Space-time trade-offs for some ranking and searching queries

We study space/time tradeoffs for querying some combinatorial structures. In the first, given an arrangement of n lines in general position in the plane, a query for a real number t asks about Rank(t), the number of vertices of the arrangement with x-coordinates ≤ t. We show that for K = O(n/logn), after a preprocessing step that uses space S = O(n/(K logK)) the query can be answered in time O(n logK). The second query involves the Cartesian sum of vectors a = (a1, . . . , an) and b = (b1, . . . , bn). For a given real t, it asks about Rank(t), the number of sums ai + bj which are ≤ t. We show that for some positive constant c and K ≤ c(logn)/(log logn), after a preprocessing step that uses space S = O(n/K), the query may be answered in time O((n/K) logK). Both results fit neatly between two obvious extremes.