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By Jeff Erickson

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The output is a list of the positions and orientations of (4n − 1)/3 tiles. Your algorithm should run in O(4n ) time. 3. Prove that the original recursive Tower of Hanoi algorithm is exactly equivalent to each of the following non-recursive algorithms; in other words, prove that all three algorithms move the same disks, to and from the same needles, in the same order. ). 9 On the other hand, nobody knows whether finding the minimum number of multiplications is NP-hard, either. 11 Lecture 1: Recursion [Fa’10] Algorithms (a) Follow these four rules: • • • • Never move the same disk twice in a row.

9 Lecture 3: Backtracking [Fa’10] Algorithms ’Tis a lesson you should heed, Try, try again; If at first you don’t succeed, Try, try again; Then your courage should appear, For, if you will persevere, You will conquer, never fear; Try, try again. — Thomas H. Palmer, The Teacher’s Manual: Being an Exposition of an Efficient and Economical System of Education Suited to the Wants of a Free People (1840) When you come to a fork in the road, take it. — Yogi Berra 3 Backtracking In this lecture, I want to describe another recursive algorithm strategy called backtracking.

These complex numbers have several useful properties for any integers n and k: • There are exactly n different nth roots of unity: ωkn = ωkn mod n . 0 • If n is even, then ωnk+n/2 = −ωkn ; in particular, ωn/2 n = −ωn = −1. k k • 1/ωkn = ω−k n = ωn = (ωn ) , where the bar represents complex conjugation: a + bi = a − bi • ωn = ωkkn . Thus, every nth root of unity is also a (kn)th root of unity. These properties imply immediately that if n is a power of 2, then the set of all nth roots of unity is collapsible!

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Algorithms by Jeff Erickson

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