Question

Give a pseudo-code description of an insertion into a hash table that uses quadratic probing to resolve collisions, assuming we also use the trick of replacing deleted items with a special "deactivated item" object.

Short answer

Initialize variables, compute initial index, use quadratic probing to find the correct slot, and insert the key-value pair.

Step-by-step solution

- Initialize Variables

Start by initializing the necessary variables. You need the hash table, the key-value pair to insert, and any constants needed for the quadratic probing.

- Compute Initial Hash

Compute the initial hash index for the key using the hash function: `index = hash(key) % table_size`

- Quadratic Probing Setup

Set up your quadratic probing variables: initialize `i` to 0, which represents the number of probes.

- Check Slot Status

While the current slot at `index` in the hash table is occupied and not deactivated, increment `i` and compute the next slot using quadratic probing: `index = (initial_index + i^2) % table_size`

- Insert Key-Value Pair

Once you find an empty or deactivated slot, insert the key-value pair at that slot.

- End Procedure

End the procedure as the insertion is complete.

Key concepts

hash tables

Hash tables are an essential data structure in computer science used for storing key-value pairs efficiently. The main idea is to use a hash function to compute an index (or hash) from the key, which determines where the key-value pair should be placed in an array. This allows for fast data retrieval because you can access elements in constant time, on average.

Each key in a hash table is unique. The hash function needs to distribute keys uniformly across the table to minimize collisions. An ideal hash function maps different keys to different locations, reducing the likelihood that multiple keys will share the same array index.

In real-world applications, perfect hash functions are rare. Thus, effective methods to handle collisions are crucial for maintaining the performance benefits of hash tables.

collision resolution

When two keys hash to the same index, a collision occurs. Collision resolution techniques are strategies to handle these situations without losing data. There are several common methods for resolving collisions:

• Chaining: Uses linked lists to store multiple elements at the same array index. When a collision occurs, the new element is added to the list at that index.
• Open Addressing: Searches for the next available slot in the array when a collision occurs. Types of open addressing include linear probing, quadratic probing, and double hashing.

Quadratic probing is a variation of open addressing where the interval between probes increases quadratically. This means that if the initial hash position is occupied, the algorithm looks at positions given by: initial_index + 1^2, initial_index + 2^2, initial_index + 3^2, and so on. This reduces clustering compared to linear probing.

quadratic probing

Quadratic probing helps resolve collisions in a hash table by examining slots at quadratic increments away from the initial hashed position. Here’s a step-by-step explanation of how it works, using the context of the provided exercise:

• Initialization: Prepare the necessary variables. These include the hash table, key-value pair to insert, and constants for probing.
• Compute Initial Hash: Use a hash function to find the initial array index:
  \[ index = \text{hash(key)} \bmod \text{table\textunderscore size} \]
• Set Up Probing: Initialize a variable, i, to count the number of probes. Start with \(i = 0\).
• Check Slot Status: While the slot at the current index is occupied (and not deactivated), increment \(i\) and compute the next slot:
  \[ \text{index} = (\text{initial\textunderscore index} + i^2) \bmod \text{table\textunderscore size} \]
• Insert Key-Value Pair: When an empty or deactivated slot is found, insert the key-value pair.
• End Procedure: The insertion process is complete.

This method avoids primary clustering by spreading out probes more broadly across the hash table, improving performance over linear probing.