data structures using java1 chapter 5 recursion. data structures using java2 chapter objectives...

Data Structures Using Java 1

Chapter 5

Recursion

Data Structures Using Java 2

Chapter Objectives

• Learn about recursive definitions• Explore the base case and the general case of a

recursive definition• Discover what a recursive algorithm is• Learn about recursive methods• Explore how to use recursive methods to

implement recursive algorithms• Learn how recursion implements backtracking

Data Structures Using Java 3

Recursive Definitions

• Recursion– Process of solving a problem by reducing it to

smaller versions of itself

• Recursive definition– Definition in which a problem is expressed in

terms of a smaller version of itself– Has one or more base cases

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Recursive Definitions• Recursive algorithm

– Algorithm that finds the solution to a given problem by reducing the problem to smaller versions of itself

– Has one or more base cases– Implemented using recursive methods

• Recursive method– Method that calls itself

• Base case– Case in recursive definition in which the solution is

obtained directly – Stops the recursion

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Recursive Definitions

• General solution– Breaks problem into smaller versions of itself

• General case– Case in recursive definition in which a smaller

version of itself is called– Must eventually be reduced to a base case

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Tracing a Recursive Method

Recursive method– Has unlimited copies of itself– Every recursive call has

• its own code• own set of parameters• own set of local variables

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Tracing a Recursive Method

After completing recursive call• Control goes back to calling environment• Recursive call must execute completely

before control goes back to previous call• Execution in previous call begins from point

immediately following recursive call

Data Structures Using Java 8

Recursive Definitions

• Directly recursive: a method that calls itself• Indirectly recursive: a method that calls another

method and eventually results in the original method call

• Tail recursive method: recursive method in which the last statement executed is the recursive call

• Infinite recursion: the case where every recursive call results in another recursive call

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Designing Recursive Methods

• Understand problem requirements

• Determine limiting conditions

• Identify base cases

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Designing Recursive Methods

• Provide direct solution to each base case

• Identify general case(s)

• Provide solutions to general cases in terms of smaller versions of itself

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Recursive Factorial Function

public static int fact(int num)

{

if(num == 0)

return 1;

else

return num * fact(num – 1);

}

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Recursive Factorial Trace

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Recursive Implementation: Largest Value in Array

public static int largest(int list[], int lowerIndex, int upperIndex)

{ int max; if(lowerIndex == upperIndex) //the size of the sublist is 1 return list[lowerIndex]; else { max = largest(list, lowerIndex + 1, upperIndex); if(list[lowerIndex] >= max) return list[lowerIndex]; else return max; }}

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Execution of largest(list, 0, 3)

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Recursive Fibonacci

public static int rFibNum(int a, int b, int n){ if(n == 1) return a; else if(n == 2) return b; else return rFibNum(a, b, n - 1) + rFibNum(a, b, n - 2);}

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Execution of rFibonacci(2,3,5)

Data Structures Using Java 17

Towers of Hanoi Problem with Three Disks

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Towers of Hanoi: Three Disk Solution

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Towers of Hanoi: Three Disk Solution

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Towers of Hanoi: Recursive Algorithm

public static void moveDisks(int count, int needle1, int needle3, int needle2)

{ if(count > 0) { moveDisks(count - 1, needle1, needle2, needle3); System.out.println("Move disk “ + count + “ from “ + needle1 + “ to “ + needle3 + ".“); moveDisks(count - 1, needle2, needle3, needle1); }}

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Decimal to Binary: Recursive Algorithm

public static void decToBin(int num, int base)

{ if(num > 0) { decToBin(num/base, base); System.out.println(num % base); }}

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Execution of decToBin(13,2)

Data Structures Using Java 23

Sierpinski Gasket

Suppose that you have the triangle ABC.

Determine the midpoints P,Q, and R of the sides AB, AC, and BC, respectively.

Draw the lines PQ,QR, and PR.

This creates three triangles APQ, BPR, and CRQ of similar shape as the triangle ABC.

Process of finding midpoints of sides, then drawing lines through midpoints on triangles APQ, BPR, and CRQ is called a Sierpinski gasket of order or level 0, level 1, level 2, and level 3, respectively.

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Sierpinski Gaskets of Various Orders

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Programming Example:Sierpinski Gasket

• Input: non-negative integer indicating level of Sierpinski gasket

• Output: triangle shape displaying a Sierpinski gasket of the given order

• Solution includes– Recursive method drawSierpinski– Method to find midpoint of two points

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Recursive Algorithm to Draw Sierpinski Gasket

private void drawSierpinski(Graphics g, int lev, Point p1, Point p2, Point p3){ Point midP1P2; Point midP2P3; Point midP3P1; if(lev > 0) { g.drawLine(p1.x, p1.y, p2.x, p2.y); g.drawLine(p2.x, p2.y, p3.x, p3.y); g.drawLine(p3.x, p3.y, p1.x, p1.y); midP1P2 = midPoint(p1, p2); midP2P3 = midPoint(p2, p3); midP3P1 = midPoint(p3, p1); drawSierpinski(g, lev - 1, p1, midP1P2, midP3P1); drawSierpinski(g, lev - 1, p2, midP2P3, midP1P2); drawSierpinski(g, lev - 1, p3, midP3P1, midP2P3); }}

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Programming Example: Sierpinski Gasket Input

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Programming Example: Sierpinski Gasket Input

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Recursion or Iteration?

• Two ways to solve particular problem– Iteration– Recursion

• Iterative control structures: uses looping to repeat a set of statements

• Tradeoffs between two options– Sometimes recursive solution is easier– Recursive solution is often slower

Data Structures Using Java 30

8-Queens Puzzle

Place 8 queens on a chessboard (8 X 8 square board) so that no two queens can attack each other. For any two queens to be non-attacking, they cannot be in the same row, same column, or same diagonals.

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Backtracking Algorithm

• Attempts to find solutions to a problem by constructing partial solutions

• Makes sure that any partial solution does not violate the problem requirements

• Tries to extend partial solution towards completion

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Backtracking Algorithm

• If it is determined that partial solution would not lead to solution– partial solution would end in dead end– algorithm backs up by removing the most

recently added part and then tries other possibilities

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Solution to 8-Queens Puzzle

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4-Queens Puzzle

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4-Queens Tree

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8 X 8 Square Board

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Chapter Summary

• Recursive Definitions

• Recursive Algorithms

• Recursive methods

• Base cases

• General cases

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Chapter Summary

• Tracing recursive methods

• Designing recursive methods

• Varieties of recursive methods

• Recursion vs. Iteration

• Backtracking

• N-Queens puzzle