2.1 Transitive Closure

Note that, in some sense, the O(n2) result is optimal if we approach this problem by storing the transitive closure matrix explicitly (we don’t necessarily have to do this, however, to answer queries). This is because in the worst case, an update can modify Ω(n2) edges in the transitive closure. Next are two results for which the product of the bulk update and query times is O(mn), where m is the number of edges. The first of the two results allows for some amount of trade-off between the two operations.