Introduction to Algorithms
Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 10.1, Problem 1E
Program Plan Intro

To model the given stack operations on an initially empty stack S stored in array S[1..6] as per figure 10.1 of the textbook.

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Consider a fixed-size circular array-based implementation of the LRU quasi-stack. Explain step-by-step the algorithm to keep the last referenced page at the top of the stack. Note that the page being referenced may, but does not need to, be already in the stack. Your algorithm must handle both cases. What is the complexity of updating the stack on each new page reference? That is, how many operations are required to add a new page number at the top, or to move an already existing page number from some location in the stack to the top?
You are allowed to operate on a stack WORK and a temporary stack TEMP (ifneeded) supporting their ADT operations of PUSH (S,X), POP (S, X) andEMPTYSTACK (S) only, where X represents an element/variable to be pushed in orpopped out of the stack and S represents a stack. You are also permitted to use onevariable if needed to carry out the operations.i) Given n distinct random numbers that are to be pushed into WORK, how canyou find the minimum element that was pushed into it? You are permitted to use alone variable.ii) Given n distinct random numbers that are to be pushed into WORK, how canyou find the maximum element that was pushed into it, all the while ensuring thatthe elements stored in WORK are in their descending order with the maximumelement beginning at the bottom of stack? You are permitted to use a lone variableand a temporary stack TEMP.iii) Given an array A[1: n] of distinct random numbers how can you obtain thesorted list in the array, making use of stacks alone?
Please assume that you have been given an implementation of a stack that supports both push and pop in O(1) time. With this information, you would like to implement a queue with these stacks.  (a) In what way can you efficiently implement a queue using two of these stacks? “Efficiently” in this case means in a way which will allow you to do part B. (b)  Please prove that the amortized cost of each dequeue and enqueue operation is O(1) for your stack-based queue by using the aggregate amortized analysis technique.
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