超图的指令集与语言

中文摘要 AI 辅助

我们提出了IsalHG，一种将任何有限、连通且超边度有界的超图结构表示为紧凑指令字母表Σ_HG上字符串的方法。编码由一个小型虚拟机执行，该虚拟机包含一个稀疏超图、节点引用的循环双链表（CDLL）和k个遍历指针。指令可移动指针或插入超边及新节点。每个字符串都能解码为有效超图。贪心的HypergraphToString（h2s）算法将超图编码为字符串，回溯变体可生成规范字符串w*。规范字符串相等性可直接判定超图同构。我们验证了150个连通随机均匀超图和命名组合设计上的往返属性s2h(h2s(H))≅H，并将规范算法与三个实际可用的精确基线（nauty、Traces和bliss）在(n,c)网格上进行基准测试。所有四种方法在600个同构判定上均一致。在运行时间上，Levi基线比测试单元中的规范算法快三到五个数量级。我们贡献了表示框架、规范完备性猜想以及超图同构的首个原生与Levi基准测试。

英文摘要

We present IsalHG, a method for representing the structure of any finite, connected hypergraph of bounded hyperedge arity as a string over a compact instruction alphabet $Σ_{\mathrm{HG}}$. The encoding is executed by a small virtual machine comprising a sparse hypergraph, a circular doubly-linked list (CDLL) of node references, and $k$ traversal pointers, where $k$ bounds the hyperedge arity. Instructions either move a pointer through the CDLL or insert a hyperedge, optionally together with new nodes, into the hypergraph. Every string over $Σ_{\mathrm{HG}}$ decodes to a valid hypergraph; the alphabet is closed. A greedy \emph{HypergraphToString} (h2s) algorithm encodes any connected hypergraph into a string; a backtracking variant seeded at nodes of lexicographically maximal structural tuple produces a \emph{canonical string} $w^{*}$, which we conjecture to be a complete isomorphism invariant. Canonical-string equality then decides hypergraph isomorphism natively, without the standard reduction to the Levi incidence graph followed by a graph-isomorphism engine. We verify the round-trip property $s2h(h2s(H)) \cong H$ on 150 connected random uniform hypergraphs and on named combinatorial designs, and we benchmark the canonical algorithm against the three practically available exact baselines -- nauty, Traces, and bliss operating on the 2-coloured Levi graph -- across a $(n, c)$ grid with ten seeds per cell. All four methods agree on every one of 600 isomorphism verdicts, consistent with the completeness conjecture. On wall-clock time the Levi baselines dominate every tested cell by three to five orders of magnitude (geometric-mean ratio $311\times$ to $117{,}672\times$), which we report as measured. We contribute the representation framework, a conjecture of canonical completeness, and the first native-versus-Levi benchmark for hypergraph isomorphism.

发表机构

• University of Málaga（马拉加大学）

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