By Georgiev P., Pardalos P., Theis F.

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The list of ranks is equally likely to be any one of the n! permutations. Equivalently, the ranks form a uniform random permutation: each of the possible n! permutations appears with equal probability. Essential idea of probabilistic analysis: We must use knowledge of, or make assumptions about, the distribution of inputs. • • The expectation is over this distribution. This technique requires that we can make a reasonable characterization of the input distribution. Lecture Notes for Chapter 5: Probabilistic Analysis and Randomized Algorithms 5-3 Randomized algorithms We might not know the distribution of inputs, or we might not be able to model it computationally.

N]. Termination: The for loop of lines 1–4 terminates when i = n + 1, so that i − 1 = n. By the statement of the loop invariant, A[1 . i − 1] is the entire array A[1 . n], and it consists of the original array A[1 . n], in sorted order. Note: We have received requests to change the upper bound of the outer for loop of lines 1–4 to length[A] − 1. That change would also result in a correct algorithm. The loop would terminate when i = n, so that according to the loop invariant, A[1 . n − 1] would consist of the n − 1 smallest values originally in A[1 .

Maintenance: Consider an iteration for a given value of i. By the loop invariant, A[1 . i − 1] consists of the i smallest values in A[1 . n], in sorted order. Part (b) showed that after executing the for loop of lines 2–4, A[i] is the smallest value in A[i . n], and so A[1 . i] is now the i smallest values originally in A[1 . n], in sorted order. Moreover, since the for loop of lines 2–4 permutes A[i . n], the subarray A[i + 1 . n] consists of the n − i remaining values originally in A[1 .

### A bilinear algorithm for sparse representations by Georgiev P., Pardalos P., Theis F.

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