一种从子串组合重构字符串的新代数方法
A New Algebraic Approach for String Reconstruction from Substring Compositions
AI总结：
针对子串组合多重集下的二进制字符串重构问题，提出基于二元多项式代数性质的新算法，在特定条件下免回溯，将复杂度降至 O(n^2) 并构造更大重构码。
摘要：
我们研究了根据二进制字符串的子串组合多重集（称为子串组合多重集）重构该二进制字符串的问题，该问题最初由 Acharya 等人提出并研究。我们提出了一种从子串组合多重集重构字符串的新算法，该算法依赖于问题等价二元多项式表述的代数性质。随后，我们刻画了待重构二进制字符串的特定代数条件，这些条件保证算法在重构过程中不需要任何回溯，因此时间复杂度具有多项式上界。更具体地说，在无回溯情况下，我们算法的时间复杂度为 O(n^2)，而 Acharya 等人的算法时间复杂度为 O(n^2 log(n))，其中 n 是二进制字符串的长度。此外，结果表明，新算法能够唯一重构且无需回溯的二进制字符串集合更大，由此得到的重构码码本相比 Pattabiraman 等人此前已知的构造在规模上大一个线性因子，同时具有 O(n^2) 的重构复杂度。
英文摘要：
We consider the problem of binary string reconstruction from the multiset of its substring compositions, i.e., referred to as the substring composition multiset, first introduced and studied by Acharya et al. We introduce a new algorithm for the problem of string reconstruction from its substring composition multiset which relies on the algebraic properties of the equivalent bivariate polynomial formulation of the problem. We then characterize specific algebraic conditions for the binary string to be reconstructed that guarantee the algorithm does not require any backtracking through the reconstruction, and, consequently, the time complexity is bounded polynomially. More specifically, in the case of no backtracking, our algorithm has a time complexity of $O(n^2)$ compared to the algorithm by Acharya et al., which has a time complexity of $O(n^2\log(n))$, where $n$ is the length of the binary string. Furthermore, it is shown that larger sets of binary strings are uniquely reconstructable by the new algorithm and without the need for backtracking leading to codebooks of reconstruction codes that are larger, by a linear factor in size, compared to the previously known construction by Pattabiraman et al., while having $O(n^2)$ reconstruction complexity.
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