Order graphs were introduced by Arroyuelo et al.\ (TODS 2024) as a generalization of the ring index, to support worst-case optimal joins on relations of arity $d$ while storing few columns: the index stores one column per edge, so its size is proportional to the number of edges of an order graph. It was known that the minimum number of edges lies between $Ω(2^d\sqrt d)$ and $O(2^d d)$, and the lower bound had been claimed to be $42$ for $d=6$. We first exhibit an optimal order graph with only $40$ edges for $d=6$, and show that the proof of that lower bound holds only for unions of cycles, whereas order graphs can be smaller than the smallest set of cycles. We give a corrected statement and proof that holds for every order graph: there must be at least $\max_m\binom dm\lceil m/2\rceil$ edges. This bound is $40$ (hence tight) for $d=6$. We then settle the asymptotic question: the minimum size is $Θ(2^d\sqrt d)$. This is obtained via an explicit construction, based on symmetric chain decompositions of the Boolean lattice, with at most $(d-1)\binom{d}{\lfloor d/2\rfloor}+2^d$ edges. The upper bound is within a factor $4+O(d^{-1/2})$ of the lower bound, and both numerically and asymptotically better than trie switching. Finally, we cast the problem as a set cover with connectivity constraints, which yields an exact ILP formulation and a greedy heuristic. With them we find optimal order graphs for $d\le 6$ in the bidirectional and for $d \le 7$ in the unidirectional model, showing that our lower bounds are tight there, and improve the best known order graphs for $d=7$ and $8$.

翻译：暂无翻译