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Recursively Defined Sequence #1: Circular Tower of Hanoi This is an expanded version of question 20 in section 5.6 of the textbook. Suppose that instead of being lined up in a row, the three poles for the original Tower of Hanoi are placed in a circle. The disks are moved one by one from one pole to another, but only in a clockwise direction. As with the original puzzle, larger disks cannot be moved on top of smaller disks. Define the sequence Mn = minimum number of moves needed to transfer a pile of n disks from one pole to the pole adjacent to it when going in a clockwise direction. For the rest of the question, we will call that destination pole the "next" pole. Obviously, M1 = 1.

Recursively Defined Sequence #1: Circular Tower of Hanoi This is an expanded version of question 20 in section 5.6 of the textbook. Suppose that instead of being lined up in a row, the three poles for the original Tower of Hanoi are placed in a circle. The disks are moved one by one from one pole to another, but only in a clockwise direction. As with the original puzzle, larger disks cannot be moved on top of smaller disks. Define the sequence Mn = minimum number of moves needed to transfer a pile of n disks from one pole to the pole adjacent to it when going in a clockwise direction. For the rest of the question, we will call that destination pole the "next" pole. Obviously, M1 = 1.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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Discrete Math recursive problem.

The textbook the question is referencing is from Discrete Mathematics with Applications by Susan Epp, 5th Edition

Recursively Defined Sequence #1: Circular Tower of Hanoi This is an expanded version of question 20 in section 5.6 of the textbook. Suppose that instead of being lined up in a row, the three poles for the original Tower of Hanoi are placed in a circle. The disks are moved one by one from one pole to another, but only in a clockwise direction. As with the original puzzle, larger disks cannot be moved on top of smaller disks. Define the sequence M, = minimum number of moves needed to transfer a pile of n disks from one pole to the pole adjacent to it when going in a clockwise direction. For the rest of the question, we will call that destination pole the "next" pole. Obviously, M1 = 1. 1. Explain why Mn < 4Mn-1+1 for every integer n >1. This explanation must include: • A description with drawings of a recursive algorithm for moving k+1 disks from one pole to the next if you know how to move k disks from one pole to the next. • A recursive definition of the cost of this algorithm derived from the drawing of the algorithm. You can annotate the drawing of the algorithm with the costs of all the steps, just like in the course slides. • Finally, an explanation of the inequality above. (in particular, why is the relationship stated with <as opposed to =?) 2. The expression 4M,-1+1 is not the minimum number of moves needed to transfer a pile of k disks from one pole to the next. In the textbook, you can find an explanation of why M3 + 4M2 + 1. It provides a faster algorithm for moving 3 disks (which you can also derive yourself without the textbook). o Generalize this new algorithm into a recursive algorithm for moving k+2 disks from one pole to the next if you know how to move k disks from one pole to the next. Give a description with drawings. • Give a recursive definition of the cost of this algorithm derived from the drawing of the algorithm. You can annotate the drawing of the algorithm with the costs of all the steps, just like in the course slides.
Transcribed Image Text:Recursively Defined Sequence #1: Circular Tower of Hanoi This is an expanded version of question 20 in section 5.6 of the textbook. Suppose that instead of being lined up in a row, the three poles for the original Tower of Hanoi are placed in a circle. The disks are moved one by one from one pole to another, but only in a clockwise direction. As with the original puzzle, larger disks cannot be moved on top of smaller disks. Define the sequence M, = minimum number of moves needed to transfer a pile of n disks from one pole to the pole adjacent to it when going in a clockwise direction. For the rest of the question, we will call that destination pole the "next" pole. Obviously, M1 = 1. 1. Explain why Mn < 4Mn-1+1 for every integer n >1. This explanation must include: • A description with drawings of a recursive algorithm for moving k+1 disks from one pole to the next if you know how to move k disks from one pole to the next. • A recursive definition of the cost of this algorithm derived from the drawing of the algorithm. You can annotate the drawing of the algorithm with the costs of all the steps, just like in the course slides. • Finally, an explanation of the inequality above. (in particular, why is the relationship stated with <as opposed to =?) 2. The expression 4M,-1+1 is not the minimum number of moves needed to transfer a pile of k disks from one pole to the next. In the textbook, you can find an explanation of why M3 + 4M2 + 1. It provides a faster algorithm for moving 3 disks (which you can also derive yourself without the textbook). o Generalize this new algorithm into a recursive algorithm for moving k+2 disks from one pole to the next if you know how to move k disks from one pole to the next. Give a description with drawings. • Give a recursive definition of the cost of this algorithm derived from the drawing of the algorithm. You can annotate the drawing of the algorithm with the costs of all the steps, just like in the course slides.
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