CH03 - Recurrence Equations vs. Recursive Procedures...

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TECH Computer Science Recursion and Induction 4 For advanced algorithm development, recursion is an essential design technique Recursive Procedures What is a Proof? Induction Proofs Proving Correctness of Procedures Recurrence Equations Recursion Trees Recurrence Equations vs. Recursive Procedures Recurrence Equations: 4 defines a function over the natural numbers, say T(n), in terms of its own value at one or more integers smaller than n. 4 T(n) is defined inductively. 4 There are base cases to be defined separately. 4 Recurrence equation applies for n larger than the base cases Recursive Procedures: 4 a procedure calls a unique copy of itself 4 converging to a base case (stopping the recursion) e.g. Fibonacci Function Recurrence Equation: e.g. Fibonacci Function 4 fib(0) = 0; fib(1) = 1; // base cases 4 fib(n) = fib(n-1) + fib(n-2) // all n>2 Recursive Procedure: int fib(int n) 4 int f, f1, f2; 4 1. if (n < 2) 4 2. f = n; // base cases 4 3. else 4 4. f1 = fib( n-1 ); 4 5. f2 = fib( n-2 ); 4 6. f = f1 + f2; 4 7. return f; The working of recursive procedure a unique copy for each call to itself 4 individual procedure invocation at run time 4 i.e. activation frame e.g. The working of fib(n) 4 main () 4 int x = fib(3); Activation Tree 4 Each node corresponds to a different procedure invocation, just at the point when it is about to return. 4 A preorder traversal visits each activation frame in order of its creation Analysis for algorithm without loops In a computation without loops, but possible with recursive procedure calls: 4 The time that any particular activation frame is on the top of the frame stack is O(L), 4 where L is the number of lines in the procedure that contain either a simple statement or a procedure call. 4 The total computation time is θ (C), 4 where C is the total number of procedure calls that occur during the computation.
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Designing Recursive Procedures // Think Inductively converging to a base case (stopping the recursion) 4 identify some unit of measure (running variable) 4 identify base cases assume p solves all sizes 0 through 100 4 assume p99 solve sub-problem all sizes 0 through 99 4 if p detect a case that is not base case it calls p99 p99 satisfies: 4 1. The sub-problem size is less than p’s problem size 4 2. The sub-problem size is not below the base case 4 3. The sub-problem satisfies all other preconditions of p99 (which are the same as the preconditions of p) Recursive Procedure design e.g. Problem: 4 write a delete(L, x) procedure for a list L 4 which is supposed to delete the first occurrence of x. 4 Possibly x does not occur in L. Strategy: 4 Use recursive Procedure 4 The size of the problem is the number of elements in list L 4 Use IntList ADT 4 Base cases: ?? 4
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CH03 - Recurrence Equations vs. Recursive Procedures...

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