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CPE 130 Algorithms and Data Structures Recursion Asst. Prof. Dr. Nuttanart Facundes
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Leonardo Fibonacci In 1202, Fibonacci proposed a problem about the growth of rabbit populations.
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Leonardo Fibonacci In 1202, Fibonacci proposed a problem about the growth of rabbit populations.
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Leonardo Fibonacci In 1202, Fibonacci proposed a problem about the growth of rabbit populations.
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The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month month1234567 rabbits112358 13131313
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The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month month1234567 rabbits112358 13131313
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The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month month1234567 rabbits112358 13131313
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The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month month1234567 rabbits112358 13131313
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The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month month1234567 rabbits112358 13131313
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The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month month1234567 rabbits112358 13131313
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The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month month1234567 rabbits112358 13131313
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Inductive Definition or Recurrence Relation for the Fibonacci Numbers Stage 0, Initial Condition, or Base Case: Fib(1) = 1; Fib (2) = 1 Inductive Rule For n>3, Fib(n) = Fib(n-1) + Fib(n-2)n01234567Fib(n)%112358 13131313
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Inductive Definition or Recurrence Relation for the Fibonacci Numbers Stage 0, Initial Condition, or Base Case: Fib(0) = 0; Fib (1) = 1 Inductive Rule For n>1, Fib(n) = Fib(n-1) + Fib(n-2)n01234567Fib(n)0112358 13131313
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Recursion case study: Fibonacci Numbers Fibonacci numbers are a series in which each number is the sum of the previous two numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.
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Figure 6-9
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Algorithm for Fibonacci Numbers The algorithm for calculating Fibonacci numbers: intfib(int n) { if (n < 2) return n; if (n < 2) return n;else return fib(n – 1) + fib(n – 2); return fib(n – 1) + fib(n – 2);}
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Great Pyramid at Gizeh
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The ratio of the altitude of a face to half the base b a
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Notre Dame in Paris: Phi
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Golden Ratio: the divine proportion = 1.6180339887498948482045… “Phi” is named after the Greek sculptor Phidias How is the Golden Ratio related to Fibonacci numbers?
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Tower of Hanoi
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Recursion case study: Tower of Hanoi We need to do 2 n – 1 moves to achieve the task, while n = the number of disks.
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Solution for 2 disks
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Solution for 3 disks, Part I
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Solution for 3 disks, Part II
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Tower of Hanoi: The Algorithm 1.Move n – 1 disks from source to auxiliaryGeneral Case 2.Move one disk from source to destinationBase Case 3.Move n – 1 disks from auxiliary to destinationGeneral Case
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