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The price that the quadratic probing algorithm pays for its improved performance is that it is more likely to result in an infinite loop. That happens in Example 8.7 with the next insertion. The string "Hof" hashes initially to index 3. After eight collisions, the linear probing algorithm found a free cell at index 0 ( = 11 mod 11). But the probe sequence used on this item by the quadratic probing algorithm is the same as for "Ast": 3, 4, 7, 1, 8, 6, 6, 8 1, 7, 4, 3, 4, . . . . This is computed from the unmodulated quadratic sequence 3, 4, 7, 12, 19, 28, 39, 52, 67, 84, 103, 124, 147, . . . . This continues indefinitely, probing only the six indexes 3, 4, 7, 1, 8, and 6, all of which have already been used. So even though the table is only 81% full, the insertion fails. That can t happen with linear probing. SEPARATE CHAINING Instead of devising a more effective collision resolution algorithm, we can avoid collisions altogether by allowing more than one item per table component. This method is called separate chaining, because is uses linked lists ( chains ) to hold the multiple items. In this context, the table components are usually called buckets. EXAMPLE 8.8 Separate Chaining

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For Brutus is an honourable man;

Thus,$ipping one or more zeros of the system function about the unit circle does not change the magnitude of the frequency response.

the magnitude of the frequency response will not be changed if it is cascaded with the allpass filter

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

Here is the Concordance class:

This allpass filter flips the zero at z = 0.2 in H(z) to its reciprocal location, z = 5, and the new filter has a system function

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A stable and causal linear shift-invariant system with a rational system function of the form given in Eq. (5.2) has all of its poles inside the unit circle, lakI < 1. The zeros, however, may lie anywhere in the z-plane. In some applications, it is necessary to constrain a system so that its inverse, G(z) = l/H(z), is also stable and causal. I This requires that the zeros of H (z) lie inside the unit circle, JBk < I. A stable and causal filter that has a stable and causal inverse is said to have minimum phase. Equivalently, we have the following definition:

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

1 2 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

Definition: A rational system function with all of its poles and zeros inside the unit circle is said to be have minimum phase.

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A minimum phase system is uniquely defined by the magnitude of its Fourier transform, IH(eJm)(.The procedure to find H (z) from 1 H (ejm)\is as follows. Given \ H (ejW)l,we find 1 H (ej")12, which is a function of cos(ko). Then, by replacing cos(kw) with C(zk zpk), we have

CHAP. 8]

public class Concordance { private Map<String,String> map = new HashMap<String,String>(); public Concordance(String file) { int lineNumber = 0; try { Scanner input = new Scanner(new File(file)); while (input.hasNextLine()) { String line = input.nextLine(); ++lineNumber; StringTokenizer parser = new StringTokenizer(line, ",.;:()-! ' "); while (parser.hasMoreTokens()) { String word = parser.nextToken().toUpperCase(); String listing = map.get(word); if (listing == null) { listing = "" + lineNumber; } else { listing += ", " + lineNumber; } map.put(word,listing); } } input.close(); } catch(IOException e) { System.out.println(e); } } public void write(String file) { try { PrintWriter output = new PrintWriter(file); for (Map.Entry<String,String> entry : map.entrySet()) { output.println(entry); } output.close(); } catch(IOException e) { System.out.println(e); } } }

Finally, the minimum phase system is then formed from the poles and zeros of G ( z )that are inside the unit circle.

Expressing cos w in terms of complex exponentials,

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

The hash table is defined at line 2. It has type java.util.HashMap<String,String>, which means that both its key and value fields have type String. Its constructor at line 4 takes the name of the input file as its argument. It uses a java.util.Scanner object to read the file line by line. Each line is parsed by a java.util.StringTokenizer object named parser, defined at line 11. Note that the parser uses as

A stable and causal system may always be factored into a product of a minimum phase system with an allpass system: (5.17) H ( z ) = Hmin(z).Hap(z)

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