Reduction

1
Reduction

• Reduction is a technique for solving problems by
  rephrasing the problem in terms of another
  problem whose solution is known.
• We say that we reduce the problem into another
  known problem.Reduction ExamplesP(x) ? Q1(x),
  Q2(x), Qk(x)
• Where Qi(x) is simpler than P(x)
• Sorting ? Finding minimum (selection sort)
• Polynomial multiplication? Integer multiplication
  and addition

2
Recursion As Reduction

• P(x) ? P(x1), P(x2), P(xk)
• When xi ltlt x (xi is smaller than x)
• Recursion is a fundamental programming technique
  that can provide an elegant solution to certain
  kinds of problems.
• In recursion we solve a problem by reducing it to
  a similar, smaller problem whose input might be
  different.

3
Recursion In Programs

• public static void f()
• f()
• public static void f(int n)
• f(n-1)//reduction!
• public static void f(int n)
• if (n 0)//base case!
• return
• f(n-1)//reduction!
• Effective Recursion Rules
• Self call

4
Factorial Example

• If n is a positive integer, we define n! - the
  factorial of n - as follows
• n!123n
• A simple implementation for a method that
  computes the factorial of a number would be with
  a loop that goes over the expression and keep
  track of the partial multiplication.

5
Factorial Example

• public static long factorial(long n)
• long fact 1
• for (int i1 iltn i)
• fact i
• return fact
• Factorial Reduction
• The previous solution is simple, but we can
  find a simpler and more elegant solution with
  the following observation
• For every integer n, such that n gt 1
• n! 12...n (12...n-1)n (n-1)! n
• So we have
• 1! 1
• n! (n-1)! n , for ngt1

6
From (n-1)! to n!

• We can use these observations to derive a new
  implementation for computing n!
• If n is 1, there is no problem, we know that
  1!1.
• If ngt1 we cant say immediately the result, but
  if we only knew how to compute (n-1)! we could
  compute n! using the fact that n!(n-1)!
  n.How to find (n-1)! ?
• Note, that we can make the reduction for any
  arbitrary n, such that n gt 1. So we can reduce
  the problem from n! to (n-1)! to (n-2)! etc.
  until we will face the problem of computing 1!
  which is known.

7
Factorial Recursive Computation
5! 5 4! 4 3!
3 2! 2 1!
1!
24
6
2
1
8
Factorial Recursive Code

• public static long factorial(long n)
• if (n1)
• return 1
• return nfactorial(n-1)
• More elegant
• public static long factorial(long n)
• return ((n1) ? 1 nfactorial(n-1))

9
Recursive Methods

• A method in Java that invokes itself is called a
  recursive method.
• A recursive method has a base case which solve
  the task without a recursive call and the
  recursive part which solve the task by reducing
  it and using a recursive call.
• If there is no base case the sequence of
  reductions will not come to an end. We call this
  case an infinite recursion.

10
isNumeric() Code
/ Checks if a given string consists only
from digit symbols. / public static boolean
isNumeric(String s) for (int i0
ilts.length() i) char c
s.charAt(i) if (clt0 cgt9)
return false return true
11
Recursive isNumeric()

• Observe the following facts
• An empty string doesnt contain non-digit
  characters, therefore the method should return
  true for the empty string.
• If the first character of s is non-digit then the
  method should return false.
• If the first character of s is a digit symbol,
  then the s is numeric if and only if the rest of
  s (without the first character) is numeric.

12
isNumeric() Recursive
/ Checks if a given string consists only
from digit symbols. / public static boolean
isNumeric(String s) if (s.length()0)
return true char c s.charAt(0)
return (cgt0 clt9
isNumeric(s.substring(1))
13
Fibonacci Code
1 1 2 3 5 8 13 21 34 55 89 144...

• /
• Returns the n number in fibonacci
• series
• /
• public static int fibonacci(int n)
• if (nlt1)
• return 1
• return fibonacci(n-1) fibonacci(n-2)

14
String Reversal

• public static String reverse (String string)
• if (string.length() 0)
• return new String()
• int last string.length() - 1
• return string.charAt(last) reverse(string.subs
  tring(0, last))
• hello world
• dlrow olleh

15
Binary Search

• Searching for 32
• from 0, middle 4, to 9
• from 5, middle 7, to 9
• from 5, middle 5, to 6
• 6

1 3 7 18 20 27 32
50 70 87
0 1 2 3 4 5 6
7 8 9
16
Binary Search

• public class Searcher
• public static int binarySearch (int values ,
  int target)
• return binarySearch(values , 0 ,
  values.length-1,target)
• public static int binarySearch (int values, int
  from, int to, int target)
• if (from gt to) return -1
• int middle (from to) / 2
• if (valuesmiddle target) return middle
• if (valuesmiddle gt target)
• return binarySearch(values, from, middle -1,
  target)
• else
• return binarySearch(values,middle 1, to,
  target)

17
I/O - Text Files

• System
• The System class contains several
• useful class fields and methods.
• It cannot be instantiated.
• err - The "standard" error output stream
• in - The "standard" input stream
• out - The "standard" output stream.
•  

18
Text Files

• import java.io.
• public class Copy
• public static void main(String args)
• try
• File inputFile new File("c\\read.txt"
  )
• File outputFile new
  File("c\\write.txt")
• FileReader in new FileReader(inputFile
  )
• FileWriter out new FileWriter(outputFi
  le)
• int c
• while ((c in.read()) ! -1)
• out.write(c)
• in.close()
• out.close()
• catch(IOException e)
• System.out.print("Error " e)
• System.exit(1)

19
Text Files

• import java.io.
• import java.util.
• public class TextFile
• public static void main(String args)
• // write to file
• try
• FileWriter fi new FileWriter("c\\out.txt")
  
• PrintWriter os new PrintWriter(fi)
• os.println("First line to write!")
• os.println("Second line!")
• os.close()
• catch(IOException e)
• System.out.print("Error while writing "
  e)
• System.exit(1)

20
Text Files

• // read from file
• try
• FileReader fr new FileReader("c\\out.txt")
  
• BufferedReader is new BufferedReader(fr)
• String s1 is.readLine()
• while (s1 ! null)
• StringTokenizer st new StringTokenizer(s1)
• for(int i0 st.hasMoreTokens() i)
• System.out.print(" " i ") "
  st.nextToken())
• System.out.println()
• s1 is.readLine()
• is.close()
• catch(IOException e)
• System.out.print("Error while reading"
  e)
• System.exit(2)

21
Directory / Folders

• import java.io.
• public class DirectoryFiles
• public static void main(String args)
• File fd new File("c\\NewFolder")
• if(fd.exists())
• if(fd.isDirectory())
• File fileList fd.listFiles()
• for(int i0 i lt fileList.length i)
• System.out.println(fileListi.getName
  ())
• else
• System.out.println("This is not a
  Directory !")
• else
• System.out.println("Folder dose not exist
  !")