By Herbert S. Wilf

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Now let’s discuss Quicksort. In contrast to the sorting method above, the basic idea of Quicksort is sophisticated and powerful. Suppose we want to sort the following list: 26, 18, 4, 9, 37, 119, 220, 47, 74. 1) The number 37 in the above list is in a very intriguing position. Every number that precedes it is smaller than it is and every number that follows it is larger than it is. What that means is that after sorting the list, the 37 will be in the same place it now occupies, the numbers to its left will have been sorted, but will still be on its left, and the numbers on its right will have been sorted, but will still be on its right.

The situation is rather similar to what happens in the theory of ordinary diﬀerential equations. Therefore, if we omit initial or boundary values, then the solutions are determined only up to arbitrary constants. 28 1. 24), the next level of diﬃculty occurs when we consider a first-order recurrence relation with a variable multiplier, such as xn+1 = bn+1 xn (n ≥ 0; x0 given). 26) Now {b1 , b2 , . } is a given sequence, and we are being asked to find the unknown sequence {x1 , x2 , . }. In an easy case like this, we can write out the first few xs and then guess the answer.

For what values of a and b is it true that no matter what the initial values x0 , x1 are, the solution of the recurrence relation xn+1 = axn + bxn−1 (n ≥ 1) is guaranteed to be o(1) (n → ∞)? 5. Suppose x0 = 0, x1 = 1, and for all n ≥ 2, it is true that xn+1 ≤ xn + xn−1 . Is it true that ∀n : xn ≤ Fn ? Prove your answer. 34 1. Mathematical Preliminaries 6. Generalize the result of Exercise 5, as follows. Suppose x0 = y0 and x1 = y1 , where yn+1 = ayn + byn−1 (∀n ≥ 1). If furthermore, xn+1 ≤ axn + bxn−1 (∀n ≥ 1), can we conclude that ∀n : xn ≤ yn ?