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“Not all recursive solutions are better than iterative solutions…”

“… recursion, however, can provide elegantly simple solutions to problems of great complexity”

- Textbook, p. 49

CompSci 105 SS 2005

Principles of Computer Science

Lecture 7: Searching via Recursion

Lecturer: Santokh Singh

Assignment 1

due Today!!!

Date:14.1.05

2public static int fact( int n ) {

if ( n == 0 ) {

return 1;

}

else {

return n * fact( n-1 );

}

}

Textbook, p. 57-58

n = 0 A: fact(n-1) = ?return = ?

n = 1A: fact(n-1) = ?return = ?

n = 3A: fact(n-1) = ?return = ?

n = 2A: fact(n-1) = ?return = ?

A

A

A

A

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Proving recursive algorithms

1. Prove each base cases works

2. Prove each recursive case works*

3. Prove all recursive calls meet a base casepublic static int fact( int n ) {

if ( n == 0 ) {

return 1;

}

else {

return n * fact( n-1 );

}

} Proof by inductionTextbook, p. 751-5

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Invariants

public static int fact( int n ) {

if ( n == 0 ) {

return 1;

}

else {

// Invariant:

return n * fact( n-1 );

}

}Textbook, p. 56

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Reversing a String

Textbook, p. 59ff

god“ ”• If string is empty

do nothing• Otherwise,

Output last character

Reverse substring of first (n-1) characters

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//General Description: Converts a decimal number to // Binary representation. //Precondition:    Input is a positive decimal // number. //Postcondition:    Returns Binary representation // of input. //----------------------------------------------------- private static String binVal(long decVal){

String retString = "";     if(decVal / 2 > 0) retString = binVal(decVal / 2);      retString += (decVal % 2);            return retString; }

Recursive Decimal to Binary Conversionby student “Charlie” during his CS 105 here

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Loop invariants

Is there something that we want to be true every time the while test is executed?

// Computes the sum of

// the first n items.

int sum = 0;

int j = 0;

while (j < n) {

sum += item[j];

j++;

}

Textbook, p. 10

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sum contains the sum of the first j items

j is in the range 0..n-1

Textbook, p. 10

// Computes the sum of

// the first n items.

int sum = 0;

int j = 0;

while (j < n) {

sum += item[j];

j++;

}

Loop Invariants:

9

sum contains the sum of the first j items

j is in the range 0..n-1

Textbook, p. 10

// Computes the sum of

// the first n items.

int sum = 0;

int j = 0;

while (j < n) {

sum += item[j];

j++;

}

Loop Invariants:

31item:

j:

n: 2

0 1

sum:

10

sum contains the sum of the first j items

j is in the range 0..n

Textbook, p. 10

// Computes the sum of

// the first n items.

int sum = 0;

int j = 0;

while (j <= n) {

sum += item[j];

j++;

}

Loop Invariants:

an error

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Searching in a sorted array

Binary search

Searching in an unsorted array

Finding the smallest item

Finding the kth smallest item

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Divide and Conquer

Search Phone Directory

Search 1st Half of Phone Directory

Search 2nd Half of Phone Directory

Textbook, p. 50

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Binary Search

Search Phone Directory

Search 1st Half of Phone Directory

Search 2nd Half of Phone Directory

Textbook, pp. 77ff

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bSearch( anArray, value)

if (anArray is of size 1)

Is anArray’s item = to value?

else

Find midpoint of anArray

Which half of anArray has value?

bSearch( halfOfAnArray, value)

Binary Search Algorithm

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bSearch( anArray, value)

if (anArray is of size 1)

Is anArray’s item = to value?

else

Find midpoint of anArray

Which half of anArray has value?

bSearch( halfOfAnArray, value)

Binary Search Algorithm

Textbook, pp. 78

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731 98anArray:

value:

Textbook, pp. 78

0 1 2 3 4

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731 98anArray:

first:

last:

value:

Textbook, pp. 78

0 1 2 3 4

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731 98anArray:

first:

last:

mid:

value:

Textbook, pp. 80

0 1 2 3 4

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bSearch( anArray, value, first, last )

if (last == first)

Is anArray’s item = to value?

else

mid = (first + last) / 2

if (value is in first half )

bSearch( anArray,value,first,mid)

else

bSearch( anArray,value,mid+1,last)

Textbook, pp. 78

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Searching in a sorted array

Binary search

Searching in an unsorted array

Finding the smallest item

Finding the kth smallest item

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Unsorted array: Find Smallest

Find smallest

Find smallest inFirst half

Find smallest inSecond half

938 710 1 2 3 4

Textbook, pp. 76

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Unsorted array: Find kth Smallest

Find kth smallest

Find kth smallest inFirst half

Find kth smallest inSecond half

938 710 1 2 3 4

Textbook, pp. 81ff

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Please remember to write

YOUR name and login/ID

on your assignment 1

answers’ booklet.