A collision occurs when two keys are mapped to the same index in a hash table. Gener- ally, there are two ways for handling collisions: open addressing and separate chaining.

Open addressing is the process of finding an open location in the hash table in the event of a collision. Open addressing has several variations: linear probing, quadratic probing, and double hashing.

Linear Probing

When a collision occurs during the insertion of an entry to a hash table, linear probing finds the next available location sequentially. For example, if a collision occurs at hashTable[k % N], check whether hashTable[(k+1) % N] is available. If not, check hashTable[(k+2)% N] and so on, until an available cell is found,

To search for an entry in the hash table, obtain the index, say k, from the hash function for the key. Check whether hashTable[k % N] contains the entry. If not, check whether hashTable[(k+1) % N] contains the entry, and so on, until it is found, or an empty cell is reached.

To remove an entry from the hash table, search the entry that matches the key. If the entry is found, place a special marker to denote that the entry is available. Each cell in the hash table has three possible states: occupied, marked, or empty. Note a marked cell is also available for insertion.

Linear probing tends to cause groups of consecutive cells in the hash table to be occupied. Each group is called a cluster. Each cluster is actually a probe sequence that you must search when retrieving, adding, or removing an entry. As clusters grow in size, they may merge into even larger clusters, further slowing down the search time. This is a big disadvantage of linear probing.

Quadratic Probing

Quadratic probing can avoid the clustering problem that can occur in linear probing. Linear probing looks at the consecutive cells beginning at index k. Quadratic probing, on the other hand, looks at the cells at indices (k + j²) , N, for j Ú 0, that is, k , N, (k + 1) , N, (k + 4) , n, (k + 9) , N, and so on, as shown below

Quadratic probing works in the same way as linear probing except for a change in the search sequence. Quadratic probing avoids linear probing’s clustering problem, but it has its own clustering problem, called secondary clustering; that is, the entries that collide with an occu- pied entry use the same probe sequence.

Linear probing guarantees that an available cell can be found for insertion as long as the table is not full. However, there is no such guarantee for quadratic probing.

Double Hashing

Another open addressing scheme that avoids the clustering problem is known as double hashing. Starting from the initial index k, both linear probing and quadratic probing add an increment to k to define a search sequence. The increment is 1 for linear probing and j2 for quadratic probing. These increments are independent of the keys. Double hashing uses a secondary hash function h‘(key) on the keys to determine the increments to avoid the clustering problem. Specifically, double hashing looks at the cells at indices (k + j * h‘(key)) , N, for j Ú 0, that is, k , N, (k + h‘(key)) , N, (k + 2 * h‘(key)) , N, (k + 3 * h‘(key)) , N, and so on.

For example, let the primary hash function h and secondary hash function h’ on a hash table of size 11 be defined as follows:

h(key) = key %  11; h'(key) = 7 – key % 7;
For a search key of 12, we have
h(12) = 12 % 11 = 1;
h'(12) = 7 – 12 % 7 = 2;

Suppose the elements with the keys 45, 58, 4, 28, and 21 are already placed in the hash table as shown in Figure 27.6. We now insert the element with key 12. The probe sequence for key 12 starts at index 1. Since the cell at index 1 is already occupied, search the next cell at index 3 (1 +

• * 2). Since the cell at index 3 is already occupied, search the next cell at index 5 (1 +
• * 2). Since the cell at index 5 is empty, the element for key 12 is now inserted at this

The indices of the probe sequence are as follows: 1, 3, 5, 7, 9, 0, 2, 4, 6, 8, 10. This sequence reaches the entire table. You should design your functions to produce a probe sequence that reaches the entire table. Note the second function should never have a zero value, since zero is not an increment.