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downvote 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 M,< 4M-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. o 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. o Finally, an explanation of the inequality above. (in particular, why is the relationship stated with < as opposed to =7) 2. The expression 4M,+ 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. o Cive a rocursive demitiof the st or his a go it im de ved from the drawing of the algorithm. You can ar ne ta 'e the (d win o the algrhnth the costt al the steps, just like in the course slides.

downvote 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 M,< 4M-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. o 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. o Finally, an explanation of the inequality above. (in particular, why is the relationship stated with < as opposed to =7) 2. The expression 4M,+ 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. o Cive a rocursive demitiof the st or his a go it im de ved from the drawing of the algorithm. You can ar ne ta 'e the (d win o the algrhnth the costt al the steps, just like in the course slides.

C++ Programming: From Problem Analysis to Program Design
C++ Programming: From Problem Analysis to Program Design
8th Edition
ISBN:9781337102087
Author:D. S. Malik
Publisher:D. S. Malik
Chapter10: Classes And Data Abstraction
Section: Chapter Questions
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otherwise l'll downvote 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. 1. Explain why M <4M-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. o 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. o Finally, an explanation of the inequality above. (in particular, why is the relationship stated with < as opposed to =?) 2. The expression 4M, p+ 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). • 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. eas provide allready ploaded answer o Give a rocursive deiti of th st on his a go it'im de ved from the drawing of the algorithm. You can ar ne ta e the d win o the alghnth the cost ot al the steps, just like in the course slides.
Transcribed Image Text:otherwise l'll downvote 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. 1. Explain why M <4M-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. o 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. o Finally, an explanation of the inequality above. (in particular, why is the relationship stated with < as opposed to =?) 2. The expression 4M, p+ 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). • 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. eas provide allready ploaded answer o Give a rocursive deiti of th st on his a go it'im de ved from the drawing of the algorithm. You can ar ne ta e the d win o the alghnth the cost ot al the steps, just like in the course slides.
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