Tournament Sort

Tournament Sort is an efficient sorting algorithm that improves upon the ideas of selection sort and can be seen as a variant of heap sort. It organizes the input into a tournament tree structure—much like in sports competitions—to determine the order of elements.

Definition

Tournament Sort, sometimes called "tree selection sort," is an optimization of selection sort. It leverages a complete binary tree (similar to that used in heap sort) to quickly locate the next element that should be selected in the sorted order. The technique uses a priority queue structure to facilitate the comparisons, ensuring that each round identifies the smallest element among a set.

Concept

The name "Tournament Sort" originates from elimination tournaments. Imagine many competitors facing off in pairs, with the winner from each match advancing to the next round. In a tournament, even though the champion is clearly the best among all, the competitor eliminated in the final round isn’t necessarily the second best since they might have faced a particularly strong opponent earlier.

In sorting, the input array’s elements are treated like competitors. They are paired off, and in each round, the smaller (or larger, depending on the order) element wins its match and moves up in the tree. While the final winner is the smallest element overall, identifying the next smallest requires re-evaluating the tournament structure.

How It Works

Consider the process using a "minimum tournament tree" as an example. Imagine the following steps:

1. The unsorted elements are placed at the leaf nodes of a binary tree.
2. In each match (represented by a red edge in a diagram if you were to visualize it), the smaller element wins and moves up to the parent node.
3. After one complete "tournament," the winner (the smallest element) is found at the root.

The comparisons among $n$ elements yield about $\frac{n}{2}$ winners in the first round. If there is an unmatched element (when the count is odd), it advances to the next round without competition. The algorithm then removes the winning element (by conceptually replacing it with $\infty$) and conducts another tournament to find the next smallest element. This process repeats until the entire array is sorted.

Note: Although a visual diagram can be highly illustrative, you can imagine the tree as a set of match-ups where the path from a leaf to the root marks the sequence of wins leading to that element’s selection.

Key Properties

Stability

Tournament Sort is not a stable algorithm, meaning that the relative order of equal elements might not be preserved during sorting.

Time Complexity

Tournament Sort has an overall time complexity of $O(n \log n)$ in the best, average, and worst scenarios. The initial tournament tree setup takes $O(n)$ time while each removal of the smallest element takes $O(\log n)$ time.

Space Complexity

The algorithm requires additional space of $O(n)$, primarily for building and maintaining the tournament tree.

Implementation in Python

Below is a Python implementation of the Tournament Sort algorithm. This version uses a helper function to determine the winner in a pairwise comparison and then proceeds to build and update the tournament tree accordingly.

# Define maximum size and infinity constant as needed
MAXN = 1000
INF = float('inf')
 
n = 0
a = [0] * MAXN
tmp = [0] * (MAXN * 2)
 
 
def winner(pos1, pos2):
    """
    Compare two positions in the tournament tree and return the winning position.
    """
    # If the index exceeds the number of elements, use the index itself as a placeholder.
    u = pos1 if pos1 >= n else tmp[pos1]
    v = pos2 if pos2 >= n else tmp[pos2]
    if tmp[u] <= tmp[v]:
        return u
    return v
 
 
def create_tree():
    """
    Build the initial tournament tree from the input array 'a'.
    """
    # Place the array elements at the leaves of the tree.
    for i in range(n):
        tmp[n + i] = a[i]
    # Build the tournament tree by comparing pairs of leaves.
    for i in range(2 * n - 1, 1, -2):
        k = i // 2
        j = i - 1
        tmp[k] = winner(i, j)
    # The value at the root of the tree is the smallest element.
    value = tmp[tmp[1]]
    # Mark the winning leaf as removed.
    tmp[tmp[1]] = INF
    return value
 
 
def recreate():
    """
    Update the tournament tree after removing the minimum element.
    """
    i = tmp[1]
    while i > 1:
        k = i // 2
        if i % 2 == 0:
            j = i + 1
        else:
            j = i - 1
        tmp[k] = winner(i, j)
        i = k
    value = tmp[tmp[1]]
    tmp[tmp[1]] = INF
    return value
 
 
def tournament_sort():
    """
    Sort the array 'a' using the Tournament Sort algorithm.
    """
    value = create_tree()
    for i in range(n):
        a[i] = value
        value = recreate()

This implementation builds the tournament tree, identifies the minimum element, and repeatedly restructures the tree after each removal until the array is completely sorted.

Conclusion

Tournament Sort is an intriguing way to apply a sports tournament metaphor to sorting. By organizing comparisons in a binary tree structure, it efficiently finds the minimum element repeatedly, yielding the sorted order. Although not stable, its consistent $O(n \log n)$ performance and conceptual clarity make it a useful technique for understanding advanced sorting methods.