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The hash function values are computed at line 19, where CAPACITY is 11 and MASK is 2147483647, expressed in hexadecimal as 0x7FFFFFFF. The operation n & MASK simply removes the sign from whatever integer n has. This is the right thing to do in Java before using the remainder operator to compute an array index because (unlike C++) Java may give a negative result to m % CAPACITY if m is negative. So the resulting value returned by the hash() function in this example is guaranteed to be in the range of 0 to 10. The first five strings hash into index values 3, 4, 2, 8, and 5, so they would be stored in those locations in the hash table. But the sixth string ("Tag") also hashes to 3, causing a collision with "Rad", which would already be stored in component 3. The most common algorithm to apply when such collisions occur is to simply put the new item in the next available component. That would be component 6 in this example, since "Uhr" would already be in component 4 and "Hut" would already be in component 5. This collision resolution algorithm is called linear probing.

law1 < 1 la(l>l < 1 + a @ )

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HASH TABLE PERFORMANCE A hash table of size 101 that contain six elements will perform very well. It is very unlikely to have any collisions, so the access functions are immediate, running in constant time O(1). This is direct access, just like an array. But a hash table of size 101 that contains 100 elements is likely to perform very poorly because there will have been many collisions in the process of storing its elements. For example, if the string "Lob" had 60 collisions before a free component was found for it, then each time it is accessed, it will take 60 probes to find it. That kind of performance is close to O(n) not much better than a linked list. The solution to the problem described here is to prevent the hash table from becoming too full. This is done by resizing it whenever it reaches a threshold size. The measure of fullness depends upon two parameters: The size of the hash table is the actual number of elements in the table; the capacity of the table is the number of components that it has. The ratio of these two parameters is called the load factor. In the first example cited in this section, the size was 6 and the capacity was 101, resulting in a load factor of 6/101 = 5.94 percent. In the second example, the size was 100, resulting in a load factor of 100/101 = 99.01 percent. The HashMap class automatically resizes its hash table when the load factor reaches a specific threshold value. This threshold value can be set when the hash table is created, using the constructor

These constraints define a stability triangle in the coefficient plane as shown in Fig. 5- 1. Thus, a causal secondorder system will be stable if and only if the coefficients a ( ] )and a(2) lie inside this triangle. This result is of special interest, because second-order systems are the basic building blocks for higher-order systems. If the coefficients lie in the shaded region above the parabola defined by the equation

The HashMap class uses a hash function just like the one in Example 8.4 to implement its accessor methods: containsKey(), get(), put(), remove(), and entrySet(). Its sets the hash table size at 101 initially. With that knowledge, we can see why the six strings in the previous examples were stored in the order indicated. EXAMPLE 8.5 Mapping Keys into a Hash Table of Size 101

Fig. 5-1. The stability triangle, which is defined by the lines la(2)I < 1 and la(l)( < 1 +a(2). The shaded region above theparabolaa2(1)4a(2) = 0 contains the values of a ( l ) and a(2) that correspond to complex roots.

public HashMap(int initialCapacity, float loadFactor)

This program is identical to the one in Example 8.4 except that the hash table CAPACITY is 101 instead of 11:

For a linear shift-invariant system with a system function H ( z ) , the inverse system is defined to be the system that has a system function G ( z ) such that H ( z ) . G(z) = 1 In other words, the cascade of H ( z ) with G ( z ) produces the identity system. In terms of H ( z ) , the inverse is simply 1 G(z) = H(z)

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