B.3 Dynamic Hashing Figure 6: Rejection Method B.1 Rejection Method

Any single hash function chosen can encounter some bad worst-case inputs that cause linear-time rather than constant-time performance. The remedy devised by Carter and Wegman 2] is to choose a hash function randomly from a good collection H of hash functions and get constant expected performance independent of any particular input sequence. We say that H is c-universal if for every pair of inputs x 6 = y in f1; 2; : : :; n ? 1g the total number of h 2 H such that h(x) = h(y) is no more than c jHj=m; that is, only a fraction of c=m of the hash functions in H cause a collision on any pair of inputs. Proposition 3 Let H be a c-universal class of hash functions, the expected cost of an insert, delete, or access operation is O(1 + cc), where is the load factor of the table. We can use the c-universal class of hash functions H = fh a;b j h a;b (x) = ((ax + b) mod n) mod m; a; b 2 f0; : : :; n ? 1gg ; where (dn=me=(n=m)) 2 = O(1). When the number of elements changes dynamically, the table may have to be expanded or contracted from time to time, but the cost of the rebuilding can be amortized so that the operations still run in amortized constant expected time. More complicated techniques for implementing the table lookup method in constant expected time are dynamic perfect hashing and its variants 4, 5, 6]. Corollary 1 The expected number of iterations in algorithm bucket rejection is 2. A comprehensive treatment of table doubling can be found in 3]. Suppose we want to implement a dynamic table that supports insertion and deletion. In order to use the power of the random-access model, the table is implemented as an array. The size of the table cannot be determined in advance, so dynamic allocation and deallocation of the array is necessary. A trivial algorithm allocates an (n + 1)-element array when an element is inserted into an n-element array, but this causes worst-case update cost proportional to the size of the array. Since the number of elements in the table is not necessarily the same as the size of the table, let us use to denote the load factor of the table, or its fraction of occupancy. Initially, the table T has size zero. …