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Comp. Sci.
August 2004
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Dominance Tree Sorts

"Dominance tree sorts" is a neologism. If anyone recognizes this class of sorts I would take it kindly if they would point to the literature (web preferred) where this class of sorts is described and named. The idea doesn't seem to be explored in Knuth, though something may be hidden in the exercises. Dominance tree sorts aare selection sorts but not tournament sorts.

Suppose we have a set S of elements and an ordering relationship R defined over S. For example, S might be an array of integers, and R might be >, the greater than relationship. Let x and y be two elements of S. Element x is said to dominate y if (x R y). Again from the example, let x be 8 and y be 5. Then x dominates y.

A dominance tree T over S is a tree such that (a) every element of S appears exactly once in T, and (b) for every element (except the root) the element's parent dominates it. Here are some examples of dominance trees over the set {1,2,3,4,5}

          A          B        C

          5          5        5
          |          |        |
       -------     -----      4
       | | | |     |   |      |
       1 2 3 4     4   2      3
                   |          |
                  ---         2
                  | |         |
                  1 3         1
Dominance trees satisfy the heap property; accordingly they are heaps, taking the term "heap" in its most general sense. Binary heaps are dominance trees, but not all dominance trees are binary heaps. Example B is a binary heap, example C is a sorted list, and in example A all we know is that '5' is maximal.

A set K of comparisons (xRy) is a minimum comparison set for a dominance tree T if it consists exclusively of the comparisons in the child/parent linkages. Thus, in our example, the minimum comparison sets are:

K(A): 5>1, 5>2, 5>3, 5>4
K(B): 5>4, 5>2, 4>1, 4>3
K(C): 5>4, 4>3, 3>2, 2>1

Note that a minimum comparison set always has n-1 members where n is the cardinality of S.

An algorithm is an efficient dominance tree construction (EDTC) algorithm if it specifies a minimum comparison sequence that in turn specifies a dominance tree.

One of the important features of EDTC algorithms is that if an element 'loses' in a comparison it cannot appear in a later comparison in the comparison's sequence. Another is that there is no EDTC algorithm that can guarantee that the root element has less than log2(n) children. In particular there are no EDTC algorithms for creating heaps.

The general plan for dominance tree sorts runs as follows: For a given set S we first construct a dominance tree T. We then emit the root of the tree. The children of the root are compared until a new root is established. In each comparison the 'loser' becomes a new child of the 'winner'. EDTC algorithms are used in both the initial construction phase and in each emission step. There are quite a number of variations depending on which EDTC algorithms are used.

There are two major EDTC algorithms that occur to me. The first is tournament pairing, also used in the tournament sort and in heapsort. In the tournament pairing algorithm the set is divided into pairs that are compared. The winners go on to the second round where they are again divided into pairs that are compared. This is continued until a final winner is determined. This requires n-1 comparisons, log2(n) rounds, and the winner, the root of the tree, will have log2(n) children.

The second is sequential comparison. The first element in a list is compared with the second. The winner is compared with the third, and so on. This also requires n-1 comparisons. The number of children of the root depends on the location of the root element in the list. If the ordering of the list elements is random, the average number of children will n/2. On the other hand, the number of children will be 1 if the list is ordered.

It turns out that we get a very efficient (as measured by the number of comparisons used) algorithm if we use tournament pairing to construct the initial dominance tree and sequential comparison in the update after emission of the root. Each element has a list associated with it of elements that it dominates. During construction when one element dominates another the losing element is appended to the winning elements list. Here is a very simple example. Let the set to be sorted be {2,4,3,1}. The actions are:

    Construction Phase
        compare (2,4); append 4 to 2's list
        compare (3,1); append 3 to 1's list
        compare (2,1); append 2 to 1's list; 1 is the root
    Emission Phase
        emit (1); compare (3,2); append 3 to 2's list; 2 is root
        emit (2); compare (3,4); append 4 to 3's list; 3 is root
        emit (3); 4 is root
        emit (4); done 
How efficient is this sort? The tests that I ran showed it to about equivalent to merge-insert (see Knuth, Vol 3, 5.3.1). For example, sorting 32 items requires a minimum of 118 comparisons on average (log2(32!)). Merge-insert needs 121. The number needed by the above algorithm depends on the precise permutation. The average number needed is about 121.5. For large n the number of comparisons required was ~ n*(log2(n) - 1.27) in the tests I performed; this is not far off from the best possible of which is ~ n*(log2(n) - log2(e)) = n*(log2(n) - 1.44).

Why is this sort efficient? It has to do with an order preservation property in the construction and updating of the dominance lists. In the construction phase a dominance list has two special properties: (a) its length is longer than the lengths of the lists of the elements in its list, and (b) the elements in the list are in order of the length of their lists. These properties are almost always preserved by the above algorithm. In turn this implies that the lengths of the dominance lists are slightly less than O(log2(n)) on average.

Can this be improved upon? Possibly. The only reference I can find to dominance tree sorts is a web article by someone called Supercat. In the web article he in effect forces property (b) to hold. (He calls it a tournament sort which is a bit of a misnomer.) He gives the following results for a test case sorting one million items.

Number of comparisons by a perfect sort (lg(n!)) : 18,488,865
Avg # of comparisons for Tournament Sort: 18,646,425 (100.85% of perfect)
Avg # of comparisons for Borland C's qsort: 22,439,507 (121.37% of perfect)

This gives (n = 1000000) n*(log2(n) - 1.284) which is tad better than the simpler algorithm given here. On the other hand Supercat's version requires addition comparisons of the element weights (integers rather than keys).

Is this a practical sort? Possibly not. It requires extra space O(n) for the lists and O(n) for the list headers. Each comparison involves adding an item to a linked list, so the cost of the inner loop operations will be higher than it is in simpler sorts. On the other hand memory is often plentiful and the inner loop costs should be no worse than those for an ordinary tree sort.

Dominance tree sorts have an obvious relationship with fibonacci heaps and sorts based on fibonacci heaps. However the fibonacci heapify operation uses in effect uses sequential comparison rather than tournament pairing comparison; the resulting sort is considerably less efficient. A question of interest is whether there is a version of fibonacci heaps that is consistent with tournament pairing.

References:

URL: http://www.halfbakery.com/idea/Tournament_20Sort
Title: Tournament sort

URL: http://algorithm.myrice.com/resources/technical_artile/fibonacci_heap/fibonacci.htm
Title: The Fibonacci Heap

Knuth, The Art of Computer Programming, Vol 3, Searching and Sorting.


This page was last updated August 24, 2004.

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Comp. Sci.
August 2004
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