11. Hashing

Overview

Hashing transforms input data into a fixed-size string of bytes.

• Improves the speed of searching

Conceptual

1. Digit Addition

   • cats is stored at $3 + 1 + 20 + 19 = 43$
   • zzzzzzzzzzz is stored at $26 * 10 = 260$
     • Approximately $50,000/260$ words at the same index

2. Polynomial Hashcode

   • cats is stored at $327^3 + 127^2 + 2027^1 + 1927^0 = 60,337$
   • zzzzzzzzzzz is stored at $2627^9 + 2627^8 + 2627^7 + \dots + 2627^0$
     • Such array is impossible - Integer overflow
     • Most array indices are empty - Waste of memory

3. Modulus (%)

   • Information loss
   • Collisions are inevitable due to the Pigeonhole Principle

Workarounds for Collisions

Load factor ($\alpha$)

• Performance degrades seriously if beyond two-thirds full

$$ \alpha = \frac{n}{N} = \frac{\text{number of items}}{\text{length of the array}}$$

1. Open Addressing

• Linear Probing: searches sequentially through the table until an empty slot is found.

  • Problem: Primary Clustering

• Quadratic Probing: searches using a quadratic function.

  • Problem: Secondary Clustering

• Double Hashing:

  • Hash function 1 $\rightarrow$ Initial index
  • Hash function 2 $\rightarrow$ Step size

2. Separate Chaining (Closed Addressing)

• Separate Chaining creates a linked list at each slot to store all elements that hash to that particular slot.
  • Load factor ($\alpha$) can rise above 1

Rehashing

• Rehashing: creates a new array twice as large and rehashes every item.
  • Amortized constant time

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