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Data Str & Algorithm HW Solutions 22

Data Str & Algorithm HW Solutions 22 - cf n and so...

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22 Chap. 3 Algorithm Analysis SWAP(array[i][j], array[tempr][tempc]); } } } Finding the corner takes O ( n 2 m 2 ) time, which is the square of the number of pieces. Filling in the rest of the pieces also takes O ( n 2 m 2 ) time, the number of pieces squared. Thus, the entire algorithm takes O ( n 2 m 2 ) time. 3.20 If an algorithm is Θ( f ( n )) in the average case, then by de fi nition it must be Ω( f ( n )) in the average case. Since the average case cost for an instance of the problem requires at least cf ( n ) time for some constant c , at least one instance requires at least as much as the average cost (this is an example of applying the Pigeonhole Principle). Thus, at least one instance costs at least
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Unformatted text preview: cf ( n ) , and so this means at least one instance (the worst case) is Ω( f ( n )) . 3.21 If an algorithm is Θ( f ( n )) in the average case, then by de f nition it must be O ( f ( n )) in the average case. Therefore, the average case cost for an instance of the problem requires at most cf ( n ) time for some constant c . By the pigeonhole principle, some instance (the best case) must therefore cost at most cf ( n ) time. Therefore, the best case must be O ( f ( n )) ....
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