April 2024
Intermediate to advanced
832 pages
29h 31m
English
Content preview from Design and Analysis of Algorithms
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Tractable and Non-tractable Problems 487
Proof If n is prime then steps 1 and 3 can never return COMPOSITE. By Lemma 2.1, the for loop also
cannot return COMPOSITE. Therefore the algorithm will identify n as PRIME either in step 4 or in step 6.
The converse of Lemma 2 requires a little more work. If the algorithm returns PRIME in step 4 then n
must be prime since otherwise step 3 would have found a non-trivial factor of n. So the only remaining case
is when the algorithm returns PRIME in step 6. For the purpose of subsequent analysis we assume this to be
the case.
Time Complexity Analysis and Improvements
It is straightforward to calculate the ...
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