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In 19 we discussed the binary search tree, which allows various operations on a set of elements In this chapter we discuss the hash table, which supports only a subset of the operations allowed by binary search trees The implementation of hash tables is frequently called hashing, which performs insertions, deletions, and finds in constant average time Unlike with the binary search tree, the average-case running time of hash table operations is based on statistical properties rather than the expectation of random-looking input This improvement is obtained at the expense of a loss of ordering information among the elements: Operations such as findMin and findMax and the printing of an entire table in sorted order in linear time are not supported Consequently, the hash table and binary search tree have somewhat different uses and performance properties In this chapter, we show: several methods of implementing the hash table, analytical comparisons of these methods, some applications of hashing, and comparisons of hash tables and binary search trees

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The hash table is to implement a set in constant time per

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The hash table supports the retrieval or deletion of any named item We want to be able to support the basic operations in constant time, as for the stack and queue Because the accesses are much less restricted, this support seems like an impossible goal That is, surely when the size of the set increases, searches in the set should take longer However, that is not necessarily the case

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A hash function the item an integer suitable to index an array where the item is stored lf the hash function were one to one, we could access the item by its array index

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Suppose that all the items we are dealing with are small nonnegative integers, ranging from 0 to 65,535 We can use a simple array to implement each operation as follows First, we initialize an array a that is indexed from 0 to 65,535 with all 0s To perform insert ( i) , we execute a [ i I ++ Note that a [ i I represents the number of times that i has been inserted To perform find ( i ) , we verify that a [ i ] is not 0 To perform remove ( i ) , we make sure that a [ i 1 is positive and then execute a [ i 1 -- The time for each operation is clearly constant; even the overhead of the array initialization is a constant amount of work (65,536 assignments) There are two problems with this solution First, suppose that we have 32-bit integers instead of 16-bit integers Then the array a must hold 4 billion items, which is impractical Second, if the items are not integers but instead are strings (or something even more generic), they cannot be used to index an array The second problem is not really a problem at all Just as a number 123 4 is a collection of digits 1, 2 3, and 4, the string "junk"is a collection of characters j , u n and ' k Note that the number 12 3 4 is just 1 1O3 + 2 10' + 3 10' + 4 lo0 Recall from Figure 131 that a character can typically be represented in 7 bits as a number between 0 and 127 Because a character is basically a small integer, we can interpret a string as an integer One possible representation is j 128% u 128' + ' n 1 128' + k 128O This approach allows the simple array implementation discussed previously The problem with this strategy is that the integer representation described generates huge integers: The representation for junk" yields 224,229,227, and longer strings generate much larger representations This result brings us back to the first problem: How do we avoid using an absurdly large array We do so by using a function that maps large numbers (or strings interpreted as numbers) into smaller, more manageable numbers A function that maps an item into a small index is known as a hash function If x is an arbitrary (nonnegative) integer, then x%tableSize generates a number between 0 and tablesize- 1 suitable for indexing into an array of size tablesize If s is a string, we can convert s to a large integer x by using the method suggested previously and then apply the mod operator (%)to get a suitable index Thus, if tablesize is 10,000, " junk"would be indexed to 9,227 In Section 202 we discuss implementation of the hash function for strings in detail

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