Hash functions: An empirical comparison

Hash tables are popular data structures for storing key-value pairs. A hash function is used to map the key value (usually a string) to array index. The functions are different from cryptographic hash functions, because they should be much faster and don't need to be resistant to preimage attack. Hashing in large databases is also left out from this article; the benchmark includes medium-size hash tables such as:

• symbol table in a parser,
• IP address table for filtering network traffic,
• the dictionary in a word counting program or a spellchecker.

There are two classes of the functions used in hash tables:

• multiplicative hash functions, which are simple and fast, but have a high number of collisions;
• more complex functions, which have better quality, but take more time to calculate.

Hash table benchmarks usually include theoretical metrics such as the number of collisions or distribution uniformity (see, for example, hash function comparison in the Red Dragon book). Obviously, you will have a better distribution with more complex functions, so they are winners in these benchmarks.

The question is whether using complex functions gives you a faster program. The complex functions require more operations per one key, so they can be slower. Is the price of collisions high enough to justify the additional operations?

Multiplicative hash functions

Any multiplicative hash function is a special case of the following algorithm:

UINT HashMultiplicative(const CHAR *key, SIZE_T len) {
 UINT hash = INITIAL_VALUE;
 for(UINT i = 0; i < len; ++i)
 hash = M * hash + key[i];
 return hash % TABLE_SIZE;
}

(Sometimes XOR operation is used instead of addition, but it does not make much difference.) The hash functions differ only by values of INITIAL_VALUE and multiplier (M). For example, the popular Bernstein's function uses INITIAL_VALUE of 5381 and M of 33; Kernighan and Ritchie's function uses INITIAL_VALUE of 0 and M of 31.

A multiplicative function works by adding together the letters weighted by powers of multiplier. For example, the hash for the word TONE will be:

 INITIAL_VALUE * M^4 + 'T' * M^3 + 'O' * M^2 + 'N' * M + 'E'

Let's enter several similar strings and watch the output of the functions:

 Bernstein Kernighan
 (M=33) (M=31)
 too b88af17 1c154
 top b88af18 1c155
 tor b88af1a 1c157
 tpp b88af39 1c174
a000 7c9312d6 2cd22f
a001 7c9312d7 2cd230
a002 7c9312d8 2cd231
a003 7c9312d9 2cd232
a004 7c9312da 2cd233
a005 7c9312db 2cd234
a006 7c9312dc 2cd235
a007 7c9312dd 2cd236
a008 7c9312de 2cd237
a009 7c9312df 2cd238
a010 7c9312f7 2cd24e
 a 2b606 61
 aa 597727 c20
 aaa b885c68 17841

Too and top are different in the last letter only. The letter P is the next one after O, so the values of hash function are different by 1 (1c154 and 1c155, b88af17 and b88af18). Ditto for a000..a009.

Now let's compare top with tpp. Their hashes will be:

 INITIAL_VALUE * M^3 + 'T' * M^2 + 'O' * M + 'P'
 INITIAL_VALUE * M^3 + 'T' * M^2 + 'P' * M + 'P'

The hashes will be different by M * ('P' - 'O') = M. Similarly, when the first letters are different by x, their hashes will be different by x * M^2.

When there are less than 33 possible letters, Bernstein's function will pack them into a number (similar to Radix40 packing scheme). For example, hash table of size 33³ will provide perfect hashing (without any collisions) for all three-letter English words written in small letters. In practice, the words are longer and hash tables are smaller, so there will be some collisions (situations when different strings have the same hash value).

If the string is too long to fit into the 32-bit number, the first letters will still affect the value of the hash function, because the multiplication is done modulo 2^32 (in a 32-bit register), and the multiplier is chosen to have no common divisors with 2^32 (in other words, it must be odd), so the bits will not be just shifted away.

There are no exact rules for choosing the multiplier, only some heuristics:

• the multiplier should be large enough to accommodate most of the possible letters (e.g., 3 or 5 is too small);
• the multiplier should be fast to calculate with shifts and additions [e.g., 33 * hash can be calculated as (hash << 5) + hash];
• the multiplier should be odd for the reason explained above;
• prime numbers are good multipliers.

Complex hash functions

These functions do a good job of mixing together the bits of the source word. The change in one input bit changes a half of the bits in the output (see Avalanche_effect), so the result looks completely random:

 Paul Hsieh One At Time
 too 3ad11d33 3a9fad1e 
 top 78b5a877 4c5dd09a 
 tor c09e2021 f2aa9d35 
 tpp 3058996d d5e9e480 
a000 7552599f ed3859d8 
a001 3cc1d896 fef7fd57 
a002 c6ff5c9b 08a610b3 
a003 dcab7b0c 1a88b478 
a004 780c7202 3621ebaa 
a005 7eb63e3a 47db8f1d 
a006 6b0a7a17 b901717b 
a007 cb5cb1ab caec1550 
a008 5c2a15c0 e58d4a92 
a009 33339829 f75aee2d 
a010 eb1f336e bd097a6b 
 a 115ea782 ca2e9442 
 aa 008ad357 7081738e 
 aaa 7dfdc310 ae4f22ec

To achieve this behavior, the hash functions perform a lot of shifts, XORs, and additions. But do we need a complex function? What is faster: tolerating the collisions and resolving them with chaining, or avoiding them with a more complex function?

Test conditions

The benchmark uses separate chaining algorithm for collision resolution. Memory allocation and other "heavy" functions were excluded from the benchmarked code. The RDTSC instruction was used for benchmarking. The test was performed on Pentium-M and Core i5 processors.

The benchmark inserts some keys in the table, then looks them up in the same order as they were inserted. The test data include:

• the list of common words from Wiktionary (500 items);
• the list of Win32 functions from Colorer syntax highlight scheme (1992 items);
• 500 names from a000 to a499 (imitates the names in auto-generated source code);
• the list of common words with a long prefix and postfix;
• all variable names from WordPress 2.3.2 source code in wp-includes folder (1842 names);
• list of all words in Sonnets by W. Shakespeare (imitates a word counting program; 3228 words);
• list of all words in La Peau de chagrin by Balzac (in French, UTF-8 encoding);
• search engine IP addresses (binary).

Results

Core i5 processor

Pentium-M processor

Each cell includes the execution time, then the number of collisions in square brackets. Execution time is expressed in thousands of clock cycles (a lower number is better). Avg column contains the average normalized execution time (and the number of collisions).

The function by Kernighan and Ritchie is from their famous book "The C programming Language", 3rd edition; Weinberger's hash and the hash with multiplier 65599 are from the Red Dragon book. The latter function is used in gawk, sdbm, and other Linux programs. x17 is the function by Peter Kankowski (multiplier = 17; 32 is subtracted from each letter code).

As you can see from the table, the function with the lowest number of collisions is not always the fastest one.

Results on a large data set (list of all words in English Wikipedia, 12.5 million words, from the benchmark by Georgi 'Sanmayce'):

Core i5 processor

Pentium-M processor

• Adler-32 (slow filling, not suitable as a hash function);
• TwoChars (bad for machine-generated names and variable names that are similar to each other, disastrous for large data sets such as Wikipedia).

The number of collisions depending on the hash table size (for the same data set, thanks to Ace for the idea):

Red Dragon Book proposes the following formula for evaluating hash function quality:

where b_j is the number of items in j-th slot, m is the number of slots, and n is the total number of items. The sum of b_j(b_j + 1) / 2 estimates the number of slots your program should visit to find the required value. The denominator (n / 2m)(n + 2m − 1) is the number of visited slots for an ideal function that puts each item into a random slot. So, if the function is ideal, the formula should give 1. In reality, a good function is somewhere between 0.95 and 1.05. If it's more, there is a high number of collisions (slow!). If it's less, the function gives less collisions than the randomly distributing function, which is not bad.

Here are the results for some of our functions:

Conclusion

Complex functions by Paul Hsieh and Bob Jenkins are tuned for long keys, such as the ones in postfix and prefix tests. Note that they do not provide the best number of collisions for these tests, but do have the best time, which means that the functions are faster than the others because of loop unrolling. At the same time, they are suboptimal for short keys (words and sonnets tests).

For a word counting program, a compiler, or another application that typically handles short keys, it's often advantageous to use a simple multiplicative function such as x17 or Larson's hash. However, these functions perform badly on long keys.

Novak showed bad results on the large data set. Jesteress has a high number of collisions in numbers test.

Murmur2, Meiyan, SBox, and CRC32 provide good performance for all kinds of keys. They can be recommended as general-purpose hashing functions on x86.

Hardware-accelerated CRC (labeled iSCSI CRC in the table) is the fastest hash function on the recent Core i5/i7 processors. However, the CRC32 instruction is not supported by AMD and earlier Intel processors.

Download the source code (152 KB, MSVC++)

Variations

XORing high and low part

For table size less than 2^16, we can improve the quality of hash function by XORing high and low words, so that more letters will be taken into account:

 return hash ^ (hash >> 16);

Subtracting a constant

x17 hash function subtracts a space from each letter to cut off the control characters in the range 0x00..0x1F. If the hash keys are long and contain only Latin letters and numbers, the letters will be less frequently shifted out, and the overall number of collisions will be lower. You can even subtract 'A' when you know that the keys will be only English words.

Using larger multipliers for a compiler

Paul Hsieh noted that large multipliers may provide better results for the hash table in a compiler, because a typical source code contains a lot of one-letter variable names (i, j, s, etc.), and they will collide if the multiplier is less than the number of letters in the alphabet.

The test confirms this assumption: the function by Kernighan & Ritchie (M = 33) has lower number of collisions than x17 (M = 17), but the latter is still faster (see Variables column in the table above).

Setting hash table size to a prime number

A test showed that the number of collisions will usually be lower if you use a prime, but the calculations modulo prime take much more time than the calculations for a power of 2, so this method is impractical. Even replacing division with multiplication by reciprocal values do not help here:

Implementing open addressing vs. separate chaining

With open addressing, most hash functions show awkward clustering behavior in "Numbers" test:

You can avoid the worst case by using chaining for collision resolution. However, chaining requires more memory for the next item pointers, so the performance improvement does not come for free. A custom memory allocator should be usually written, because calling malloc() for a large number of small structures is suboptimal.

Some implementations (e.g., hash table in Python interpreter) store a full 32-bit hash with the item to speed up the string comparison, but this is less effective than chaining.

About the author

Peter is the developer of Aba Search and Replace, a tool for replacing text in multiple files. He likes to program in C with a bit of C++, also in x86 assembly language, Python, and PHP.

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