Cross-Referenced Dictionaries and the Limits of Write Optimization.

Dictionaries remain the most well studied class of data structures. A dictionary supports insertions, deletions, membership queries, and usually successor, predecessor, and extract-min. In a RAM, all such operations take O(log N) time on N elements. Dictionaries are often cross-referenced as follows. Consider a set of tuples {〈 ai, bi, ci ...〉 }. A database might include more than one dictionary on such a set, for example, one indexed on the a's, another on the b's, and so on. Once again, in a RAM, inserting into a set of L cross-referenced dictionaries takes O(L log N) time, as does deleting. The situation is more interesting in external memory. On a Disk Access Machine (DAM), B-trees achieve O(logB N) I/Os for insertions and deletions on a single dictionary and K-element range queries take optimal O(logB N + K/B) I/Os. These bounds are also achievable by a B-tree on cross-referenced dictionaries, with a slowdown of an L factor on insertion and deletions. In recent years, both the theory and practice of external memory dictionaries has been revolutionized by write optimization techniques. A dictionary is write optimized if it is close to a B-tree for query time while beating B-trees on insertions. The best (and optimal) dictionaries achieve a substantially improved insertion and deletion cost of [EQUATION], 0 ≤ ε ≤ 1, amortized I/Os on a single dictionary while maintaining optimal O(log1+Bε N + K/B)- I/O range queries. Although write optimization still helps for insertions into cross-referenced dictionaries, its value for deletions would seem to be greatly reduced. A deletion into a cross referenced dictionary only specifies a key a. It seems to be necessary to look up the associated values b, c... in order to delete them from the other dictionaries. This takes Ω(logB N) I/Os, well above the per-dictionary write optimization budget of [EQUATION] I/Os. So the total deletion cost is [EQUATION] I/Os. In short, for deletions, write optimization offers an advantage over B-trees in that L multiplies a lower order term, but when L = 2, write optimization seems to offer no asymptotic advantage over B-trees. That is, no known query-optimal solution for pairs of cross-referenced dictionaries seem to beat B-trees for deletions. In this paper, we show a lower bound establishing that a pair of cross-referenced dictionaries that are optimal for range queries and that supports deletions cannot match the write optimization bound available to insert-only dictionaries. This result thus establishes a limit to the applicability of write-optimization techniques on which many new databases and file systems are based.