Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In computational complexity theory, a numeric algorithm runs in pseudo-polynomial time if its running time is polynomial in the numeric value of the input (which is exponential in the length of the input – its number of digits). An NP-complete problem with known pseudo-polynomial time algorithms is called weakly NP-complete. An NP-complete problem is called strongly NP-complete if it is proven that it cannot be solved by a pseudo-polynomial time algorithm unless P=NP. The strong/weak kinds of NP-hardness are defined analogously. Consider the problem of testing whether a number n is prime, by naively checking whether no number in {2,3,..., n/2} divides n evenly. This approach can take up to n/2-1 divisions, which is indeed linear in n but not in the size of n. For example, the number n = 2,000,000,000 would require approximately 1 billion divisions, even though the length of n is only 10 digits.

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