Introduction to algorithms
Data Structures and Algorithms
Data Structures and Algorithms
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We present a simplified derivation of the fact that the complexity-theoretic lower bound of comparison-based sorting algorithms, both for the worst-case and for the average-case time measure, is Ω(nlogn). The standard proofs typically are directly presented over decision trees. The proof for the average-case however relies on differential calculus, which presents a main hurdle in undergraduate presentations. Here we present a simplified derivation of this result based on the well-known Kraft inequality for binary trees. This inequality enables one to derive the worst-case lower bound via a very simple argument. It also yields the average-case lower bound, via a similar argument as for the worst-case and also involves an elegant and simple inequality obtained by the Danish mathematician Jensen. The Jensen inequality essentially implies that, for convex functions, the function value of a mean of numbers is bounded by the mean of the function values of these numbers. This approach removes the need to present the results based on differential calculus and makes the material easily accessible for undergraduate Computer Science students.