These should be treated as informal definitions or intuitive defintions rather than mathematical precise definitions. Making some of these definitions mathematically precise is rather a pain.

• A algorithm A runs in pseudo-polynomial time if it runs in time polynomial in the following modified way of measuring the input size: Each integer x contributes |x|+1 to the input size, instead of the usual $1 + \log_2 (\vert x\vert +1)$.

• An optimization problem P is self-reducible if there is a polynomial time reduction from P to the obvious decision version of P.

• A problem P is NP-hard in the strong sense, or equivalently is stronly NP-hard, if there exists some polynomial p(n) such that P is NP-hard even if restricted to instances I where the largest number in I is bounded by p of the number of numbers in I.

• The worst-case (average case) time complexity of a problem P is O(f(n)) if there is a an algorithm A that solves P and has worst-case (average case) time complexity O(f(n)).

• The worst-case (average case) time complexity of a problem P is o(f(n)) if there is a an algorithm A that solves P and has worst-case (average case) time complexity o(f(n)).

• The worst-case (average case) time complexity of a problem P is $\Omega(f(n))$ if for every algorithm A that solves P it is the case that the worst-case (average case) time complexity of A is $\Omega(f(n))$.

• The worst-case (average case) time complexity of a problem P is $\omega(f(n))$ if for every algorithm A that solves P it is the case that the worst-case (average case) time complexity of A is $\omega(f(n))$.

• The worst-case (average case) time complexity of a problem P is $\Theta(f(n))$ iff it is O(f(n)) and $\Omega(f(n))$.

• A problem has an information theoretic lower bound of $\Omega(f(n))$ if every comparison based algorithm that solves the problem uses $\Omega(f(n))$ comparisons. An algorithm is comparison based if the only way that it learns about the input is by comparing various numbers in the input.

• An algorithm runs in strongly polynomial time if it runs in time polynomial in a modified input size, where we measure the size of each number as one, if each standard operation on the numbers is only counted as one step.