The tournament sort is a sorting algorithm that works by...

The tournament sort is a sorting algorithm that works by building an ordered binary tree. We represent the elements to be sorted by vertices that will become the leaves. We build up the tree one level at a time as we would construct the tree representing the winners of matches in a tournament. Working left to right, we compare pairs of consecutive elements, adding a parent vertex labeled with the larger of the two elements under comparison. We make similar comparisons between labels of vertices at each level until we reach the root of the tree that is labeled with the largest element. The tree constructed by the tournament sort of 22, 8, 14, 17, 3, 9, 27, 11 is illustrated in part (a) of the figure. Once the largest element has been determined, the leaf with this label is relabeled by ??, which is defined to be less than every element. The labels of all vertices on the path from this vertex up to the root of the tree are recalculated, as shown in part (b) of the figure. This produces the second largest element. This process continues until the entire list has been sorted. Complete the tournament sort of the list 22, 8, 14, 17, 3, 9, 27, 11. Show the labels of the vertices at each step.

The tournament sort is a sorting algorithm that works by building an ordered binary tree. We represent the elements to be sorted by vertices that will become the leaves. We build up the tree one level at a time as we would construct the tree representing the winners of matches in a tournament. Working left to right, we compare pairs of consecutive elements, adding a parent vertex labeled with the larger of the two elements under comparison. We make similar comparisons between labels of vertices at each level until we reach the root of the tree that is labeled with the largest element. The tree constructed by the tournament sort of 22, 8, 14, 17, 3, 9, 27, 11 is illustrated in part (a) of the figure. Once the largest element has been determined, the leaf with this label is relabeled by ??, which is defined to be less than every element. The labels of all vertices on the path from this vertex up to the root of the tree are recalculated, as shown in part (b) of the figure. This produces the second largest element. This process continues until the entire list has been sorted.

Complete the tournament sort of the list 22, 8, 14, 17, 3, 9, 27, 11. Show the labels of the vertices at each step.

Key Concepts

Tournament Sort

Tournament sort is a comparison-based sorting algorithm that organizes elements into a structure modeled after a sports tournament. It systematically determines the maximum element by building a binary tree where each internal node represents the winner from a pairwise comparison, and then iteratively removes the largest element by updating the tree.

Tournament Tree

A tournament tree is a complete binary tree used in tournament sort, where the leaves represent the original unsorted elements. Each internal node is assigned the result of a match (comparison) between its two children, culminating in the root node holding the overall maximum element.

Pairwise Comparison

Pairwise comparison is the fundamental operation in tournament sort. At each level of the tree, two elements are compared, with the larger one advancing as the winner. This process is repeated level-by-level and forms the backbone of the tournament structure.

Relabeling and Tree Recalculation

After the tournament identifies the maximum element, its corresponding leaf is relabeled with a value smaller than any element. The tree is then updated by recalculating the winning nodes along the path from that leaf to the root. This ensures that the next round of comparisons yields the next maximum element, ultimately sorting the entire list.

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