A signed tree model of a graph $G$ is a compact binary structure consisting of a rooted binary tree whose leaves are bijectively mapped to the vertices of $G$, together with 2-colored edges $xy$, called transversal pairs, interpreted as bicliques or anti-bicliques whose sides are the leaves of the subtrees rooted at $x$ and at $y$. We design an algorithm that, given such a representation of an $n$-vertex graph $G$ with $p$ transversal pairs and a source $v \in V(G)$, computes a shortest-path tree rooted at $v$ in $G$ in time $O(p \log n)$. A wide variety of graph classes are such that for all $n$, their $n$-vertex graphs admit signed tree models with $O(n)$ transversal pairs: for instance, those of bounded symmetric difference, more generally of bounded sd-degeneracy, as well as interval graphs. As applications of our Single-Source Shortest Path algorithm and new techniques, we - improve the runtime of the fixed-parameter algorithm for first-order model checking on graphs given with a witness of low merge-width from cubic [Dreier and Toruńczyk, STOC '25] to quadratic; - give an $O(n^2 \log n)$-time algorithm for All-Pairs Shortest Path (APSP) on graphs given with a witness of low merge-width, generalizing a result known on twin-width [Twin-Width III, SICOMP '24]; - extend and simplify an $O(n^2 \log n)$-time algorithm for multiplying two $n \times n$ matrices $A, B$ of bounded twin-width in [Twin-Width V, STACS '23]: now $A$ solely has to be an adjacency matrix of a graph of bounded twin-width and $B$ can be arbitrary; - give an $O(n^2 \log^2 n)$-time algorithm for APSP on graphs of bounded twin-width, bypassing the need for contraction sequences in [Twin-Width III, SICOMP '24; Bannach et al. STACS '24]; - give an $O(n^{7/3} \log^2 n)$-time algorithm for APSP on graphs of symmetric difference $O(n^{1/3})$.

翻译：图的符号树模型是一种紧凑的二元结构，由一棵根二叉树组成，其叶子与图的顶点建立双射关系，并附带称为横向对的二色边$xy$，这些边被解释为双团或反双团，其两侧分别是以$x$和$y$为根的子树的所有叶子。我们设计了一种算法，给定具有$p$个横向对的$n$顶点图$G$的此类表示以及源点$v \in V(G)$，能够在$O(p \log n)$时间内计算出以$v$为根的$G$中的最短路径树。多种图类均满足以下性质：对于所有$n$，其$n$顶点图允许使用$O(n)$个横向对的符号树模型表示，例如有界对称差图类、更一般的有界sd-退化图类以及区间图。作为我们单源最短路径算法及新技术的应用，我们：- 将给定低合并宽度见证的图上的一阶模型检测固定参数算法运行时间从立方阶[Dreier and Toruńczyk, STOC '25]改进至平方阶；- 针对给定低合并宽度见证的图，提出一种$O(n^2 \log n)$时间的全对最短路径算法，推广了孪生宽度图类的已知结果[Twin-Width III, SICOMP '24]；- 扩展并简化了[Twin-Width V, STACS '23]中关于两个有界孪生宽度$n \times n$矩阵$A, B$相乘的$O(n^2 \log n)$时间算法：现仅需$A$为有界孪生宽度图的邻接矩阵，$B$可为任意矩阵；- 针对有界孪生宽度图提出$O(n^2 \log^2 n)$时间的全对最短路径算法，避免了[Twin-Width III, SICOMP '24; Bannach et al. STACS '24]中对收缩序列的依赖；- 针对对称差为$O(n^{1/3})$的图提出$O(n^{7/3} \log^2 n)$时间的全对最短路径算法。