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vb.net code to generate barcode RECURSION in Java
RECURSION GS1  13 Decoder In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Make EAN13 In Java Using Barcode generation for Java Control to generate, create UPC  13 image in Java applications. [CHAP. 9
Recognize GS1  13 In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. Barcode Generator In Java Using Barcode generator for Java Control to generate, create barcode image in Java applications. basis step and part (ii) is called the inductive step. The assumption that the preceding statements are true is called the inductive hypothesis. The recursive factorial function is correct. To prove this fact, we first verify the basis. The call f(0) returns the correct value 1 because of the first part: Recognize Bar Code In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. EAN13 Drawer In C#.NET Using Barcode generation for VS .NET Control to generate, create UPC  13 image in .NET applications. if (n < 2) { return 1; } EAN13 Supplement 5 Maker In VS .NET Using Barcode creation for ASP.NET Control to generate, create GTIN  13 image in ASP.NET applications. Drawing EAN / UCC  13 In .NET Using Barcode generation for Visual Studio .NET Control to generate, create European Article Number 13 image in .NET applications. Next, we assume that the function returns the correct value for all integers less than some n > 0. Then the second part EAN13 Generator In VB.NET Using Barcode generator for VS .NET Control to generate, create European Article Number 13 image in .NET applications. UPCA Supplement 5 Generator In Java Using Barcode generator for Java Control to generate, create Universal Product Code version A image in Java applications. return n*f(n1); Linear 1D Barcode Creation In Java Using Barcode encoder for Java Control to generate, create Linear Barcode image in Java applications. Generating Matrix 2D Barcode In Java Using Barcode drawer for Java Control to generate, create Matrix Barcode image in Java applications. will return the correct value n! because (by the inductive hypothesis) the call f(n1) will return (n 1)! and n! = n (n 1). Note that we are using the strong principle of mathematical induction here (also called the second principle of mathematical induction). In this version, the inductive hypothesis allows us to assume that all the preceding statements are true. In the weak (or first ) principle, we are allowed to assume that only the single preceding statement is true. But since these two principles are equivalent (i.e., they are both valid methods of proof), it is usually better to apply strong induction. The Euclidean algorithm is correct. We can use (strong) induction to prove this fact. (See page 322.) If m and n are equal, then that number is their greatest common divisor. So the function returns the correct value in that case because of the part Generating RM4SCC In Java Using Barcode printer for Java Control to generate, create Royal Mail Barcode image in Java applications. Generating Code128 In VB.NET Using Barcode drawer for Visual Studio .NET Control to generate, create Code128 image in VS .NET applications. if (m == n) { return n; } If m and n are not equal, then the function returns either gcd(m,nm) or gcd(mn,n). To see that this too is the correct value, we need only realize that all three pairs (m,n), (m,nm), and (mn,n) will always have the same greatest common divisor. This fact is a theorem from Recognize Barcode In VS .NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. Generating Code 39 Extended In .NET Using Barcode generation for .NET framework Control to generate, create Code 3 of 9 image in .NET applications. number theory. COMPLEXITY ANALYSIS The complexity analysis of a recursive algorithm depends upon the solubility of its recurrence relation. The general technique is to let T(n) be the number of steps required to carry out the algorithm on a problem of size n. The recursive part of the algorithm translates into a recurrence relation on T(n). Its solution is then the complexity function for the algorithm. The recursive factorial function runs in O(n) time. Let T(n) be the number of recursive calls made from the initial call f(n) to the function in Example 9.2 on page 165. Then T(0) = T(1) = 0, because if n < 2, no recursive calls are made. If n > 1, then the line Code 128C Maker In Java Using Barcode printer for BIRT Control to generate, create Code128 image in BIRT reports applications. EAN / UCC  13 Generation In ObjectiveC Using Barcode creation for iPhone Control to generate, create EAN13 image in iPhone applications. return n*f(n1); Create Code 3 Of 9 In ObjectiveC Using Barcode printer for iPhone Control to generate, create Code 39 Full ASCII image in iPhone applications. Read EAN13 Supplement 5 In Visual Basic .NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. executes, making the recursive call f(n1). Then the total number of recursive calls is 1 plus the number of calls that are made from f(n1). That translates into the recurrence relation T(n) = 1 + T(n 1) The solution to this recurrence is T(n) = n 1, for n > 0 This conclusion is obtained in two stages: First we find the solution; then we use induction to prove that it is correct. The simplest technique for finding the solution to a recurrence relation is to make a table of values and look for a pattern. This recurrence relation says that each value of T(n) is 1 more than the previous value. So the solution f(n) = n 1 is pretty obvious. CHAP. 9] RECURSION
Now to prove that T(n) = n 1 for all n > 0, let f(n) = n 1 and apply the (weak) principle of mathematical induction. The basis case is where n = 1. In that case, T(n) = T(1) = 0 and f(n) = f(1) = (1) 1 = 0. For the inductive step, we assume that T(n) = f(n) for some n > 0 and then deduce from that assumption that T(n +1) = f(n +1): T(n +1) = 1 + T(n) = 1 + f(n) = 1 + (n 1) = n f(n +1) = (n +1) 1 = n That completes the proof. Now that we have determined that the complexity function for this recursive implementation of the factorial function T(n) = n 1, we can conclude that this implementation will run in O(n) time. This means that its execution time will be proportional to the size of its argument n. If it takes 3 milliseconds to compute 8!, then it should take about 6 milliseconds to compute 16!. DYNAMIC PROGRAMMING In most cases, recursion is very inefficient because of its frequent function calls. So an iterative implementation may be better if it is not too complex. Another alternative is to implement the recurrence relation by storing previously computed values in an array instead of recomputing them with recursive function calls. This method is called dynamic programming. EXAMPLE 9.14 Dynamic Programming Implementation of the Fibonacci Function public static int fib(int n) { if (n < 2) { 3 return n; 4 } 5 int[] f = new int[n]; 6 f[0] = 0; 7 f[1] = 1; 8 for (int i=2; i<n; i++) { // store the Fibonacci numbers 9 f[i] = f[i1] + f[i2]; 10 } 11 return f[n1] + f[n2]; 12 } This implementation uses a dynamic array f[] of n integers to store the first n Fibonacci numbers. THE TOWERS OF HANOI We have seen important examples of functions that are more naturally defined and more easily understood by using recursion. For some problems, recursion is the only reasonable method of solution. The Towers of Hanoi puzzle is a classic example of a problem whose solution demands recursion. The game consists of a board with three vertical pegs labeled A, B, and C, and a Figure 9.5 The Towers of Hanoi puzzle sequence of n disks with holes in their centers. (See Figure 9.5.) The radii of the disks are in an arithmetic progression (e.g., 5cm, 6cm, 7cm, 8cm, . . .) and are mounted on peg A. The rule is that no disk may be above a smaller disk on the same peg. The objective of the game is to move all the disks from peg A to peg C, one disk at a time, without violating the rule.

