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The search algorithms that we have been considering are based on an abstract comparison operation. A significant exception to this assertion is the key-indexed search method in , where we store the item with key i in table position i, ready for immediate access. Key-indexed search uses key values as array indices rather than comparing them and depends on the keys being distinct integers falling in the same range as the table indices. In this chapter, we consider hashing, an extension of key-indexed search which handles more typical search apps where we do not happen to have keys with such fortuitous properties. The end result is a completely different approach to search from the comparison-based methods—rather than navigating through dictionary data structures by comparing search keys with keys in items, we try to reference items in a table directly by doing arithmetic operations to transform keys into table addresses.

Search algorithms that use hashing consist of two separate parts. The first step is to compute a hash function that transforms the search key into a table address. Ideally, different keys would map to different addresses, but often two or more different keys may hash to the same table address. Thus, the second part of a hashing search is a collision-resolution process that deals with such keys. One of the collision-resolution methods that we shall study uses linked lists and is thus immediately useful in dynamic situations where the number of search keys is difficult to predict in advance. The other two collision-resolution methods that we shall examine achieve fast search times on items stored within a fixed array. We shall also examine a way to improve these methods to handle the case where we cannot predict the table size in advance.

Hashing is a good example of a time–space tradeoff. If there were no memory limitation, then we could do any search with only one memory access by simply using the key as a memory address, as in key-indexed search. This ideal often cannot be achieved, however, because the amount of memory required is prohibitive when the keys are long. On the other hand, if there were no time limitation, then we could get by with only a minimum amount of memory by using a sequential search method. Hashing provides a way to use a reasonable amount of both memory and time to strike a balance between these two extremes. In particular, we can strike any balance we choose, merely by adjusting hash table size, not by rewriting code or choosing different algorithms.

Hashing is a classical computer-science problem: The various algorithms have been studied in depth and are widely used. We shall see that, under generous assumptions, it is not unreasonable to expect to support the search and insert symbol-table operations in constant time, independent of the size of the table.

This expectation is the theoretical optimum performance for any symbol-table implementation, but hashing is not a panacea, for two primary reasons. First, the running time does depend on the length of the key, which can be a liability in practical apps with long keys. Second, hashing does not provide efficient implementations for other symbol-table operations, such as select or sort. We shall examine these and other matters in detail in this chapter.

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