visual c++ programming: concepts and projects

Visual C++ Programming: Concepts and Projects

Chapter 10A: Recursion (Concepts)

Objectives

In this chapter, you will:• Discover how to recognize a recursive problem• Code a recursive method• Use recursion to draw fractal images• Analyze recursion• Create and code menus

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Factorial Numbers

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• Factorial numbers– Example: 5! = 5 x 4 x 3 x 2 x 1– Can be easily computed using a loop

Factorial Numbers (continued)

• Factorial numbers– Can also be computed as separate steps from the

bottom up– Solving the problem from the bottom up will lead

us to a new solution strategy called recursion

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Recursion

• Recursion• Refers to re-occurring events of a similar nature• Mathematical example: factorial numbers• Everyday example: a chain of phone calls• A problem (finding an apartment) must be solved by a

chain of similar events (phone calls)• The events repeat themselves until an answer is

obtained• The answer follows the same path back

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Recursion (continued)

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Recursion (continued)

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Recursion

• Recursive solutions• Have a recursive case

• A call to continue the same operation at another level

• Have a base case• A stopping point

• Use backtracking• To go back to the previous operation at a higher level

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Recursion (continued)

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• Algorithm for computing factorial numbers

Recursion (continued)

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Recursion (continued)

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Recursion (continued)

• Recursion and factorial numbers– Recursive definition of a factorial number (n!)– For all positive integers n, n! is defined as follows• If n = 0 or 1, then n! = 1 (this is the base case)• If n > 1, then n! = n x (n-1)! (this is the recursive case)

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Recursion (continued)

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– Recursive definitions can be translated easily into recursive methods• Example: Factorial() method

Recursion (continued)

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Recursion (continued)

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Recursion vs. Iteration

• Iteration is usually faster– Does not require background overhead to keep

track of each method call• Recursion is more elegant– More closely matches the mathematical definition

of the operation it is performing• Recursion can more easily solve some

problems– Example: generating fractal images

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Recursion vs. Iteration (continued)

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Creating Fractal Images

• Draw a line – Then draw three lines half the size (at 90 degree

angles to one another) from the end of the line you just drew

• Apply this strategy recursively to every line you draw

• Do this until you have reached a predetermined stopping point (the base case)

• Example: recursive crosses

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Creating Fractal Images (continued)

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Creating Fractal Images (continued)

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Creating Fractal Images (continued)

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Creating Fractal Images (continued)

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Creating Fractal Images (continued)

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Creating Fractal Images (continued)

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Computer-Generated Fractal Images

• Drawing lines with Graphics class method DrawLine()

• Use DrawLine() to create a line– Parameters• Pen• x1, y1 (coordinates of the start of the line)• x2, y2 (coordinates of the end of the line)

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Computer-Generated Fractal Images (continued)

• The starting coordinates of the line are (x, y)• The ending coordinates are (newX, newY)

• In a recursive solution, there are many levels of line-drawing tasks

• The coordinates of the end of the previous line become the coordinates of the starting point of the next one

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Recursive DrawBranch() Method

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Recursive DrawBranch() Method (continued)

• DrawBranch() is recursive• Each level of recursion draws three new lines

from the end of the previous one• The DrawBranch() method is called three

times to do this• The base case is the last level of recursion

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Recursive DrawBranch() Method (continued)

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Recursive DrawBranch() Method (continued)

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Recursive DrawBranch() Method (continued)

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• Algorithm for method DrawBranch()

Recursive DrawBranch() Method (continued)

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• A single branch with five levels of recursion looks like a pattern of crosses

• You will develop a program in the Tutorial to implement this

Recursive DrawBranch() Method (continued)

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Summary

• Recursion involves a method calling itself• Recursive methods must have:– A base case– A recursive case

• Recursive backtracking– After the base case is reached, the method

returns to the previous one– Backtracking continues until the program returns

to the original method call

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Summary (continued)

• Some problems have easy and elegant recursive solutions– Tasks with mathematically inductive definitions

are naturally recursive– Example: factorial numbers– Example: fractal images

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