day7 - COP 3503 Computer Science II CLASS NOTES DAY#7...

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COP 3503 – Computer Science II CLASS NOTES - DAY #7 Problem Solving Techniques We now shift our focus from run-time complexity to introducing several general techniques which can be applied to problem solving. Different problems require different approaches in defining their solutions. To this end the algorithm designer needs to have a good understanding of many different techniques that can be applied toward solving the problem at hand. This set of notes introduces the problem solving techniques of recursion , divide and conquer , dynamic programming , and backtracking . Recursion A method or function either directly or indirectly calls itself. Can be a powerful problem solving tool. Many algorithms and mathematical properties are naturally expressed in a recursive manner. Some programming languages are inherently recursive (Lisp, Scheme). If properly done, a recursive function F calling itself, does not lead to an infinite cycle of the function calling itself. Instead, each recursive call is made on a different, generally simpler, instance of the problem. Day 7 - 1 Example: Consider the factorial function, n!. 0! = 1 n! = n * (n-1) * … * 3 * 2 *1 To determine 6! using this definition requires that we calculate: 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720 Defining the factorial function recursively, we have: 1 if n = 0 fact (n) = 1 if n = 1 n * fact(n-1) if n >= 2 Now 6! = 6 * 5! = 6 * (5 * 4!) = 6 * (5 * (4 * 3!)) = 6 * (5 * (4 * (3 * 2!))) = 6 * (5 * (4 * (3 * (2 * 1!)))) = 6 * (5 * (4 * (3 * (2 * (1 * 0!))))) = 6 * 5 * 4 * 3 * 2 * 1 * 1

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A recursive function is a function which is defined in terms of itself. For example, it is common to express the factorial function, f(n) = n! where n is an integer , as the recursive function: - = 1 n ) 1 n ( f n 1 n 1 ) n ( f This definition states that f(n) equals 1 whenever n is less than or equal to 1. For example, f(-3) = f(0) = f(1) = 1. However, when n is more than 1, f(n) is defined recursively, since the definition of f now contains an occurrence of f on the right side. Note that the use of f on the right side does not constitute a circular definition since the parameter of f on the right side is smaller than the one on the left side. For example, the from the definition we have: f(2) = 2f(1) and the definition also gives us f(1) = 1, substitution then provides the answer that f(2) = 2(1) = 2. Similarly, for f(3) the definition gives us: f(3) = 3f(2), from above we know that f(2) = 2f(1) and f(1) = 1, thus f(3) = 3f(2) = 3(2(f(1))) = 3(2(1))) = 6. For a recursive definition of f(n) (assuming direct recursion) to be a complete specification of the function f , it must meet the following requirements: The definition must include a base component in which f(n) is defined directly (i.e., without recursion) for one or more values of n . For simplicity, we assume that the domain of f is the nonnegative integers and that the base covers the case 0 n k for some constant k . (Note: it is possible to have recursive definitions in which the base covers the case k instead, but these definitions occur infrequently.) In the recursive component all occurrences of f on the right side must have a parameter smaller than n so that repeated application of the recursive
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day7 - COP 3503 Computer Science II CLASS NOTES DAY#7...

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