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Here is how part of a definition for a HashTable class might look, using separate chaining:

Fig. 5-4.

Constraints on the zeros of the system function of an FIR system with generalized linear phase and a real unit sample response. Types I11 and IV filters must have a zero at z = 1 , whereas types I1 and 111 filters must have a zero a t z = -1.

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Another collision resolution algorithm that usually performs better than linear probing is called quadratic probing. This algorithm jumps over items in its probing, with the result that the used components are more uniformly distributed with fewer large clusters. That improves performance because the resulting probe chains are shorter. EXAMPLE 8.7 Quadratic Probing

public class HashTable { private static final int MASK = 0x7FFFFFFF; // 2^32-1

The cases of z = 1 and z = - 1 deserve special attention. Evaluating the system function at z = -1 for a type I1 filter, we have

which will be true only if H(-1) = 0. Therefore, a type I1 linear phase filter must have a zero at z = -1. Similarly, evaluating H ( z ) at z = - 1 for a type 111 filter, we have

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This program is the same as the program in Example 8.6 except for the modified hash() function shown here.

CHAP. 8]

which, because N is even, requires that there be a zero at z = -1. Because the system function evaluated at z = - 1 is equal to the frequency response at w = n, H (ejW)~,,, = 0 Types I1 and 111 filters

(5.13)

38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

which will be true only if H ( z ) is zero at z = 1. Therefore, types 111and IV linear phase filters must have a zero at z = 1, which implies that H (eio)lo,o = 0 Types 111 and IV filters

(5.14)

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public class TestQuadraticProbing { private static final int MASK = 0x7FFFFFFF; // 2^32-1 private static final int CAPACITY = 11; private static int size = 0; private static boolean[] used = new boolean[CAPACITY]; public static void main(String[] args) { printHash("Rad"); printHash("Uhr"); printHash("Ohr"); printHash("Tor"); printHash("Hut"); printHash("Tag"); printHash("Eis"); printHash("Ast"); printHash("Zug"); printHash("Hof"); printHash("Mal"); } private static void printHash(String word) { System.out.printf("hash(%s) = %d, load = %d%%%n", word, hash(word), 100*size/CAPACITY); } private static int hash(Object object) { ++size;

int capacity = 101; int size = 0; float load = 0.75F; LinkedList[] buckets;

An allpass filter has a frequency response with a constant magnitude,

This unit magnitude constraint constrains the poles and zeros of a rational system function to occur in conjugate reciprocal pairs:

int h = (object.hashCode() & MASK) % CAPACITY; if (used[h]) { int h0 = h; int jump = 1; while (used[h]) { System.out.printf("%d, ", h); h = (h0 + jump*jump)%CAPACITY; // squared increment ++jump; } } used[h] = true; return h; } }

HashTable() { buckets = new LinkedList[capacity]; for (int i = 0; i < capacity; i++) { buckets[i] = new LinkedList(); } } HashTable(int capacity, float load) { this(); this.capacity = capacity; this.load = load; } Object put(Object key, Object value) { int h = hash(key); LinkedList bucket=buckets[h]; Object oldValue = null; for (ListIterator it = bucket.iterator(); it.hasNext(); ) { Map.Entry entry = it.next(); if (entry.getKey().equals(key)) { break; } } if (entry.getKey().equals(key)) { oldValue = entry.setValue(value); } else { bucket.add(new Entry(key,value)); } return oldValue; } // more methods... }

Thus, if H ( z ) has a pole at z = ak, H ( z ) must have a zero at the conjugate reciprocal location z = l/a,*. If h ( n ) is real-valued, the complex roots in Eq. (5.15)occur in conjugate pairs, and if these conjugate pairs are combined to form second-order factors, the system function may be written as

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