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 as few columns as possible: 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 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 $40$ for $d=6$, so our $40$-edge graph is optimal. 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.
Large sparse matrices---adjacency matrices of web and social graphs, feature matrices of machine-learning pipelines, the relations of graph databases---are routinely compressed to save space. Classical compressors, however, force the matrix to be decompressed before it can be used, so the space savings often yield a computational overhead. A recent line of work studies \emph{computation-friendly} compression: formats that are not only small, but let us carry out matrix--vector products, and in some cases matrix--matrix products, sums and closures, directly on the compressed representation {\em in time proportional to that compressed size}. This survey organizes that literature around the kind of redundancy each format exploits: spatial clustering of the nonzeros ($k^2$-trees and their variants), similarity between rows (Boldi--Vigna-style referencing and set-difference compression), repeated substrings (grammar compression), and dense bipartite substructures (bicliques). For each family we describe the representation, the algorithms, the theoretical guarantees and the empirical performance. Further, we show that the last three families are instances of a single general scheme, which we call a {\em sparse straight-line factorization} of the matrix, whose forward and backward substitutions evaluate right and left products in time proportional to the size of the representation, and that the product of two such factorizations is again one, which gives matrix--matrix products with a compressed output. An appendix reviews formats that must be decompressed (at least partially) before operating, for completeness. We close with a cross-family comparison and a list of open problems.