lect17-linear-time-sort - Lecture Notes CMSC 251 Using...

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Lecture Notes CMSC 251 Using Decision Trees for Analyzing Sorting: Consider any sorting algorithm. Let T ( n ) be the maximum number of comparisons that this algorithm makes on any input of size n . Notice that the running time fo the algorithm must be at least as large as T ( n ) , since we are not counting data movement or other computations at all. The algorithm defines a decision tree. Observe that the height of the decision tree is exactly equal to T ( n ) , because any path from the root to a leaf corresponds to a sequence of comparisons made by the algorithm. As we have seen earlier, any binary tree of height T ( n ) has at most 2 T ( n ) leaves. This means that this sorting algorithm can distinguish between at most 2 T ( n ) different final actions. Let’s call this quantity A ( n ) , for the number of different final actions the algorithm can take. Each action can be thought of as a specific way of permuting the oringinal input to get the sorted output. How many possible actions must any sorting algorithm distinguish between? If the input consists of n distinct numbers, then those numbers could be presented in any of n ! different permutations. For each different permutation, the algorithm must “unscramble” the numbers in an essentially different way, that is it must take a different action, implying that A ( n ) n ! . (Again, A ( n ) is usually not exactly equal to n ! because most algorithms contain some redundant unreachable leaves.) Since A ( n ) 2 T ( n ) we have 2 T ( n ) n ! , implying that T ( n ) lg( n !) . We can use Stirling’s approximation for n ! (see page 35 in CLR) yielding: n ! 2 πn ± n e ² n T ( n ) lg ± 2 ± n e ² n ² =l g 2 + n lg n - n lg e Ω( n log n ) . Thus we have, the following theorem. Theorem: Any comparison-based sorting algorithm has worst-case running time Ω( n log n ) . This can be generalized to show that the average-case time to sort is also Ω( n log n ) (by arguing about the average height of a leaf in a tree with at least n ! leaves). The lower bound on sorting can be generalized to provide lower bounds to a number of other problems as well. Lecture 17: Linear Time Sorting (Tuesday, Mar 31, 1998) Read: Chapt. 9 of CLR. Linear Time Sorting: Last time we presented a proof that it is not possible to sort faster than Ω( n log n ) time assuming that the algorithm is based on making 2-way comparisons. Recall that the argument was based on showing that any comparison-based sorting could be represented as a decision tree, the decision tree must have at least n ! leaves, to distinguish between the n ! different permutations in which the keys could be input, and hence its height must be at least lg( n !) Ω( n lg n ) .
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This note was uploaded on 01/13/2012 for the course CMSC 351 taught by Professor Staff during the Fall '11 term at University of Louisville.

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lect17-linear-time-sort - Lecture Notes CMSC 251 Using...

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