Non-comparison sorting algorithms offer a fresh approach to organizing data. These methods, including counting sort, radix sort, and bucket sort, bypass traditional element comparisons. Instead, they leverage the inherent properties of the data to achieve efficient sorting.

These algorithms shine when dealing with integer arrays or data within specific ranges. By exploiting the distribution of elements, they can achieve linear time complexity in certain scenarios. This makes them powerful tools for sorting large datasets with known characteristics.

Non-Comparison Sorting Algorithms

Principles of non-comparison sorting

• Counting Sort
  • Assumes input elements are integers within a specific range $[0, k]$
  • Counts occurrences of each distinct element in the input array (histogram)
  • Uses count to determine position of each element in sorted output array
    • Enables direct placement of elements without comparisons
  • Suitable for sorting arrays with small range of distinct elements (exam scores, age data)
• Radix Sort
  • Sorts elements based on individual digits or radix (base of number system)
  • Performs stable sort on each digit, from least significant to most significant
    • Stable sort maintains relative order of elements with equal digits
  • Uses stable sorting algorithm (counting sort) as subroutine to sort each digit
  • Efficient for sorting integers or strings represented as sequences of digits or characters (zip codes, phone numbers)
• Bucket Sort
  • Distributes elements into set of buckets based on value range
    • Buckets represent subintervals of the input range (years, price ranges)
  • Sorts each bucket individually using another sorting algorithm or recursively applying bucket sort
  • Concatenates sorted buckets to obtain final sorted output
  • Suitable when input is uniformly distributed over a range (floating-point numbers between 0 and 1)

Implementation for integer arrays

• Counting Sort Implementation
  1. Create count array to store count of each distinct element
  2. Iterate through input array and increment count for each element
  3. Modify count array to store actual position of each element in sorted output
     • Each count represents the number of elements less than or equal to the current element
  4. Build sorted output array using modified count array
     • Traverse input array in reverse order to maintain stability
     • Place each element at its correct position in output array based on count
  5. Copy sorted output array back to original input array
• Radix Sort Implementation
  1. Determine maximum number of digits in input array
  2. For each digit position (starting from least significant):
     • Use counting sort to sort elements based on current digit
     • Maintain stability by considering previous digit positions
  3. Output sorted array after processing all digit positions
• Bucket Sort Implementation
  1. Determine number of buckets and range each bucket represents
     • Number of buckets typically chosen as square root of input size
  2. Distribute elements into respective buckets based on value
     • Elements with similar values are grouped together in buckets
  3. Sort each bucket individually using another sorting algorithm (insertion sort, quick sort)
  4. Concatenate sorted buckets to obtain final sorted output

Complexity of non-comparison algorithms

• Counting Sort
  • Time Complexity: $O(n + k)$
    • $n$: number of elements in input array
    • $k$: range of input values (difference between maximum and minimum values)
    • Iterates through input array once and count array once
  • Space Complexity: $O(n + k)$
    • Requires additional space to store count array and output array
    • Count array size depends on range of input values
• Radix Sort
  • Time Complexity: $O(d \cdot (n + k))$
    • $d$: number of digits in the maximum element
    • $n$: number of elements in input array
    • $k$: range of digit values (radix)
    • Performs counting sort for each digit position
  • Space Complexity: $O(n + k)$
    • Requires additional space to store output array and temporary count array for each digit
    • Count array size depends on radix (10 for decimal, 256 for ASCII)
• Bucket Sort
  • Time Complexity: $O(n + k)$ on average
    • $n$: number of elements in input array
    • $k$: number of buckets
    • Assumes elements are uniformly distributed across buckets
    • Sorting each bucket takes $O(n_i \log n_i)$ time, where $n_i$ is size of $i$-th bucket
  • Space Complexity: $O(n + k)$
    • Requires additional space to store buckets and output array
    • Bucket size depends on distribution of input elements

Non-comparison vs comparison-based sorting

• Advantages of Non-Comparison Sorting Algorithms
  • Achieves linear time complexity for integer sorting ($O(n + k)$)
    • Outperforms comparison-based algorithms ($O(n \log n)$ lower bound)
  • Efficient for sorting arrays with small range of distinct elements
    • Counting sort suitable for arrays with limited range (grades, ages)
  • Handles large datasets that fit within specific range
    • Radix sort effective for sorting large integers or strings
• Limitations of Non-Comparison Sorting Algorithms
  • Requires prior knowledge of range of input elements
    • Counting sort and bucket sort need to determine range beforehand
  • Higher space complexity compared to comparison-based algorithms
    • Additional space needed for count array, buckets, or output array
  • Not suitable for sorting non-integer or complex data types without modification
    • Requires mapping elements to integers or adapting the algorithm
  • Performance degrades when range of elements is large compared to number of elements
    • Large range leads to increased space and time complexity
  • May not be in-place, requiring additional space for output array
    • In-place sorting is often preferred to minimize space usage