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. Use the tournament sort to sort the list 17, 4, 1, 5, 13, 10, 14, 6.

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.

Use the tournament sort to sort the list 17, 4, 1, 5, 13, 10, 14, 6.

Key Concepts

Binary Tree

A binary tree is a hierarchical data structure in which each node has at most two children. In tournament sort, binary trees are used to systematically organize and compare elements. The leaves represent the initial unsorted elements, and internal nodes store the results of comparisons between child nodes, ultimately leading to a root that reflects the overall winner.

Tournament Sort

Tournament sort is a comparison-based sorting algorithm that mimics the structure of an elimination tournament. Elements are paired and compared, with the larger element moving up the tree until the overall maximum is determined. After removing the maximum, the tree is updated to find the next largest element, and this process is repeated until all elements are sorted.

Winner Tree

A winner tree is a type of binary tree used in tournament sort where each internal node is assigned the element that wins the comparison between its two children. The root of the tree holds the ultimate winner—the largest (or smallest, depending on the comparison criteria) element. This approach simplifies the process of repeatedly selecting the next candidate after removing the current winner.

Tree Update Mechanism

Once the maximum element is identified in the winner tree, its corresponding leaf is replaced with a marker value that is less than any remaining element. This ensures that the eliminated element no longer competes. The tree is then updated by recalculating the results of comparisons along the path from the modified leaf up to the root, allowing a new winner to emerge for the next iteration of the sorting process.

Transcript

00:01 So they want to use the tournament sort to sort this list of numbers that they provide here.

00:06 So we just kind of go through what that algorithm is.

00:11 So they tell us we start from left to right and then we just compare each of the pairs that come up.

00:16 So we do 17 verse 4.

00:18 So we're going to choose the largest of these two, which is going to be 17.

00:23 Then we're going to compare the next set.

00:25 1 and 5, so 5 is larger.

00:27 And then we're going to compare 13 and 10, and so there 13 is larger.

00:33 And then we would compare 14 and 6, so 14 is larger.

00:38 Now we're going to compare each pairs going left to right for this second round here.

00:47 So we're going to do 17 verse 5, well 17 is larger.

00:50 And then 13 verse 14, well 14 is larger.

00:54 Then lastly, we're going to compare 17 versus 14.

00:58 And this is going to give that 17 is larger.

01:02 So what we've learned from this is that 17 is the largest value in our set.

01:09 So now what we're going to do is go ahead and copy this graph and then replace 17 with negative infinity.

01:20 So let me go ahead and copy this real quick.

01:23 So let's erase all these 17s.

01:29 Like i was saying, we're replacing them.

01:32 With a negative infinity, or at least we'll replace the one at the very bottom with negative infinity.

01:36 So now we're going to repeat this, but notice we only need to do this now for the ones where we erase 17 from.

01:44 So we do negative infinity versus four.

01:46 So we're assuming negative infinity is the largest or the smallest possible element that we can have.

01:51 So that means four is larger than it.

01:53 And then we do four verse five, well five's larger.

01:56 And then five verse 14, will 14 is larger.

01:59 So what this tells us is that 17 and then 14 would be our largest.

02:05 And now we're going to repeat this and then continue this process.

02:11 And this time we would place 14 with negative infinity.

02:17 So erase all of these 14s.

02:21 And then at the very bottom we'll put a negative infinity.

02:25 All right.

02:26 So negative infinity for six, remember negative infinity we're assuming is smaller than all other values.

02:30 So we'd have six.

02:30 Winning and 6 versus 13 will 13 wins and then 5 verse 13 13 wins so that means we'd have 17 14 13 for our ordering so far and now again we're going to replace 13 with negative infinity and then repeat this all right so erase all the 13s and now we put negative infinity down here so negative infinity down here so negative 20 verse 10, 10 ones, 10 verse 6, 10 ones, 5 verse 10, 10 ones.

03:14 So, so far we have 17, 14, 13, and then 10.

03:23 Now we're going to copy this one more time, and then replace 10 with negative infinity.

03:31 So erase all of the tens, and now we replace that with negative infinity.

03:40 So if we have the same number, it doesn't matter which one moves up...

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