version of the simplified spectrum in which every cut through a given peptide was produced, so that the spectrum possessed all possible b-ions and y-ions cutting the peptide. In reality, not every cut will be produced in a spectrum, which may also contain errors. As a result, it is difficult or impossible to recover an entire peptide from a single spectrum.

In the more practical case of a mass spectrum, where intensity is plotted against ions' mass-charge ratios, inferring the protein is also greatly complicated by the presence of erratic peaks in the spectrum.

Problem

For a weighted alphabet $\mathscr{A}$ and a collection $L$ of positive real numbers, the spectrum graph of $L$ is a digraph constructed in the following way. First, create a node for every real number in $L$. Then, connect a pair of nodes with a directed edge $(u, v)$ if $v > u$ and $v - u$ is equal to the weight of a single symbol in $\mathscr{A}$. We may then label the edge with this symbol.

In this problem, we say that a weighted string $s = s_1 s_2 \cdots s_n$ matches $L$ if there is some increasing sequence of positive real numbers $(w_1, w_2, \ldots, w_{n+1})$ in $L$ such that $w(s_1) = w_2 - w_1$, $w(s_2) = w_3 - w_2$, ..., and $w(s_n) = w_{n+1} - w_{n}$.

Given: A list $L$ (of length at most 100) containing positive real numbers.

Return: The longest protein string that matches the spectrum graph of $L$ (if multiple solutions exist, you may output any one of them). Consult the monoisotopic mass table.

Sample Dataset

3524.8542
3623.5245
3710.9335
3841.974
3929.00603
3970.0326
4026.05879
4057.0646
4083.08025

Sample Output

WMSPG

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How can our question be rephrased in terms of the spectrum graph?