Cs 59000 Ctt Current Topics in Theoretical Cs Lecture 12

Our goal is to store S (a set of keys) in such a way that we can quickly (i.e. with few memory accesses) answer queries of the form “Is i ∈ S?”. So we want to encode S in a table via some encoding scheme S → E(S). An algorithm for the membership problem takes a query “Is i ∈ S” and makes a few ‘probes’ into the table E(S), where a probe is just an index (cell) in the table. Based on the values seen at the probed locations the algorithm should correctly answer ‘yes’ or ‘no’. This is a fundamental problem in data structures. Yao [3] showed that if the keys are stored explicitly (so we have n keys of size log m) then the best one can do is to store the set S in sorted order on a given query perform binary search to find the answer. This means that E(S) has size n log m bits, and our search could require reading log n keys, so log n log m bits, which is somewhat inefficient. In the so-called ‘cell-probe model’ introduced by Yao [3], a cell contains a number of bits and the time complexity of the scheme is counted in terms of the number of cells probed (as opposed to the number of bits seen). Fredman et al [2] conceived a scheme which improved on the sorted-storage approach, in that it required only a constant number c of probes. They stored their data in a table of size n cells (of size log m bits each) but the number of probes needed is only a constant c (so the algorithm sees c log m bits). Another well-studied model is the ‘bit-probe’ model, where a cell contains only one bit. So in this model the goal is to store the set S in a small bit-string (hopefully close to informationtheoretic optimal) but such that any query can be answered with only a few bit-probes. What is the information-theoretic minimum number of bits needed to store sets S of size ≤ n from U = [m]? Since there are ∑n i=1 ( n k ) distinct sets, it follows that these sets can be uniquely represented with Ω(n log m) bits. So the goal of a storing scheme is to get as close to n log m bits of storage as possible, but also be able to efficiently answer any query. In today’s lecture we’ll construct a scheme where the data structure has O(n log m) bits and queries can be answered correctly with high probability by making only one bit-probe into the encoding.