Question

Describe a radix-sort method for lexicographically sorting a sequence \(S\) of triplets \((k, l, m),\) where \(k, l,\) and \(m\) are integers in the range \([0, N-1],\) for some \(N \geq 2 .\) How could this scheme be extended to sequences of \(d\) -tuples \(\left(k_{1}, k_{2}, \ldots, k_{d}\right),\) where each \(k_{i}\) is an integer in the range \([0, N-1] ?\)

Short answer

Sort triplets by \( m \), then \( l \), and finally \( k \. Extend method to \( d \)-tuples by sorting from least to most significant position.

Step-by-step solution

Understanding Radix Sort Basics

Radix sort is a non-comparative sorting algorithm that processes each digit of the numbers. It sorts numbers digit by digit starting from the least significant digit (LSD) to the most significant digit (MSD).

Identify the Digits/Positions

Triplets \( (k, l, m) \) can be broken down into three positions: \( k \), \( l \), and \( m \). In radix sort, treat each position individually.

Sort by Least Significant Digit

Start by sorting the sequence based on the least significant digit \( m \). Use counting sort (or any stable sorting algorithm) for this step.

Sort by Second Least Significant Digit

Next, sort the sequence based on the second least significant digit \( l \). Once again, use counting sort to ensure stability.

Sort by Most Significant Digit

Finally, sort the sequence based on the most significant digit \( k \). Ensure stability by using counting sort.

Extending to d-tuples

For a \( d \)-tuple \( (k_1, k_2, \ldots, k_d) \), apply the same method. Start by sorting according to \( k_d \) and proceed to sort incrementally to \( k_1 \).

Key concepts

Lexicographic Sorting

Lexicographic sorting refers to the method of sorting sequences in dictionary order. This is similar to how words are listed in a dictionary. In radix sort, this principle is applied to sort sequences like triplets \( (k, l, m) \) or any d-tuple \( (k_1, k_2, \ldots, k_d) \). Each element in the sequence is treated as an individual position and sorted in a defined order.

For example, in a sequence of triplets \( (2, 1, 3), (1, 3, 2), (2, 2, 1) \), we first look at the last digits (3, 2, 1) and sort them. Then we look at the second-to-last digits, and so on, till we reach the first digits. This ensures that our sequence is sorted in lexicographic order, which means treating the sequence as if it were a string of characters sorted from left to right.

Non-Comparative Sorting Algorithm

Radix Sort is notable for being a non-comparative sorting algorithm. This means that it doesn’t compare the elements being sorted against each other to determine their order.

Instead, it processes the elements based on their positional values, and sorts them digit by digit. This can make radix sort more efficient than comparison-based algorithms like QuickSort or MergeSort. Typical steps in radix sort include:

• Identify the maximum number of digits in the longest number or sequence.
• Sort the numbers digit by digit starting from the least significant digit (LSD) to the most significant digit (MSD).
• Use a stable sorting algorithm like counting sort to maintain the order.

Because the steps don’t involve direct comparison, but rather positional sorting, radix sort can handle large datasets efficiently, particularly those with a lot of digits.

Stable Sorting

A sorting algorithm is described as stable when it preserves the relative ordering of records with equal keys. This means that if two elements are equal according to the sorting key, they remain in the same order in the output as they were in the input.

In the context of radix sort, stability is crucial. Each pass of the sorting algorithm must maintain the relative positions of elements sorted in previous passes. For example, after sorting by the least significant digit, subsequent sorts for higher significant digits should not perturb this order.

Typical stable sorting algorithms used in radix sort include:

• Counting Sort
• Bucket Sort

Using stable sorting at each significant digit ensures that the final result is correctly lexicographically sorted. For example, if two triplets \( (k, l, m) \) and \( (k', l', m') \) have \( m = m' \), the relative order based on \( k \) and \( l \) will be preserved in the final sorted sequence.