Question

I have been working on writing a Java program to convert from infix notation to prefix notation using an operand stack and an operator stack. I have implemented a working converter based on the pseudocode in the top answer here:

conversion from infix to prefix

However I am now trying to work out the time and space complexity of the above algorithm.

I think that the space complexity must be O(n) because we just have two stacks that store the input shared between them.

Thinking about the time complexity, I’m not entirely sure, is it O(n^2) because of having to convert each subpart from infix to prefix? I’m not really sure about this part.

Basically my question is: is my space complexity result correct, and what is the time complexity of the algorithm?

Many thanks!

EDIT:
This is the pseudocode for the algorithm:

Algorithm ConvertInfixtoPrefix

Purpose: Convert and infix expression into a prefix expression. Begin 
// Create operand and operator stacks as empty stacks. 
Create OperandStack
Create OperatorStack

// While input expression still remains, read and process the next token.

while( not an empty input expression ) read next token from the input expression

// Test if token is an operand or operator 
if ( token is an operand ) 
// Push operand onto the operand stack. 
    OperandStack.Push (token)
endif

// If it is a left parentheses or operator of higher precedence than the last, or the stack is empty,
else if ( token is '(' or OperatorStack.IsEmpty() or OperatorHierarchy(token) > OperatorHierarchy(OperatorStack.Top()) )
// push it to the operator stack
    OperatorStack.Push ( token )
endif

else if( token is ')' ) 
// Continue to pop operator and operand stacks, building 
// prefix expressions until left parentheses is found. 
// Each prefix expression is push back onto the operand 
// stack as either a left or right operand for the next operator. 
    while( OperatorStack.Top() not equal '(' ) 
        OperatorStack.Pop(operator) 
        OperandStack.Pop(RightOperand) 
        OperandStack.Pop(LeftOperand) 
        operand = operator + LeftOperand + RightOperand 
        OperandStack.Push(operand) 
    endwhile

// Pop the left parthenses from the operator stack. 
OperatorStack.Pop(operator)
endif

else if( operator hierarchy of token is less than or equal to hierarchy of top of the    operator stack )
// Continue to pop operator and operand stack, building prefix 
// expressions until the stack is empty or until an operator at 
// the top of the operator stack has a lower hierarchy than that 
// of the token. 
    while( !OperatorStack.IsEmpty() and OperatorHierarchy(token) lessThen Or Equal to OperatorHierarchy(OperatorStack.Top()) ) 
        OperatorStack.Pop(operator) 
        OperandStack.Pop(RightOperand) 
        OperandStack.Pop(LeftOperand) 
        operand = operator + LeftOperand + RightOperand 
        OperandStack.Push(operand)
    endwhile 
    // Push the lower precedence operator onto the stack 
    OperatorStack.Push(token)
endif
endwhile 
// If the stack is not empty, continue to pop operator and operand stacks building 
// prefix expressions until the operator stack is empty. 
while( !OperatorStack.IsEmpty() ) OperatorStack.Pop(operator) 
OperandStack.Pop(RightOperand) 
OperandStack.Pop(LeftOperand) 
operand = operator + LeftOperand + RightOperand

OperandStack.Push(operand) 
endwhile

 // Save the prefix expression at the top of the operand stack followed by popping // the      operand stack.

print OperandStack.Top()

OperandStack.Pop()

End

Answer 1

Yep, O(n^2) looks correct – because essentially you have an outer and an inner while loop.

Edit: O(m *n) where m <= n, but still quadractic