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Join counts for 3 category data on a grid using 'rook' (north, south, east, west) neighbors. Left: each category never has a neighbour of its own type, resulting in zeros on the diagonal. Centre: random pattern shows no bias for pairing colours, resulting in approximately equal values for all join count statistics.
algorithm nested_loop_join is for each tuple r in R do for each tuple s in S do if r and s satisfy the join condition then yield tuple <r,s> This algorithm will involve n r *b s + b r block transfers and n r +b r seeks, where b r and b s are number of blocks in relations R and S respectively, and n r is the number of tuples in relation R.
In 2016, Blelloch et al. formally proposed the join-based algorithms, and formalized the join algorithm for four different balancing schemes: AVL trees, red–black trees, weight-balanced trees and treaps. In the same work they proved that Adams' algorithms on union, intersection and difference are work-optimal on all the four balancing schemes.