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Thinking Recursively xkcd #878

Thinking Recursively xkcd #878. Recursion. Recursive call is divider Instructions before happen as stack is built Instructions after happen as stack torn down. Fibonacci Sequence. Famous recursively defined sequence. Naïve Implementation. Direct code translation. Fibonacci.

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Thinking Recursively xkcd #878

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  1. Thinking Recursivelyxkcd #878

  2. Recursion • Recursive call is divider • Instructions before happen as stack is built • Instructions after happen as stack torn down

  3. Fibonacci Sequence • Famous recursivelydefined sequence

  4. Naïve Implementation • Direct code translation

  5. Fibonacci • Multiple recursive calls = combinatorial explosion:

  6. Better Fibonacci • Redefined Fibonacci function: • Parameters keep track of previous values • Call must include first two terms:

  7. Reduced Load • Parameters store old work…

  8. Recursive Helpers • This is ugly: • Can make it a recursive helper: • Call from main:

  9. Moral • Parameters are your main tool • Use them to "store" information • Use them to change where work happens

  10. Recursive Design • Recursive function design • What is base case? • What is one step? • What parameters do I need? • Do I want/need extra ones to simplify problem?

  11. Number of Digits • How many digits does an integer have? • What is base case? • What is one step? • What parameters do I need?

  12. Number of Digits • How many digits does integer have? • What is base case?Anything < 10 is 1 digit • What is one step? • What parameters do I need?

  13. Number of Digits • How many digits does integer have? • What is base case?Anything < 10 is 1 digit • What is one step?Digits(n) = 1 + Digits(n/10) • What parameters do I need?

  14. Number of Digits • How many digits does integer have? • What is base case?Anything < 10 is 1 digit • What is one step?Digits(n) = 1 + Digits(n/10) • What parameters do I need?n

  15. Number Of Digits • Code: • Assumes number >= 0

  16. Recursion With Array • Want to total an array using recursion: • What is base case?Size 0 will equal 0 • What is one step?Total(size n) = nth element + Total(size n-1) • What parameters do I need?Array, size

  17. Recursion With Array • Want to total an array using recursion: • Work backwards through array, pretending it gets smaller

  18. Palindrome • Is a string a Palindrome? (e.g. "radar") • What is base case? • What is one step? • What parameters do I need?

  19. Palindrome • Is a string a Palindrome? (e.g. "radar") • What is base case? • 0 or 1 letters is a palindrome : "d" • If first letter != last, is NOT a palindrome : "ben" • What is one step? • What parameters do I need?

  20. Palindrome • Is a string a Palindrome? (e.g. "radar") • What is base case? • 0 or 1 letters is a palindrome : "d" • If first letter != last, is NOT a palindrome : "ben" • What is one step?"madamImadam"Test all but first and last • What parameters do I need?

  21. Palindrome • Is a string a Palindrome? (e.g. "radar") • What is base case? • 0 or 1 letters is a palindrome : "d" • If first letter != last, is NOT a palindrome : "ben" • What is one step?"madamImadam"Test all but first and last • What parameters do I need? Current string

  22. Palindrome • Is a string a Palindrome? (e.g. "radar")

  23. Palindrome • Is a string a Palindrome? (e.g. "radar") • What is base case? • 0 or 1 letters is a palindrome : "d" • If first letter != last, is NOT a palindrome : "ben" • What is one step?Test all but first and last • What parameters do I need? Current string • Do I want/need extra ones to simplify problem?Indexes to point to "start" and "end"

  24. Palindrome • Extra parameters avoid making new strings: • Use as helper function:

  25. Binary Search • Binary Search • Pick middle of remaining search space • Too high? Eliminate middle and above • Too low? Eliminate middle and below

  26. Binary Search • Binary search • What is base case? • What is one step? • What parameters do I need?

  27. Binary Search • Binary search • What is base case? • What is one step? • What parameters do I need?List, value looking for, lowest possible location, highest possible location

  28. Binary Search • Binary search • What is base case?If lowest possible location > highest possible location, it can't be there • What is one step? • What parameters do I need?List, value looking for, lowest possible location, highest possible location

  29. Binary Search • Binary search • What is base case?If lowest possible location > highest possible location, it can't be there • What is one step?Find middle • If key == middle, return middle location • If key < middle, highest is now middle -1 • If key > middle, lowest is now middle + 1 • What parameters do I need?…

  30. Binary Search • Binary search – recursive helper • Non-recursive starter:

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