chapter 5
DESCRIPTION
Chapter 5. Recursion. 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 - PowerPoint PPT PresentationTRANSCRIPT
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
Data Structures Using Java 4
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
Data Structures Using Java 5
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
Data Structures Using Java 6
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
Data Structures Using Java 7
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
Data Structures Using Java 9
Designing Recursive Methods
• Understand problem requirements
• Determine limiting conditions
• Identify base cases
Data Structures Using Java 10
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
Data Structures Using Java 11
Recursive Factorial Function
public static int fact(int num)
{
if(num == 0)
return 1;
else
return num * fact(num – 1);
}
Data Structures Using Java 12
Recursive Factorial Trace
Data Structures Using Java 13
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; }}
Data Structures Using Java 14
Execution of largest(list, 0, 3)
Data Structures Using Java 15
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);}
Data Structures Using Java 16
Execution of rFibonacci(2,3,5)
Data Structures Using Java 17
Towers of Hanoi Problem with Three Disks
Data Structures Using Java 18
Towers of Hanoi: Three Disk Solution
Data Structures Using Java 19
Towers of Hanoi: Three Disk Solution
Data Structures Using Java 20
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); }}
Data Structures Using Java 21
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); }}
Data Structures Using Java 22
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.
Data Structures Using Java 24
Sierpinski Gaskets of Various Orders
Data Structures Using Java 25
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
Data Structures Using Java 26
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); }}
Data Structures Using Java 27
Programming Example: Sierpinski Gasket Input
Data Structures Using Java 28
Programming Example: Sierpinski Gasket Input
Data Structures Using Java 29
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.
Data Structures Using Java 31
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
Data Structures Using Java 32
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
Data Structures Using Java 33
Solution to 8-Queens Puzzle
Data Structures Using Java 34
4-Queens Puzzle
Data Structures Using Java 35
4-Queens Tree
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8 X 8 Square Board
Data Structures Using Java 37
Chapter Summary
• Recursive Definitions
• Recursive Algorithms
• Recursive methods
• Base cases
• General cases
Data Structures Using Java 38
Chapter Summary
• Tracing recursive methods
• Designing recursive methods
• Varieties of recursive methods
• Recursion vs. Iteration
• Backtracking
• N-Queens puzzle