In previous sections we were able to make improvements in our searchalgorithms by taking advantage of information about where items arestored in the collection with respect to one another. For example, byknowing that a list was ordered, we could search in logarithmic timeusing a binary search. In this section we will attempt to go one stepfurther by building a data structure that can be searched in(O(1)) time. This concept is referred to as hashing.

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In order to do this, we will need to know even more about where theitems might be when we go to look for them in the collection. If everyitem is where it should be, then the search can use a single comparisonto discover the presence of an item. We will see, however, that this istypically not the case.

A hash table is a collection of items which are stored in such a wayas to make it easy to find them later. Each position of the hash table,often called a slot, can hold an item and is named by an integervalue starting at 0. For example, we will have a slot named 0, a slotnamed 1, a slot named 2, and so on. Initially, the hash table containsno items so every slot is empty. We can implement a hash table by usinga list with each element initialized to the special Python valueNone. Figure 4 shows a hash table of size (m=11).In other words, there are m slots in the table, named 0 through 10.

The mapping between an item and the slot where that item belongs in thehash table is called the hash function. The hash function will takeany item in the collection and return an integer in the range of slotnames, between 0 and m-1. Assume that we have the set of integer items54, 26, 93, 17, 77, and 31. Our first hash function, sometimes referredto as the “remainder method,” simply takes an item and divides it by thetable size, returning the remainder as its hash value((h(item)=item % 11)). Table 4 gives all of thehash values for our example items. Note that this remainder method(modulo arithmetic) will typically be present in some form in all hashfunctions, since the result must be in the range of slot names.

**Table 4: Simple Hash Function Using Remainders**¶
Item	Hash Value
54	10
26	4
93	5
17	6
77	0
31	9

Once the hash values have been computed, we can insert each item intothe hash table at the designated position as shown inFigure 5. Note that 6 of the 11 slots are now occupied. Thisis referred to as the load factor, and is commonly denoted by(lambda = frac {numberofitems}{tablesize}). For this example,(lambda = frac {6}{11}).

Figure 5: Hash Table with Six Items¶

Now when we want to search for an item, we simply use the hash functionto compute the slot name for the item and then check the hash table tosee if it is present. This searching operation is (O(1)), sincea constant amount of time is required to compute the hash value and thenindex the hash table at that location. If everything is where it shouldbe, we have found a constant time search algorithm.

You can probably already see that this technique is going to work onlyif each item maps to a unique location in the hash table. For example,if the item 44 had been the next item in our collection, it would have ahash value of 0 ((44 % 11 0)). Since 77 also had a hashvalue of 0, we would have a problem. According to the hash function, twoor more items would need to be in the same slot. This is referred to asa collision (it may also be called a “clash”). Clearly, collisionscreate a problem for the hashing technique. We will discuss them indetail later.

6.5.1. Hash Functions¶

Given a collection of items, a hash function that maps each item into aunique slot is referred to as a perfect hash function. If we knowthe items and the collection will never change, then it is possible toconstruct a perfect hash function (refer to the exercises for more aboutperfect hash functions). Unfortunately, given an arbitrary collection ofitems, there is no systematic way to construct a perfect hash function.Luckily, we do not need the hash function to be perfect to still gainperformance efficiency.

One way to always have a perfect hash function is to increase the sizeof the hash table so that each possible value in the item range can beaccommodated. This guarantees that each item will have a unique slot.Although this is practical for small numbers of items, it is notfeasible when the number of possible items is large. For example, if theitems were nine-digit Social Security numbers, this method would requirealmost one billion slots. If we only want to store data for a class of25 students, we will be wasting an enormous amount of memory.

Our goal is to create a hash function that minimizes the number ofcollisions, is easy to compute, and evenly distributes the items in thehash table. There are a number of common ways to extend the simpleremainder method. We will consider a few of them here.

The folding method for constructing hash functions begins bydividing the item into equal-size pieces (the last piece may not be ofequal size). These pieces are then added together to give the resultinghash value. For example, if our item was the phone number 436-555-4601,we would take the digits and divide them into groups of 2(43,65,55,46,01). After the addition, (43+65+55+46+01), we get210. If we assume our hash table has 11 slots, then we need to performthe extra step of dividing by 11 and keeping the remainder. In this case(210 % 11) is 1, so the phone number 436-555-4601 hashes toslot 1. Some folding methods go one step further and reverse every otherpiece before the addition. For the above example, we get(43+56+55+64+01 = 219) which gives (219 % 11 = 10).

Another numerical technique for constructing a hash function is calledthe mid-square method. We first square the item, and then extractsome portion of the resulting digits. For example, if the item were 44,we would first compute (44 ^{2} = 1,936). By extracting themiddle two digits, 93, and performing the remainder step, we get 5((93 % 11)). Table 5 shows items under both theremainder method and the mid-square method. You should verify that youunderstand how these values were computed.

**Table 5: Comparison of Remainder and Mid-Square Methods**¶
Item	Remainder	Mid-Square
54	10	3
26	4	7
93	5	9
17	6	8
77	0	4
31	9	6

We can also create hash functions for character-based items such asstrings. The word “cat” can be thought of as a sequence of ordinalvalues.

We can then take these three ordinal values, add them up, and use theremainder method to get a hash value (see Figure 6).Listing 1 shows a function called

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