Trees:Tournament Trees

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Tournament Trees

Consider the following problem: suppose we have k sequences, each of which is sorted in nondecreasing order, that are to be merged into one sequence in nondecreasing order. This can be achieved by repeatedly transferring the element with the smallest key to an output array. The smallest key has to be found from the leading elements in the k sequences. Ordinarily, this would require k − 1 comparisons for each element transferred. However, with a tournament tree, this can be reduced to log2 k comparisons per element.

Winner Trees

A winner tree is a complete binary tree in which each node represents the smaller of its two children. The root represents the smallest node in the tree. Figure 3.15 illustrates a winner tree with k = 8 sequences. The winner of the tournament is the value 8 from sequence 0. The winner of the tournament is the smallest key from the 8 sequences and is transferred

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to an output array. The next element from sequence 0 is now brought into play and a tournament is played to determine the next winner. This is illustrated in Figure 3.16. It is easy to see that the tournament winner can be computed in Θ(log n) time.

Loser Trees

The loser tree is an alternative representation that stores the loser of a match at the corresponding node. The loser tree corresponding to Figure 3.15 is shown in Figure 3.17. An advantage of the loser tree is that to restructure the tree after a winner has been output, it is sufficient to examine nodes on the path from the leaf to the root rather than the siblings of nodes on this path.

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