Miltersen. Lower Bounds for Union-split--nd Related Problems on Random Access Machines. In

10 to instances with fewer don't cares. The hardest case seems to be when queries have exactly d 2 don't cares. In this case npm is extremely rich. Almost all entries in the communication matrix are one. However, perhaps not surprisingly, we have the following: 11 Claim 16. The communication matrix for npm restricted to queries with exactly d 2 don't cares contains a 1-monochromatic rectangle of size n ?2 ? d d=2 2 d=2 e ?1 2 nd. The consequence here is obvious. The total number of possible queries is ? d d=2 2 d=2. Thus, the best lower bound we can prove by the richness technique is the rather pathetic \either the query sends (log n) bits or the database sends (1) bits". Proof of Claim 16. Take the set of queries to be all possible queries with d 2 don't cares, and the rst k bits xed as zeros (k to be determined shortly). The number of such queries is d ? k d=2 2 d=2?k d d=2 2 d=2?2k : The number of cube points matched by at least one query is exactly 2 d?k. Therefore, the number of databases that are not matched by any query is (2 d ? 2 d?k) n = (1 ? 2 ?k) n 2 nd e ?n=2 k 2 nd : Now, take k = log n. Returning to the case of log n + 1 exposed bits, is it possible to improve upon the proven bounds? If all possible queries are enumerated in some predeened order, the database can store the answer to all possible queries and the query player can then simply send the index of the query using O(logn log d) bits (and the database player responds with the correct answer using one bit). Hence the bound on the query player is optimal. Finally , we ask if we can improve our lower bound on the database side to (n)? The following claim shows that our analysis cannot be improved signiicantly. Claim 17. For every integer c, there is > 0 such that for every n , the communication matrix of npm restricted to queries from Q n;d contains a 1-monochromatic rectangle of size 2 ?c(logd+1) jQ n;d j 2 nd?nlog e=2 c. Proof. We may assume that n is suuciently large so that logn c. Take all queries in Q n;d with the rst c …