Data Structures and Algorithms with Object-Oriented Design Patterns in Python

Merging Binomial Queues

Merging two binomial queues is like doing binary addition. For example, consider the addition of and :

In the next step, we add the from and the from . Combining the two s we get a which we carry  to the next column. Since there are no s left, the result does not contain any. The addition continues in a similar manner until all the columns have been added up.

Program  gives an implementation of this addition algorithm. In addition to self, the merge method of the BinomialQueue class takes a BinomialQueue and adds its subtrees to the binomial queue self.

  
Program: BinomialQueue class merge method.

Each iteration of the main loop of the algorithm (lines 11-28) computes the ``bit'' of the result--the bit is a binomial tree of order i. At most three terms need to be considered: the carry from the preceding iteration and two s, one from each of the queues that are being merged.

The method fullAdder computes the result required in each iteration. Program  defines fullAdder method. In the worst case, the fullAdder method calls the add method to combine two BinomialTrees into one. Therefore, the worst-case running time for fullAdder is

  
Program: BinomialQueue class fullAdder method.

Suppose the merge method of Program  is used to combine a binomial queue with n items with another that contains m items. Since the resulting priority queue contains n+m items, there are at most binomial trees in the result. Thus, the worst-case running time for the merge operation is

Copyright © 2003, 2004 by Bruno R. Preiss, P.Eng. All rights reserved.