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05+-+Recursion+and+Iteration - Review General Recursion...

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1/26/11 1 Recursion and Iteration EECS 280 Programming and Introductory Data Structures Review General Recursion A recursive problem is one that is defined in terms of itself. A recursive problem has two important features: 1. There be one or more trivial base cases. 2. There be a way to solve “bigger” instances by first solving “smaller” instances, then performing a simple computation. Recursive problems usually follow the structure of an inductive proof. Once the problem has been recursively stated, it is fairly simple to write a recursive program to solve it Unfortunately, these types of recursive programs require one stack frame for every recursive call; this can be substantial. Review Tail Recursion To solve the general recursive stack problem, we considered tail recursive programs which require only “constant space”. It doesn't matter how “large” the problem to be solved is, we only need a fixed number of stack frames to solve it. For example, in our tail-recursive factorial solution, we only need the first fact_helper stack frame, no matter the size of N. This is also true when solving the problems with iteration . In fact, the two versions of factorial() we've seen are very, very similar – that similarity is no accident! Recursion and Iteration Are they related? What we will show today: 1. For any tail-recursive function, you can write an iterative equivalent. In other words, tail-recursion is just iteration. 2. This equivalence is not true for general recursive programs. Note: There are some recursive programs that cannot be solved in constant space, including the tree traversal problem from Project 2. We want to show that tail-recursion is “no more powerful” than iteration (i.e. for any computation that we could express tail-recursively, we could also express it iteratively). We do this by giving a method for converting an arbitrary tail-recursive function to an iterative one. This technique is called a “proof by construction”. (Tail-Recursion == Iteration) True while (<boolean-condition>) <body-statement> On entry, the condition is evaluated. If the condition is false, the loop terminates. Otherwise, the body statement is evaluated, and the loop is re-entered. do <body-statement> while (<boolean-condition>); On entry, the body statement is evaluated followed by the condition. If the condition is false, the loop terminates. Otherwise, it is re-entered at the body statement. Reviewing Loops
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1/26/11 2 for ( <init-statement>; <boolean-condition>; <post-body> ) <body-statement> When a for-loop is entered for the first time, the init-statement is executed. Then, the condition is evaluated. If it’s false, the loop terminates. Otherwise, the body is evaluated, then the post-body is evaluated, and finally the loop is resumed at the point of evaluating the boolean condition.
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