Transcript CS 170 – Intro to Programming for Scientists and Engineers

CS 170 – INTRO TO
SCIENTIFIC AND
ENGINEERING
PROGRAMMING
The problem with rabbits…
A man puts a pair of rabbits in a place surrounded on all
sides by a wall.
How many pairs of rabbits can be produced from that pair
in a year if it is supposed that every month each pair
begets a new pair which from the second month on
becomes productive?
Fibonacci’s rabbits..
• Fibonacci numbers were invented to model the growth of
a rabbit colony
fib1 = 1
fib2 = 1
fibn = fibn-1 + fibn-2
• 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
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Recursive
Thinking
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Recursive Thinking
• Recursion reduces a problem into one or more simpler
versions of itself
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Recursive Thinking (cont.)
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Recursive Thinking (cont.)
Recursion of Process Nested Dreams
A child couldn't sleep, so her mother told a story about a little
frog,
who couldn't sleep, so the frog's mother told a story about a
little bear,
who couldn't sleep, so bear's mother told a story about a
little weasel
...who fell asleep.
...and the little bear fell asleep;
...and the little frog fell asleep;
...and the child fell asleep.
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Steps to Design a Recursive Algorithm
 Base case:
 for a small value of n, it can be solved directly
 Recursive case(s)
 Smaller versions of the same problem
 Algorithmic steps:
 Identify the base case and provide a solution to it
 Reduce the problem to smaller versions of itself
 Move towards the base case using smaller versions
Finding… a needle in a haystack
• This is a classical computational thinking problem
• Write a function: find_needle
• Inputs of your function: the number that you are looking for and an
array of numbers
• Outputs of your function: the index of the array where the number
was found, OR a message if the number is not in the array.
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Recursive Thinking (cont.)
• Consider searching for a target value in an array
• With elements sorted in increasing order
• Compare the target to the middle element
• If the middle element does not match the target
• search either the elements before the middle
element
• or the elements after the middle element
• Instead of searching n elements, we search n/2 elements
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Recursive Thinking (cont.)
Recursive Algorithm to Search an Array
if the array is empty
return -1 as the search result
else if the middle element matches the target
return the subscript of the middle element as the
result
else if the target is less than the middle element
recursively search the array elements before the
middle element and return the result
else
recursively search the array elements after the
middle element and return the result
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Recursive Algorithm for Finding the
Length of a String
if the string is empty (has no characters)
the length is 0
else
the length is 1 plus the length of the string that
excludes the first character
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Recursive
Definitions of
Mathematical
Formulas
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Recursive Definitions of Mathematical
Formulas
• Mathematicians often use recursive definitions of formulas
• Examples include:
• factorials
• powers
• greatest common divisors (gcd)
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Factorial of n: n!
• The factorial of n, or n! is defined as follows:
0! = 1
n! = n x (n -1)! (n > 0)
• The base case: n equal to 0
• The second formula is a recursive definition
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Factorial of n: n! (cont.)
 The recursive definition can be expressed by the
following algorithm:
if n equals 0
n! is 1
else
n! = n x (n – 1)!
 The last step can be implemented as:
return n * factorial(n – 1);
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Infinite Recursion and Stack Overflow
 Call factorial with a negative argument, what will
happen?
StackOverflowException
Resources
• Lecture slides CS112, Ellen Hildreth,
http://cs.wellesley.edu/~cs112/
QUESTIONS??