# FCH Algorithm

## The Algorithm

The algorithm is presented in [1].

## Memory Consumption

Now we detail the memory consumption to generate and to store minimal perfect hash functions using the FCH algorithm. The structures responsible for memory consumption are in the following:

- A vector containing all the n keys.
- Data structure to speed up the searching step:
  1. random_table: is a vector used to remember currently empty slots in the hash table. It stores n 4 byte long integer numbers. This vector initially contains a random permutation of the n hash addresses. A pointer called filled_count is used to keep the invariant that any slots to the right side of filled_count (inclusive) are empty and any ones to the left are filled.
  2. hash_table: Table used to check whether all the collisions were resolved. It has n entries of one byte.
  3. map_table: For any unfilled slot x in hash_table, the map_table vector contains n 4 byte long pointers pointing at random_table such that random_table[map_table[x]] = x. Thus, given an empty slot x in the hash_table, we can locate its position in the random_table vector through map_table.

- Other auxiliary structures
  1. sorted_indexes: is a vector of cn/(log(n) + 1) 4 byte long pointers to indirectly keep the buckets sorted by decreasing order of their sizes.

  2. function g: is represented by a vector of cn/(log(n) + 1) 4 byte long integer numbers, one for each bucket. It is used to spread all the keys in a given bucket into the hash table without collisions.

Thus, the total memory consumption of FCH algorithm for generating a minimal perfect hash function (MPHF) is: O(n) + 9n + 8cn/(log(n) + 1) bytes. The value of parameter c must be greater than or equal to 2.6.

Now we present the memory consumption to store the resulting function. We only need to store the g function and a constant number of bytes for the seed of the hash functions used in the resulting MPHF. Thus, we need cn/(log(n) + 1) + O(1) bytes.

## Papers

1. E.A. Fox, Q.F. Chen, and L.S. Heath. A faster algorithm for constructing minimal perfect hash functions. In Proc. 15th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 266-273, 1992.

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Davi de Castro Reis

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