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Turgay Korkmaz Office : NPB 3.330 Phone: (210) 458-7346 Fax: (210) 458-4437

Understand the essence and implementation of recursion, write/debug recursive functions, solve problems through recursive strategies. Learn the recursive leap of faith and wrapper functions.

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Turgay Korkmaz Office : NPB 3.330 Phone: (210) 458-7346 Fax: (210) 458-4437

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  1. Do Quiz07 CS 2123 Data Structures Ch 4 – Recursion Turgay Korkmaz Office: NPB 3.330 Phone: (210) 458-7346 Fax: (210) 458-4437 e-mail: korkmaz@cs.utsa.edu web: www.cs.utsa.edu/~korkmaz Thanks to Eric S. Roberts, the author of our textbook, for providing some slides/figures/programs.

  2. Objectives • To be able to define the concept of recursion as a programming strategy distinct from other forms of algorithmic decomposition. • To recognize the paradigmatic form of a recursive function. • To understand the internal implementation of recursive calls. • To appreciate the importance of the recursive leap of faith. • To understand the concept of wrapper functions in writing recursive programs. • To be able to write and debug simple recursive functions at the level of those presented in this chapter.

  3. *** Recursion: One of the most important “Great Ideas” • Recursion is the process of solving a problem by dividing it into smaller sub-problems of the same form. • The italicized phrase is the essential characteristic of recursion; without it, all you have is a description of stepwise refinement of the solution. • Since the recursive decomposition generates sub-problems that have the same form as the original problem, we can use the same function or method to solve the generated sub-problems at different levels. • In terms of the structure of the code, the defining characteristic of recursion is having functions that call themselves, directly or indirectly, as the decomposition process proceeds.

  4. A Simple Illustration of Recursion • Suppose that you need to raise $1,000,000. • One possible approach is to find a wealthy donor and ask for a single $1,000,000 contribution. • Individuals with that much money are difficult to find. • Donors are much more likely to donate in the $100 range. • Another strategy would be to ask 10,000 friends for $100 each. But, most of us don’t have 10,000 friends. • There are, however, more promising strategies. • You could, for example, find ten regional coordinators and charge each one with raising $100,000. • Those regional coordinators could in turn delegate the task to local coordinators, each with a goal of $10,000, continuing the process reached a manageable contribution level.

  5. Goal: $100,000 Goal: $100,000 Goal: $100,000 Goal: $100,000 Goal: $100,000 Goal: $100,000 Goal: $100,000 Goal: $100,000 Goal: $100,000 Goal: $100,000 Goal: $10,000 Goal: $10,000 Goal: $10,000 Goal: $10,000 Goal: $10,000 Goal: $10,000 Goal: $10,000 Goal: $10,000 Goal: $10,000 Goal: $10,000 Goal: $1000 Goal: $1000 Goal: $1000 Goal: $1000 Goal: $1000 Goal: $1000 Goal: $1000 Goal: $1000 Goal: $1000 Goal: $1000 Goal: $100 Goal: $100 Goal: $100 Goal: $100 Goal: $100 Goal: $100 Goal: $100 Goal: $100 Goal: $100 Goal: $100 A Simple Illustration of Recursion (cont’d) The following diagram illustrates the recursive strategy for raising $1,000,000 Goal: $1,000,000

  6. What makes this strategy recursive is that the line Get each volunteer to collect n/10 dollars. will be implemented using the following recursive call: CollectContributions(n / 10); A Pseudocodefor Fundraising Strategy void CollectContributions(int n) { if (n <= 100) { Collect the money from a single donor. } else { Find 10 volunteers. Get each volunteer to collect n/10 dollars. Combine the money raised by the volunteers. } }

  7. *** Recursive Paradigm: Writing a Recursive Function if (test for simple case) { Compute a simple solution without using recursion } else { Break the problem down into sub-problems of the same form. Solve each of the sub-problems by calling this function recursively. Reassemble the solutions to the sub-problems into a solution for the whole. } Finding a recursive solution is mostly a matter of figuring out how to break it down so that it fits the paradigm. For this, you must do two things: • Identify simple case(s) that can be solved without recursion. • Find a recursive decomposition that breaks each instance of the problem into simpler sub-problems of the same type, which you can then solve by applying the method recursively.

  8. *** The recursive formulation of Factorial • n! = n x (n – 1)! • Thus, 4! is 4 x 3!, 3! is 3 x 2!, and so on. • To make sure that this process stops at some point, mathematicians define 0! to be 1. • The conventional mathematical definition of the factorial looks like

  9. *** Recursive vs. iterative implementation int Fact(int n) { if (n == 0) { return 1; } else { return n * Fact(n-1); } } intFactIteration(int n) { int product; product = 1; for (int i = 1; i <= n; i++) { product *= i; } return product; }

  10. int Fact(int n) { if (n == 0) { return 1; } else { return n * Fact(n - 1); } } int Fact(int n) { if (n == 0) { return 1; } else { return n * Fact(n - 1); } } int Fact(int n) { if (n == 0) { return 1; } else { return n * Fact(n - 1); } } int Fact(int n) { if (n == 0) { return 1; } else { return n * Fact(n - 1); } } n n n n 3 2 0 3 1 Fact Tracing Factorial Function Local variables and return addresses are stored in a stack. int main() { printf("Enter n: “); int n = GetInteger(); printf(“%d! = %d\n”,n, Fact(n) ); return 0; } n 6 2 1 1 Enter n: 3 3! = 6 skip simulation

  11. The Recursive “Leap of Faith” • The purpose of going through the complete decomposition of factorial is to convince you that the process works and that recursive calls are in fact no different from other method calls, at least in their internal operation. • The danger with going through these details is that it might encourage you to do the same when you write your own recursive programs. As it happens, tracing through the details of a recursive program almost always makes such programs harder to write. • Writing recursive programs becomes natural only after you have enough confidence in the process that you don’t need to trace them fully. • As you write a recursive program, it is important to believe that any recursive call will return the correct answer as long as the arguments define a simpler sub-problem of the same form. • Believing that to be true—even before you have completed the code—is called the recursive leap of faith.

  12. int Fact(int n) { if (n == 0) { return 1; } else { return n * Fact(n-1); } } *** What may go wrong with recursion? • What happens when your program calls Fact(1000000) and/or FactIteration(1000000)? • The call to Fact crashes --- usually with a stack overflow error. • The reason is that the recursive solution to the Fact has 1,000,000 nested calls. Each call requires the runtime system to create an environment for the call with local variables.... our machines just don't have that kind of memory. • The call to FactIterationinvolves a single method call and no exhaustion of memory resources • In this case, however, the result will be still wrong anyway since an integer variable cannot hold very large numbers.

  13. Exercise – a simple one • Find the sum of the integers between 1 and n using both recursive and iterative solutions 1+2+3+…+n 1+2+3+…+(n-1)+n intsumRecursion(int n) { if (n == 1)return 1; return sumRecursion(n-1) + n; } • intsumIteration(int n) { • int result = 0; • for (inti = 1; i <= n; i++) • result = result + i; • return result; • }

  14. The Fibonacci function t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 0 1 1 2 3 5 8 13 21 34 55 89 144 int Fib(int n) { if (n < 2) { return n; } else { return Fib(n - 1) + Fib(n - 2); } }

  15. Steps in the calculation of Fib(5) Recursive leap of faith

  16. *** Efficiency of the Recursive implementation of Fib • Can you implement an iterative version of Fib, say intIterativeFib(int n)? • Look at the details of Fib(5) in previous slide: • you will see that it is extremely inefficient • the same Fib term is computed many times (redundant calls to Fib()) • Which one will be faster? • Should we blame Recursion! • Can we fix this?

  17. *** Wrapper function and a subsidiary function for the more general case • Suppose we have the following function intAdditiveSequence(int n, int t0, int t1) { if (n == 0) return t0; if (n == 1) return t1; return AdditiveSequence(n-1, t1, t0+t1); }intAdditiveSequence(int n, int t0, int t1) • Then we can simply implement Fib(n) as int Fib(int n) { return AdditiveSequence(n, 0, 1); } subsidiary function for the more general case Wrapper function

  18. *** Trace and Efficiency of Fib Fib(5) = AdditiveSequence(5, 0, 1) = AdditiveSequence(4, 1, 1) = AdditiveSequence(3, 1, 2) = AdditiveSequence(2, 2, 3) = AdditiveSequence(1, 3, 5) = 5 • The new implementation is entirely recursive, and it is comparable in efficiency to the standard iterative version of the Fib() function.

  19. *** Common Errors • Recursive function may not terminate if the stopping case is not correct or is incomplete • stack overflow: run-time error • Make sure that each recursive step leads to a situation that is closer to a stopping case. (problem size gets smallerandsmaller and smaller and smaller)

  20. *** Iteration vs. Recursion • In general, an iterative version of a program will execute more efficiently in terms of time and space than a recursive version. Why? • This is because the overhead involved in entering and exiting a function is avoided in iterative version. • A recursivesolution can be easy and sometimes the most natural and logical way to solve a problem (tree traversal). • Conflict: • machine efficiency vs. programmer efficiency • It is always true that recursion can be replaced with iteration and a stack data structure.

  21. Mutual Recursion • So far, the recursive functions have called themselves directly • But, the definition is broader: • To be recursive, a function must call itself at some point during its evaluation. • For example, if a function ƒ calls a function g, which in turn calls ƒ, the function calls are still considered to be recursive. • The recursive call is actually occurring at a deeper level of nesting.

  22. Mutual Recursion Example intIsEven(unsigned int n) { if (n == 0) { return 1; /* true;*/ } else { return IsOdd(n - 1); } } Boint ol IsOdd(unsigned int n) { return !IsEven(n); }

  23. *** Exercise • The following method implements the linear search of an array of integers using iteration. Implement a linear search using recursion. Which will be more efficient? intlinearSearchIterative(int data[], int n, int target) { for (inti = 0; i < n; i++) if (data[i] == target) return i; return -1; }

  24. *** Exercise • The following method implements the binarysearch of an array of integers using iteration. Implement it with recursion. intBinarySearchIterative(intdata[], int n, int target)// BSI { // data is sorted int low = 0, high = n - 1; while (low <= high) { int mid = (low + high) / 2; if (data[mid] == target) return mid; else if (data[mid] < target) low = mid + 1; else high = mid - 1; } return -1; } int BSR(int data[], int low, int high, int target);

  25. Study Other examples in Section 4.4 • A palindrome is a string that reads identically backward and forward, such as "level” or "noon". • it is easy to check whether a string is a palindrome by iterating through its characters, • Palindromes can also be defined recursively. • Insight: any palindrome longer than a single character must contain a shorter palindrome in its interior. • For example, "level" consists of the palindrome "eve" with an "l" at each end. boolIsPalindrome(string str) { intlen = strlen(str); if (len <= 1) { return true; } else { return (str[0] == str[len – 1] && IsPalindrome(SubString(str,1, len - 2))); } } // see the textbook using wrapper function

  26. As always, the key to solving this problem lies in identifying the recursive decomposition and defining appropriate simple cases. Rewrite this method so that it uses recursion instead of iteration, taking advantage of Euclid’s insight that the greatest common divisor of x and y is also the greatest common divisor of y and the remainder of x divided by y. Solution: A Recursive gcd Function Exercise: A Recursive GCD Function One of the oldest known algorithms that is worthy of the title is Euclid’s algorithm for computing the greatest common divisor (GCD) of two integers, x and y. Euclid’s algorithm is usually implemented iteratively using code that looks like this: intgcd(int x, int y) { if (y == 0) { return x; } else { return gcd(y, x % y); } } int GCD(int x, int y) { int r = x % y; while (r != 0) { x = y; y = r; r = x % y; } return y; } intgcd (int x, int y) { int temp; while (y != 0) { temp = y; y = x % y; x = temp; } return a; } intgcd (int x, int y) { if (y == 0) return x; else return gcd (y, x % y); }

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