Recursion

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Chapter 5

• Recursion

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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

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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

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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 listlowerIndex
• else
• max largest(list, lowerIndex 1,
  upperIndex)
• if(listlowerIndex gt max)
• return listlowerIndex
• 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)
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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 gt 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 gt 0)
• decToBin(num/base, base)
• System.out.println(num base)

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Execution of decToBin(13,2)
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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 ExampleSierpinski 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 gt 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

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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