Evaluating the relational join is one of the central algorithmic and most well-studied problems in database systems. A staggering number of variants have been considered including Block-Nested loop join, Hash-Join, Grace, Sort-merge (see Grafe [17] for a survey, and [4, 7, 24] for discussions of more modern issues). Commercial database engines use finely tuned join heuristics that take into account a wide variety of factors including the selectivity of various predicates, memory, IO, etc. This study of join queries notwithstanding, the textbook description of join processing is suboptimal. This survey describes recent results on join algorithms that have provable worst-case optimality runtime guarantees. We survey recent work and provide a simpler and unified description of these algorithms that we hope is useful for theory-minded readers, algorithm designers, and systems implementors.

Much of this progress can be understood by thinking about a simple join evaluation problem that we illustrate with the so-called triangle query, a query that has become increasingly popular in the last decade with the advent of social networks, biological motifs, and graph databases [36, 37]