We introduce a compact pangenome representation based on an optimal segmentation concept that aims to reconstruct founder sequences from a multiple sequence alignment (MSA). Such founder sequences have the feature that each row of the MSA is a recombination of the founders. Several linear time dynamic programming algorithms have been previously devised to optimize segmentations that induce founder blocks that then can be concatenated into a set of founder sequences. All possible concatenation orders can be expressed as a founder graph. We observe a key property of such graphs: if the node labels (founder segments) do not repeat in the paths of the graph, such graphs can be indexed for efficient string matching. We call such graphs repeat-free founder graphs when constructed from a gapless MSA and repeat-free elastic founder graphs when constructed from a general MSA with gaps. We give a linear time algorithm and a parameterized near linear time algorithm to construct a repeat-free founder graph and a repeat-free elastic founder graph, respectively. We derive a tailored succinct index structure to support queries of arbitrary length in the paths of a repeat-free (elastic) founder graph. In addition, we show how to turn a repeat-free (elastic) founder graph into a Wheeler graph in polynomial time. Furthermore, we show that a property such as repeat-freeness is essential for indexability. In particular, we show that unless the Strong Exponential Time Hypothesis (SETH) fails, one cannot build an index on an elastic founder graph in polynomial time to support fast queries.

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索引方正图的算法和复杂性

我们介绍了一种基于最佳分割概念的紧凑型全基因组表示法，该概念旨在从多序列比对（MSA）重建创始人序列。这样的创建者序列具有以下特征：MSA的每一行都是创建者的重组。先前已经设计了几种线性时间动态编程算法来优化分割，从而诱发创建者块，然后可以将这些创建者块连接到一组创建者序列中。所有可能的连接顺序都可以表示为创建者图。我们观察到此类图的关键特性：如果节点标签（创建者细分）在图的路径中不重复，则可以为此类图建立索引以进行有效的字符串匹配。当由无间隙MSA构造时，我们将此类图称为无重复创建者图；当由具有间隙的普通MSA构造时，我们将此类图称为无重复弹性创建者图。我们给出了线性时间算法和参数化的近似线性时间算法，分别构建了无重复的创建者图和无重复的弹性创建者图。我们推导了量身定制的简洁索引结构，以支持无重复（弹性）创建者图的路径中任意长度的查询。此外，我们展示了如何在多项式时间内将无重复（弹性）创建者图转换为Wheeler图。此外，我们证明了诸如重复自由度之类的属性对于可索引性是必不可少的。特别是，我们表明除非强指数时间假说（SETH）失败，