Abstract 
We present an indexing scheme for triplebased graphs that supports join queries in worstcase optimal (wco) time within compact space. This scheme, called a ring, regards each triple as a cyclic string of length 3. Each rotation of the triples is lexicographically sorted and the values of the last attribute are stored as a column, so we obtain the order of the next column by stably resorting the triples by its attribute. We show that, by representing the columns with a compact data structure called a wavelet tree, this ordering enables forward and backward navigation between columns without needing pointers. These wavelet trees further support wco join algorithms and cardinality estimations for query planning. While traditional data structures such as BTrees, tries, etc., require 6 index orders to support all possible wco joins over triples, we can use one ring to index them all. This ring replaces the graph and uses only sublinear extra space, thus supporting wco joins in almost no space beyond storing the graph itself. Experiments querying a large graph (Wikidata) in memory show that the ring offers nearly the best overall query times while using only a small fraction of the space required by several stateoftheart approaches.
We then turn our attention to some theoretical results for indexing tables of arity d higher than 3 in such a way that supports wco joins. While a single ring of length d no longer suffices to cover all d! orders, we need much fewer rings to index them all: O(2^d) rings with a small constant. For example, we need 5 rings instead of 120 orders for d=5. We show that our rings become a particular case of what we dub order graphs, whose nodes are attribute orders and where stably sorting by some attribute leads us from an order to another, thereby inducing an edge labeled by the attribute. The index is then the set of columns associated with the edges, and a set of rings is just one possible graph shape. We show that other shapes, like for example a single ring instead of several ones of length d can lead us to even smaller indexes, and that other more general shapes are also possible. For example, we handle d=5 attributes within space equivalent to 4 rings.
