CIS 403503 Accelerated DataFile Structures

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CIS 403/503Accelerated Data-File Structures

• Lecture 12
• Recurrence Stack

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Objectives

• In this lecture, we will learn Space
• Solving recurrences
• Master method
• Stack STL
• Runtime stack

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Recurrence

• An algorithm contains a recursive call to itself
• A function in terms of its value on smaller input
• T(n)
• ?(1) if n 1
• 2T(n/2) ?(n)

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Recurrence

• In general, T(n) aT(n/b) f(n)
• a - of subproblems
• n/b - size of a subproblem
• f(n) - the cost of dividing the problem and
  combining the result of subproblems

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Solve Recurrences

• Method 1 substitution
• Guess a solution
• Verify by induction
• Solve for constants
• e.g., T(n) 4T(n/2) n

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Solve Recurrences

• Method 2 iteration or unrolling
• T(n)
• n 4T(n/2)
• n4(n/24T(n/4))

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Solve Recurrences

• Method 3 recursion tree

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Solve Recurrences

• Method 4 Master Method
• T(n) a T(n/b) f(n) assuming a ? 1 and b ? 1
• Using recursion tree analysis, the total will be
• ?(nlogba) ?i0,,logbn -1aif(n/bi)

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Master Method

• Case 1 if f(n)/nlogba O(1/n?) for some ? gt 0,
  then T(n) is ?(nlogba)
• Case 2 if f(n)/nlogba ?(logkn) for some k ? 0,
  then T(n) ?(nlogbalogk1n)
• Case 3 if f(n)/nlogba ?(n?) for some ? gt 0 and
  af(n/b) ? cf(n) for c lt 1, then T(n) ?(f(n))

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Examples

• T(n) 2T(n/2)n
• T(n)4T(n/2)n
• T(n)4T(n/2)n3
• T(n)4T(n/2)n2/logn

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Stacks
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Overview

• Stack
• A data structure in which data are added and
  removed from only one end
• LIFO

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STL Stack Operations

• Void push(const T item)
• Insert item at the top
• Postcondition new item at the top
• void pop()
• Remove the item from the top
• Precondition stack is not empty
• Postcondition stack has one less item
• bool empty() const
• Check if stack is empty
• int size() const
• Number of items on the stack
• T top()
• A reference to the top of the stack
• Precondition stack is not empty
• const T top() const
• Constant reference

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Example

• includeltstackgt
• includeltiostreamgt
• using namespace std
• int main()
• stackltintgt s
• int i
• for (i1 ilt5 i)
• s.push(i)
• cout ltlt "Stack size " ltlt s.size() ltlt
  endl
• cout ltlt "Popping the stack" ltlt endl
• while(!s.empty())
• cout ltlt s.top() ltlt " "
• s.pop()
• cout ltlt endl

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Example

• includeltstackgt
• includeltiostreamgt
• using namespace std
• int main()
• stackltintgt s
• s.push(2)
• s.push(3)
• s.push(-1)
• cout ltlt s.top() ltlt endl
• //change top to 4
• s.top() 5
• while(!s.empty())
• cout ltlt s.top() ltlt " "
• s.pop()

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Uncoupling Stack Elements
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Uncouple

• //remove the 1st occurrence of target
• //true if target was removed, false otherwise
• templateltclass Tgt
• bool uncouple(stackltTgt s, const T target)
• stackltTgt tmpStk
• //assume target is on the stack
• bool foundTarget true
• //pop elements from s and push them to
  tmpStack
• //till target is found or empty stack
• while(!s.empty() s.top() ! target)
• tmpStk.push(s.top())
• s.pop()
• //locates item if s is not empty
• if(!s.empty())
• s.pop()
• else

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Runtime Stack

• Set up activation record
• List of runtime arguments
• Local variables and objects
• Return address

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Recursive Call

• includeltiostreamgt
• using namespace std
• int fact(int)
• int main()
• int factValue
• factValue fact(4)
• cout ltlt "Value fact(4) " ltlt factValue
  ltlt endl
• return 0
• int fact(int n)
• if ( n 0)
• return 1
• else
• return n fact(n-1)

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Summary

• There four methods to solve recurrences
• Substitution
• Unrollng or iteration
• Recursion trees
• Master method
• Stack is a LIFO data structure
• In C STL, a stack is infinite
• Each item is accessed throught top() and pop()

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Possible Quiz Questions

• Solve recurrences.
• Stack oprations and application