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Recursion

Recursion. A common problem Problem: Given a word, find its meaning Algorithm: Look up the word in Webster’s Ninth New Collegiate Dictionary . Problem: Given the word aardwolf, find its meaning

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Recursion

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  1. Recursion CS262

  2. A common problem Problem: Given a word, find its meaning Algorithm: Look up the word in Webster’s Ninth New Collegiate Dictionary Problem: Given the word aardwolf, find its meaning Answer: “a maned striped mammal of southern and eastern Africa that resembles the related hyenas and feeds chiefly on carrion” Sub-problem: Given the word carrion, find its meaning Answer:“dead and putrefying flesh” CS262

  3. Sub-sub-problem: Given the word putrefying, find its meaning Answer:“undergoing putrefaction” Sub-sub-sub-problem: Given the word putrefaction, find its meaning Answer:“the decomposition of organic matter” Final meaning: “a maned striped mammal of southern and eastern Africa that resembles the related hyenas and feeds chiefly on dead flesh that is undergoing decomposition” CS262

  4. A common problem-solving strategy aardwolf lookup a maned striped mammal of southern and eastern Africa that resembles the related hyenas and feeds chiefly on carrion lookup dead flesh that is putrefying lookup undergoing putrefaction lookup decomposition combine undergoing decomposition combine combine dead flesh that is undergoing decomposition a maned striped mammal of southern and eastern Africa that resembles the related hyenas and feeds chiefly on dead flesh that is undergoing decomposition CS262

  5. Recursion: A common problem-solving strategy Recursion occurs when an entity (e.g., a function or an algorithm) is defined in terms of itself. In our example, the recursive entity is the algorithm for looking up a word in a dictionary. Given a word: 1. Find the page containing the word 2. Read the definition of the word 3. If all words in the definition are understood, then stop. Otherwise, look up each unknown word in the definition and combine their meaning with the meaning of the known words to obtain the meaning of the original word. But wait, does it make sense to define something in terms of itself? How does that help?In our example, each time we look up a word, it is a different word. The hope is thateventually, the meaning of all the words in the definition will be known, without need for another lookup. CS262

  6. Recursion works because the entity is defined in terms of a variant of itself, and the variant gets easier and easier, until the variant is so easy that its solution is obvious. Recursion is a powerful problem-solving strategy that computer programs can take advantage of. CS262

  7. The factorial function • The factorial function is denoted by an exclamation point. For example, the factorial of 5 is denoted by 5! • Definition of factorial: For any positive integer n: n!=n×(n−1)×(n−2)×...×3×2×1 0!=1 CS262

  8. Two ways of implementation Iteration: Since n! = n x (n-1) x (n-2) x . . . x 2 x 1, use a for loop. Recursion: Another way of defining factorial: n! = n x (n-1)! 0! = 1. We may want to define a method: int factorial (n) { return n x factorial(n-1); } CS262

  9. Recursion considerations • Why is this definition recursive? • Write the Java implementation of this function. • Can this program go into an infinite loop? • We need to ensure that there is a base case where the recursion “bottoms out.” • What is the base case in the factorial example? CS262

  10. Factorial(3) factorial = 3 * factorial(2) factorial = 3 * (2 * factorial(1)) factorial = 3 * ( 2 * (1 * factorial(0))) factorial = 3 * ( 2 * ( 1 * 1))) multiply = 3 * ( 2 * 1) multiple = 3 * 2 multiply = 6 CS262

  11. Example • ComputeFactorial class • NRComputeFactorial CS262

  12. Example of recursion with multiple base cases • The Fibonacci numbers are computed as follows: fib(0)=0 fib(1)=1 For each n>1, fib(n) = fib(n−1) + fib(n−2) • Run and study computeFabonacci.java CS262

  13. More examples • Recursive print • RecPrint.java and RePrintTest.java • Towers of Hanoi: • TowersOfHanoi.java • TowersOfHanoi1.java • TowersOfHanoi2.java • Hanoe’s_Towers.jar • Maze.java SearchMaze.java CS262

  14. More examples • Fractals • SierpinskiTriangle.java • Koch snowflake – KochSnowflake.java • Graphics • TiledPictures.java • Recursive Binary Search • Palindrome CS262

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