RECURSION

What is recursion? a function calls itself direct recursion a function calls its invoker indirect recursion f1 f f2

Recursion is an alternative to iteration (loop) often more "elegant" and concise than a loop sometimes very inefficient recursion should be considered when a problem can be defined in terms of successive smaller problems of the same type, i.e. recursively eventually the problem gets small enough that the answer is known (base case) reducing the problem size leads to the base case sometimes the "work" of the algorithm is done after the base case is reached

Outline of a Recursive Function if (answer is known) provide the answer else make a recursive call to solve a smaller version of the same problem base case recursive case - “leap of faith”

Factorial (n) - recursive Factorial (n) = n * Factorial (n-1) for n > 0 Factorial (0) = 1 int RecFact (int n) { if (n > 0) return n * RecFact (n-1); else return 1; } base case

How Recursion Works a recursive function call is handled like any other function call each recursive call has an activation record on the stack stores values of parameters and local variables when base case is reached return is made to previous call the recursion “unwinds”

Recursive Call Tree RecFact (4) 24 int RecFact (int n) 4 { 3 2 1 6 24 int RecFact (int n) { if (n > 0) return n * RecFact (n-1); else return 1; }

Factorial (n) - iterative Factorial (n) = n * (n-1) * (n-2) * ... * 1 for n > 0 Factorial (0) = 1 int IterFact (int n) { int fact =1; for (int i = 1; i <= n; i++) fact = fact * i; return fact; }

Another Recursive Definition Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, …. 1 for n == 1 fib(n) = 1 for n == 2 fib(n-2) + fib(n-1) for n>2

Tracing fib(6) 5 6 4 4 3 3 2 3 2 2 1 2 1 2 1

When to use recursion? if recursive and iterative algorithms are of similar efficiency --> iteration if the problem is inherently recursive and a recursive algorithm is less complex to write than an iterative algorithm --> recursion recursion often simpler than iteration when it is not tail recursion or there are multiple recursive calls recursion very inefficient when values are recomputed

Euclid'sAlgorithm Finds the greatest common divisor of two nonnegative integers that are not both 0 Recursive definition of gcd algorithm gcd (a, b) = a (if b is 0) gcd (a, b) = gcd (b, a % b) (if b != 0) Write a recursive gcd function prototype? definition?

Ackermann's function A(0, n) = n + 1 A(m+1, 0) = A(m, 1) A(m+1, n+1) = A(m, A(m+1, n))

Iterative gcd int gcd (int a, int b) { int temp; while (b != 0) temp = b; b = a % b; a = temp; } return a;

iterative binary search bool binarySearch (const int a[ ], int numItems, int item) { bool found = false; int first = 0; int last = numItems – 1; while (first <= last && ! found) { mid = (first + last) / 2; if (item == a[mid]) found = true; else if (item > a[mid]) first = mid +1; else last = mid – 1; } return found;

bool binarySearch(const int a[ ], int first, int last, int item) { if (first > last) return false; // base case 1 (unsuccessful search) else int mid = (first + last) / 2; if (item == a[mid]) return true; // base case 2 (successful search) else if (item > a[mid]) return binarySearch(a, mid+1, last, item); // X return binarySearch(a, first, mid-1, item); // Y }

a = [1 5 9 12 15 21 29 31] binarySearch(a, 0, 7, 6) binarySearch(a, 0, 7, 9) 2, 1, 0, 7, 3 0, 2, 1 2, 2, 2 X Y false 0, 7, 3 0, 2, 1 2, 2, 2 X Y true base case 2 base case 1

What Does a Compiler Do? Lexical analysis Parsing Generate object code divide a stream of characters into a stream of tokens total = cost + 0.08 * cost; if ( ( cond1 && ! cond2 ) ) Parsing do the tokens form a valid program, i.e. follow the syntax rules? Generate object code

BNF (Backus-Naur form) a language used to define the syntax rules of a programming language consists of productions – rules for forming some construct of the language meta-terms – symbols of BNF terminals – appear as shown non-terminals – syntax defined by another production

Syntax Rules for a simplified boolean expression meta-terms are in red bexpr -> bterm || bterm | bterm bterm -> bfactor && bfactor | bfactor bfactor -> !bfactor|(bexpr)|true|false|ident ident -> alpha {alpha|digit|_} alpha -> a .. z | A .. Z digit -> 0 .. 9 meta-terms are in red terminals are in blue non-terminals are in black

bexpr -> bterm || bterm | bterm does a sequence of tokens form a valid bexpr? if (there is a valid bterm) if (nextToken is ||) if (there is a valid bterm) return true else return false else return true else return false

Comparing Algorithms and ADT Data Structures big Oh Notation Comparing Algorithms and ADT Data Structures

Algorithm Efficiency a measure of the amount of resources consumed in solving a problem of size n time space benchmarking – code the algorithm, run it with some specific input and measure time taken better for measuring and comparing the performance of processors than for measuring and comparing the performance of algorithms Big Oh (asymptotic analysis) provides a formula that associates n, the problem size, with t, the processing time required to solve the problem