Hash Tables

What is strength of b-tree? Can we make an array to be as fast search and insert as B-tree and LL?

Introduction of hash table Data structure that offers very fast insertion and searching, almost O(1). Relatively easy to program as compared to trees. Based on arrays, hence difficult to expand. No convenient way to visit the items in a hash table in any kind of order.

Hashing A range of key values can be transformed into a range of array index values. A simple array can be used where each record occupies one cell of the array and the index number of the cell is the key value for that record. But keys may not be well arranged. In such a situation hash tables can be used.

Converting Words to Numbers Adding the digits :- Add the code numbers for each character. E.g. cats: c = 3, a = 1, t = 20, s = 19, gives 43. What if, the Total range of word codes is from 1 to 260. 50,000 words exist. No enough index numbers. Multiplying by powers :- Decompose a word into its letters. Convert the letters to their numerical equivalents. Multiply them by appropriate powers of 27 and add the results. E.g. Leangsuksun = much larger than 260

Hash Function Need to compress the huge range of numbers. arrayIndex = hugenumber % smallRange; This is a hash function. It hashes a number in a large range into a number in a smaller range, corresponding to the index numbers in an array. An array into which data is inserted using a hash function later is called a hash table.

Collisions Two words can hash to the same array index, resulting in collision. Open Addressing: Search the array in some systematic way for an empty cell and insert the new item there if collision occurs. Separate chaining: Create an array of linked list of words, so that the item can be inserted into the linked list if collision occurs.

Open Addressing Three methods to find next vacant cell: Linear Probing :- Search sequentially for vacant cells, incrementing the index until an empty cell is found. Clustering is a problem occurring in linear probing. As the array gets full, clusters grow larger, resulting in very long probe lengths. Array can be expanded if it becomes too full.

Quadratic Probing load factor = nItems / arraySize; If load factor isn’t high, clusters can form. In quadratic probing more widely separated cells are probed. The step is the square of the step number. If index is x, the probe goes to x+1, x+4, x+9, x+16 and so on. Eliminates primary clustering, but all the keys that hash to a particular cell follow the same sequence in trying to find a vacant cell (secondary clustering).

Double Hashing Better solution. Generate probe sequences that depend on the key instead of being the same for every key. Hash the key a second time using a different hash function and use the result as the step size. Step size remains constant throughout a probe, but its different for different keys. Secondary hash function should not be the same as primary hash function. It must never output a zero. stepSize = constant – (key % constant); Requires that size of hash table is a prime number.

Separate Chaining No need to search for empty cells. The load factor can be 1 or greater. If there are more items on the lists access time is reduced. Deletion poses no problems. Table size is not a prime number. Arrays (buckets) can be used at each location in a hash table instead of a linked list.

Hash Functions A good hash function is simple and can be computed quickly. Speed degrades if hash function is slow. Purpose is to transform a range of key values into index values such that the key values are distributed randomly across all the indices of the hash table. Keys may be completely random or not so random.

Random Keys If the world were perfect, Evenly distributed NOT! A perfect hash function maps every key into a different table location. In most cases large number of keys are compressed into a smaller range of index numbers. Distribution of key values in a particular database determines what the hash function needs to be. For random keys: index = key % arraySize;

Non-random Keys Consider a number of the form 033-400-03-94-05-0-535. Every digit serves a purpose. The last 3 digits are redundant for error checking. These digits shouldn’t be considered. Every part of the remaining key should contribute to the data. Use a prime number for the modulo base.

Folding Break the key into groups of digits and add the groups. The number of digits in a group should correspond to the size of the array.

Hashing Efficiency Insertion and searching can approach O(1) time. If collision occurs, access time depends on the resulting probe lengths. Individual insert or search time is proportional to the length of the probe. This is in addition to a constant time for hash function. Relationship between probe length (P) and load factor (L) for linear probing : P = (1+1 / (1 – L2)) / 2 for successful search and P = (1 + 1 / (1 – L))/ 2

Hashing Efficiency Quadratic probing and Double Hashing share their performance equations. For successful hashing : -log2(1 - loadFactor) / loadFactor For an unsuccessful search :- 1 / (1 - loadFactor) Searching for separate chaining :- 1 + loadFactor /2 For unsuccessful search :- 1 + loadFactor For insertion :- 1 +loadfactor ?2 for ordered lists and 1 for unordered lists.

Open Addressing vs. Separate Chaining If open addressing is to be used, double hashing is preferred over quadratic probing. If plenty of memory is available and the data won’t expand, then linear probing is simpler to implement. If number of items to be inserted in hash table isn’t known, separate chaining is preferable to open addressing. When in doubt use separate chaining

External Storage Hash table can be stored in main memory. If it is too large it can be stored externally on disk, with only part of it being read into main memory at a time. In external hashing its important that the blocks do not become full. Even with a good hash function, the block might become full. This situation can be handled using variations of the collision-resolution schemes.