MSIT 127

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MSIT 127

• Data Structures and Algorithms

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RECURSION

• What is Recursion?
• it is a function or procedure that calls itself.
• it is a powerful tool for repetitive task
• Advantage(s)
• it makes algorithm compact
• Simple and easy to debug
• Disadvantage(s)
• requires more memory space
• Slow compared to iteration

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• Characteristics of recursion
• Repetitively Calls itself, thus making a Clone of
  itself
• It has a terminating condition (Base Value)
• Every Call, it moves closer to the terminating
  condition

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• Some Applications
• Factorial
• Fibonacci Numbers
• Binomial Coefficients
• Search
• Sorting

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FACTORIAL

• A factorial 5 is evaluated as
• 5! 5 4 3 2 1
• 5! 120
• Rewriting the relation,
• 5! 5 (5-1) (5-2) (5-3) (5-4)
• Thus mathematical, the factorial of n is
• n! n(n-1) (n-2) (n-3) (n-(n-1))

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Factorial

• If f is a function, then
• f(n) n f(n-1)
• (n-1) f(n-2)
• (n-2)f(n-3)
• (n-3)f(n-4)
• ..
• .n-(n-2)f(n-(n-1))

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Factorial

• Algorithm
• factorial(n)
• if n 0
• return 1
• otherwise
• return n factorial(n-1)

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FIBONACCI NUMBERS

• Fibonacci number is defined as follows
• Fib(n) 1 for n 1 n 2
• Fib(n) Fib(n-1)Fib(n-2) for n gt 2
• The Fibonacci series
• 1,1,2,3,5,8,13,21

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Fibonacci.

• Algorithm
• Fib(n)
• if n 0 or n 2
• return 0
• otherwise
• return Fib(n-1)Fib(n-2)

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BINOMIAL COEFFICIENT

• Any one coefficients that arise in the expansion
  of the polynomial (xy)2. Recursively,
• C(n,k) C(n-1,k-1) C(n-1,k)

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Binomial

• function C(n, k) if k 0 or k n
• return 1
• otherwise
• return C(n-1, k-1) C(n-1, k)

• An Exercise..
• C(6,5) ?

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RECURSIVE BINARY SEARCH

• Algorithm
• BinSearch(LB,UB,Arr,Key)
• If LBgtUB
• return 0
• otherwise
• Mid(LBUB)/2
• if KeyltArr(Mid)
• return BinSearch(LB,Mid-1,Arr,Key)
• othewise
• if KeygtArr(Mid)
• return BinSearch(Mid1,UB,Arr,Key)
• otherwise
• return Mid

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Binary Tree Search.

• IsThere(Tree, item)
• if isEmpty(tree) then
• found false
• else
• if itemltTree.data then
• foundisthere(tree.left, item)
• else
• if itemgttree.data then
• foundisthere(tree.right,item)
• else
• foundtrue
• ifend
• ifend
• Istherefound

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Bubble Sort

• Algorithm
• BubSort(Arr,Pass,ele)
• nlength of Arr
• if ( Passltn-1)
• if (eleltn-p) then
• if Arr(ele) gt Arr(ele1)
• swap(Arr(ele),Arr(ele1))
• BubSort(Arr,Pass,ele1)
• BubSort(Arr,Pass1,0)

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THE END

• THANK YOU