Construction methods of Perfect Hash Families

Conversely, we can find a function family F {f , f , f , f , such that f (x) is denoted as and for each x A, labeling the part for each partition according to the given order.

i

The proof of Proposition 2.6.7 is also included in Appendix A. Referring to the matrix provided in Example 2.6.1, it is obvious that it is an example of Proposition 2.6.7.

Next section provides a summary of construction methods of perfect hash families proposed in other studies. We also discuss some known bounds of N(n, m, w).

2.6.3 Construction methods of Perfect Hash

Families

In this section, we are more interested in the behavior of minimum N as a function of n when m and w are fixed. Bounds on N have been studied extensively (For examples, see [5][6][17][18][19]). In particular, in [19], it provides a proof that when m and w are fixed, N is Θ(log n). However, this existence is non-constructive. It is also believed that it is difficult to give explicit constructions that are as asymptotically good. Here, we introduce some explicit constructions and point out the bound on N in those constructions. Although these

constructions are not as asymptotically good as the one presented in [19], they are quite

reasonable.

There are many kinds of method to construct perfect hash families, such as using combinatorial structures and using algebraic structures. Table 2.6.2 lists the approaches included in the combinatorial and algebra structures.

Combinatorial Structures Algebra Structures

Design Theory Special Global Function Field

Error-Correcting Codes Algebraic Curves Recursive Constructions

Table 2.6.2 Construction Methods

In the combinatorial structures, we can use the design theory to construct perfect hash families. There are some set systems, such as the balanced incomplete block design (BIBD) and the separating resolvable block design (SRBD). The detail of this construction was introduced in [17]. According to their inference and proof, in the situation when m and w are fixed, the bound of N is Ω(n). Although these methods give simple constructions, they are limited in the sense that they cannot be applied to obtaining a PHF with an arbitrary m ≥w. In other words, they cannot obtain a PHF in which m is O(w). In addition, in a construction of perfect hash families using Error-Correcting Codes, the bound of N is O(n). The restriction of this method is the same as using the design theory to construct it. It also cannot construct a PHF in which m is O(w). Finally, in [17], the authors proposed two kinds of recursive construction. First, they used an already existing PHF together with a (n, k, λ) - difference matrix to obtain another PHF with larger N and n. In this construction, the bound of N is

2 1

log( )

((log )

w

O n

⎛ ⎞+

⎜ ⎟⎝ ⎠ ) . Second, the authors used three already existing PHFs and combine them into a new PHF with larger N and n. The bound value of N in the second method is about the same as the first one, but the second method has a slightly larger constant term.

In [18], the authors proposed the method of using algebra structures to construct perfect

hash families. Specifically, they used an algebraic curve to construct a PHF. In this method, the bound of N is O(log n). Details of this kind of construction are not included in this text since algebra structures are not used in this thesis. (For more related information, see [18].)

In this thesis, the PHF is constructed by the affine plane and resolvable BIBD. Before we introduce the construction mechanism, we first give some definitions for affine plane and resolvable BIBD.

An affine plane is a PBD(P, B) with some specific properties[20]. Before we state the corresponding properties, PBD(P, B) is introduced first. A pairwise balanced design, referred to as the PBD, is an ordered pair (P, B). P is a finite set of symbols, and B is a collection of subsets of P called blocks, such that each pair of distinct elements of P occurs together in exactly one block of B. The properties of an affine plane are summarized below.

(1) P contains at least one subset of 4 points, and no 3 of which are collinear.

(2) Given a line h and a point p not on h, there is exactly one line of B containing p, which is parallel to h.

Example 2.6.3 Affine plane.

P = {1, 2, 3, 4}

B = { {1, 2} {1, 3} {1. 4}

{3, 4} {2, 4} {2, 3} } □

In an affine plane (P, B), the number of points in each block is called the order of the affine plane.

Definition 2.6.8 k-power set of X P is the power set of X.

A is a -power set of X if Ak ⊆P and for each x A |x| ∈ =k.

| |

We have X -power sets of X.

k k

⎛ ⎞

⎜ ⎟

⎝ ⎠ □

Definition 2.6.9 [5]

X is a non-empty set of points, and A is a subset of the k-power set of X called blocks.

Let v, k, λ be positive integers such that v ≥ k ≥ 2. A (v, b, r, k, λ) – balanced incomplete block design (denoted as (v, b, r, k, λ) – BIBD) is a set system(X, A) such that the following properties are satisfied:

1. | X | = v,

2. Every point occurs in r blocks, and

3. Every pair of points occurs in exactly λ blocks. □

For simplicity, in the following examples, we write blocks in the form abc, rather than {a, b, c}.

Example 2.6.4

A (10, 15, 6, 4, 2) – BIBD.

X = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, and

A = {0123, 0145, 0246, 0378, 0579, 0689, 1278, 1369, 1479, 1568, 2359, 2489, 2567, 3458, 3467}.

□

Theorem 2.6.10. [5][21]

1. A (v, b, r, k,λ)–BIBD follows from elementary counting that vr = bk and λ(v–1) =

r(k–1). □

The proof of Theorem 2.6.10 is included in Appendix A.

A parallel class in (X, A) is a set of blocks that forms a partition of the point set X. A BIBD is resolvable if A can be partitioned into r parallel classes, and each of which consists of v/k disjoint blocks. Obviously, a BIBD can have a parallel class only if v ≡ 0 mod k.

Example 2.6.5 A resolvable (6, 15, 5, 2, 1) – BIBD.

Let X = {0, 1, 2, 3, 4, 5}, and r = 5. Hence there are 5 parallel classes, and each consists of 3 blocks.

So parallel classes = {01, 25, 34},

{02, 13, 45},

{03, 24, 15},

{04, 35, 12},

{05, 14, 23} □

It is well-known that an affine plane of order q is an (q², q(q+1), q+1, q, 1) – BIBD. It is also a resolvable BIBD. Thus, the following theorem can be derived: For any prime power q, there exists an affine plane of order q. That is, there exists a (q², q(q+1), q+1, q, 1) – BIBD.

Theorem 2.6.11. [5][18]

If there exists a resolvable (v, b, r, k, λ) – BIBD with , then there exists a PHF(r; v,

v/k, w). □