Data Structures and Algorithms with Object-Oriented Design Patterns in C#

## Character String Keys

Strings of characters are represented in C# as instances of the String class. A character string is simply a sequence of characters. Since such a sequence may be arbitrarily long, to devise a suitable hash function we must find a mapping from an unbounded domain into the finite range of int.

We can view a character string, s, as a sequence of n characters,

where n is the length of the string. (The length of a string can be determined using the String property Length). One very simple way to hash such a string would be to simply sum the numeric values associated with each character:

As it turns out, this is not a particularly good way to hash character strings. Given that a C# char is a 16-bit quantity, , for all . As a result, . For example, given a string of length n=5, the value of f(s) falls between zero and . In fact, the situation is even worse, in North America we typically use only the ASCII  subset of the Unicode  character set. The ASCII character set uses only the least-significant seven bits of a char. If the string is comprised of only ASCII characters, the result falls in the range between zero and 640.

Essentially the problem with a function f which produces a result in a relatively small interval is the situation which arises when that function is composed with the function . If the size of the range of the function f is less than M, then does not spread its values uniformly on the interval [0,M-1]. For example, if M=1031 only the first 640 values (62% of the range) are used!

Alternatively, suppose we have a priori knowledge that character strings are limited to length n=4. Then, we can construct an integer by concatenating the binary representations of each of the characters. For example, given , we can construct an integer with the function

where . Since B is a power of two, this function is easy to write in C#:

```static int F(String s)
{
return (int)s[0] << 21 | (int)s[1] << 14
| (int)s[2] << 7 | (int)s[3];
}```
While this function certainly has a larger range, it still has a problems--it cannot deal strings of arbitrary length.

Equation  can be generalized to deal with strings of arbitrary length as follows:

This function produces a unique integer for every possible string. Unfortunately, the range of f(s) is unbounded. A simple modification of this algorithm suffices to bound the range:

where such that w is word size of the machine. Unfortunately, since W and B are both powers of two, the value computed by this hash function depends only on the last W/B characters in the character string. For example, for and , this result depends only on the last five characters in the string--all character strings having exactly the same last five characters collide!

Writing the code to compute Equation  is actually quite straightforward if we realize that f(s) can be viewed as a polynomial in B, the coefficients of which are , , ..., . Therefore, we can use Horner's rule  (see Section ) to compute f(s) as follows:

```static int F(string s)
{
int result = 0;
for (int i = 0; i < s.Length; ++i)
result = result * B + (int)s[i];
return result;
}```
This implementation can be simplified even further if we make use of the fact that , where b=7. Since B is a power of two, in order to multiply the variable result by B all we need to do is to shift it left by b bits. Furthermore, having just shifted result left by b bits, we know that the least significant b bits of the result are zero. And since each character has no more than b=7 bits, we can replace the addition operation with an exclusive or  operation.
```static int F(String s)
{
int result = 0;
for (int i = 0; i < s.Length; ++i)
result = result << b ^ (int)s[i];
return result;
}```

Of the 128 characters in the 7-bit ASCII character set, only 97 characters are printing characters including the space, tab, and newline characters (see Appendix ). The remaining characters are control characters which, depending on the application, rarely occur in strings. Furthermore, if we assume that letters and digits are the most common characters in strings, then only 62 of the 128 ASCII codes are used frequently. Notice, the letters (both upper and lower case) all fall between and . All the information is in the least significant six bits. Similarly, the digits fall between and --these differ in the least significant four bits. These observations suggest that using should work well. That is, for , the hash value depends on the last five characters plus two bits of the sixth-last character.

We have developed a hashing scheme which works quite well given strings which differ in the trailing letters. For example, the strings "temp1", "temp2", and "temp3", all produce different hash values. However, in certain applications the strings differ in the leading letters. For example, the two Internet domain names  "ece.uwaterloo.ca" and "cs.uwaterloo.ca" collide when using Equation . Essentially, the effect of the characters that differ is lost because the corresponding bits have been shifted out of the hash value.

Program: ComparableString class GetHashCode method.

This suggests a final modification which shown in Program . Instead of losing the b=6 most significant bits when the variable result is shifted left, we retain those bits and exclusive or  them back into the shifted result variable. Using this approach, the two strings "ece.uwaterloo.ca" and "cs.uwaterloo.ca" produce different hash values.

Table  lists a number of different character strings together with the hash values obtained using Program . For example, to hash the string "fyra", the following computation is performed (all numbers in octal):

 1 4 6 `f` 1 7 1 `y` 1 6 2 `r` 1 4 1 `a` 1 4 7 7 0 6 3 4 1

 x Hash(x) (octal) "ett" 01446564 "två" 01656545 "tre" 01656345 "fyra" 0147706341 "fem" 01474455 "sex" 01624470 "sju" 01625365 "åtta" 0344656541 "nio" 01575057 "tio" 01655057 "elva" 044556741 "tolv" 065565566