Hash Codes

Whenever equality is redefined for a type, the hash code computation for that type needs to be redefined in a consistent way. This is done by overriding that type’s GetHashCode method. In order for hashing to be implemented correctly and efficiently, this method should satisfy the following goals:

• Equal keys must have the same hash code. This is necessary in order for the Dictionary<TKey, TValue> class to be able to find a given key that it has stored. On the other hand, because the number of possible keys is usually larger than the number of possible hash codes, unequal keys are also allowed to have the same hash code.
• The computation should be done quickly.
• Hash codes should be uniformly distributed even if the keys are not.

The last goal above may seem rather daunting, particularly in light of our desire for a quick computation. In fact, it is impossible to guarantee in general — provided there are more than $ 2^{32}(k - 1) $ possible keys from which to choose, no matter how the hash code computation is implemented, we can always find at least $ k $ keys with the same hash code. However, this is a problem that has been studied a great deal, and several techniques have been developed that are effective in practice. We will caution, however, that not every technique that looks like it should be effective actually is in practice. It is best to use techniques that have been demonstrated to be effective in a wide variety of applications. We will examine one of these techniques in what follows.

A guiding principle in developing hashing techniques is to use all information available in the key. By using all the information, we will be sure to use the information that distinguishes this key from other keys. This information includes not just the values of the individual parts of the the key, but also the order in which they occur, provided this order is significant in distinguishing unequal keys. For example, the strings, “being” and “begin” contain the same letters, but are different because the letters occur in a different order.

One specific technique initializes the hash code to 0, then processes the key one component at a time. These components may be bytes, characters, or other parts no larger than 32 bits each. For example, for the Nim board positions discussed in “Memoization”, the components would be the number of stones on each pile, the limit for each pile, and the total number of piles (to distinguish between a board ending with empty piles and a board with fewer piles). For each component, it does the following:

• Multiply the hash code by some fixed odd integer.
• Add the current component to the hash code.

Due to the repeated multiplications, the above computation will often overflow an int. This is not a problem — the remaining bits are sufficient for the hash code.

In order to understand this computation a little better, let’s first ignore the effect of this overflow. We’ll denote the fixed odd integer by $ x $, and the components of the key as $ k_1, \dots, k_n $. Then this is the result of the computation:

Because the above is a polynomial, this hashing scheme is called polynomial hashing. While the computation itself is efficient, performing just a couple of arithmetic operations on each component, the result is to multiply each component by a unique value ( $ x^i $ for some $ i $) depending on its position within the key.

Now let’s consider the effect of overflow on the above polynomial. What this does is to keep only the low-order 32 bits of the value of the polynomial. Looking at it another way, we end up multiplying $ k_i $ by only the low-order 32 bits of $ x^{n-i} $. This helps to explain why $ x $ is an odd number — raising an even number to the $ i $th power forms a number ending in $ i $ 0s in binary. Thus, if there are more than 32 components in the key, all but the last 32 will be multiplied by $ 0 $, and hence, ignored.

There are other potential problems with using certain odd numbers for $ x $. For example, we wouldn’t want to use $ 1 $, because that would result in simply adding all the components together, and we would lose any information regarding their positions within the key. Using $ -1 $ would be almost as bad, as we would multiply all components in odd positions by $ -1 $ and all components in even positions by $ 1 $. The effect of overflow can cause similar behavior; for example, if we place $ 2^{31} - 1 $ in an int variable and square it, the overflow causes the result to be 1. Successive powers will then alternate between $ 2^{31} - 1 $ and $ 1 $.

It turns out that this cyclic behavior occurs no matter what odd number we use for $ x $. However, in most cases the cycle is long enough that keys of a reasonable size will have each component multiplied by a unique value. The only odd numbers that result in short cycles are those that are adjacent to a multiple of a large power of $ 2 $ (note that $ 0 $ is a multiple of any integer).

The other potential problem occurs when we are hashing fairly short keys. In such cases, if $ x $ is also small enough, the values computed will all be much smaller than the maximum possible integer value $ (2^{31} - 1) $. As a result, we will not have a uniform distribution of values. We therefore want to avoid making $ x $ too small.

Putting all this together, choosing $ x $ to be an odd number between $ 30 $ and $ 40 $ works pretty well. These values are large enough so that seven key components will usually overflow an int. Furthermore, they all have a cycle length greater than $ 100 $ million.

We should always save the hash code in a private field after we compute it so that subsequent requests for the same hash code don’t result in repeating the computation. This can be done in either of two ways. One way is to compute it in an eager fashion by doing it in the constructor. When doing it this way, the GetHashCode method simply needs to return the value of the private field. While this is often the easiest way, it sometimes results in computing a hash code that we end up not using. The alternative is to compute it in a lazy fashion. This requires an extra private field of type bool. This field is used to indicate whether the hash code has been computed yet or not. With this approach, the GetHashCode method first checks this field to see if the hash code has been computed. If not, it computes the hash code and saves it in its field. In either case, it then returns the value of the hash code field.