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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Students have asked these similar questions
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?
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.
multiple choice:
consider an array based implementation of a stack and its push operation. Beginning with an array of length 1 = (2^0), consider where the array’s length will be doubled whenever an insertion(via the push operation) is attempted when the array is full. What is the amortized complexity of performing a sequence of n push operations.
a) Θ(log n)
b) Θ(n)
c) Θ(n^2)
d) Θ(1)
Chapter 10 Solutions
Introduction to Algorithms
Ch. 10.1 - Prob. 1ECh. 10.1 - Prob. 2ECh. 10.1 - Prob. 3ECh. 10.1 - Prob. 4ECh. 10.1 - Prob. 5ECh. 10.1 - Prob. 6ECh. 10.1 - Prob. 7ECh. 10.2 - Prob. 1ECh. 10.2 - Prob. 2ECh. 10.2 - Prob. 3E
Ch. 10.2 - Prob. 4ECh. 10.2 - Prob. 5ECh. 10.2 - Prob. 6ECh. 10.2 - Prob. 7ECh. 10.2 - Prob. 8ECh. 10.3 - Prob. 1ECh. 10.3 - Prob. 2ECh. 10.3 - Prob. 3ECh. 10.3 - Prob. 4ECh. 10.3 - Prob. 5ECh. 10.4 - Prob. 1ECh. 10.4 - Prob. 2ECh. 10.4 - Prob. 3ECh. 10.4 - Prob. 4ECh. 10.4 - Prob. 5ECh. 10.4 - Prob. 6ECh. 10 - Prob. 1PCh. 10 - Prob. 2PCh. 10 - Prob. 3P
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- 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?arrow_forwardS and Q are a stack and a priority queue of integers, respectively. The priority ofan element C joining the priority queue Q is computed as C mod 3. In other words,the priority numbers of the elements are either 0 or 1 or 2. Given A, B, and C to beinteger variables, what is the output of the following code? The procedures aresimilar to those used in illustrative problems 5.1 and 5.2. However, the queueprocedures are modified to appropriately work on a priority queue.1.A = 102.B = 113.C = A+B4.while (C < 110) do5. if (C mod 3) = 0 then PUSH (S,C)6. else ENQUEUE (Q,C)7. A = B8. B = C9. C = A + B10.end11.while not EMPTY_STACK (S) do12. POP (S,C)13. PRINT (C)14.end15.while not EMPTY_QUEUE (Q) do16. DEQUEUE (Q, C)17. PRINT (C)18.endarrow_forwardThe pseudocode of Figure 6.16 illustrates the basic push() and pop() operations of an array-based stack. Assuming that this algorithm could be used in a concurrent environment, answer the following questions:a. What data have a race condition?b. How could the race condition be fixed?arrow_forward
- Implement a stack S with two queues Q1 and Q2, allowing the queues to act as a stack as a whole to handle the operations PUSH (S, X) and POP(S, Y), where S is presumably a stack and X is the element to be added to S and Y is the element removed from S. Assume that ADT's ENQUEUE, DEQUEUE, and EMPTYQUEUE operations can only access queues.Show how your function functions on the set of items (a, b, and c) that the queues Q1 and Q2 will utilise as a stack.arrow_forwardInvestigate the operational mechanics of the stack in its unaltered state.arrow_forwardIf a linked stack is implemented as a doubly linked list, provide a process to flip the stack such that the top and bottom places are now the bottom and top, respectively.A connected stack S and its reversed counterpart Srev, for instance, are displayed as follows:arrow_forward
- Solve the code much needed. Given a stack, a function is_consecutive takes a stack as a parameter and thatreturns whether or not the stack contains a sequence of consecutive integersstarting from the bottom of the stack (returning true if it does, returningfalse if it does not). For example:bottom [3, 4, 5, 6, 7] topThen the call of is_consecutive(s) should return true.bottom [3, 4, 6, 7] topThen the call of is_consecutive(s) should return false.bottom [3, 2, 1] topThe function should return false due to reverse order. Note: There are 2 solutions:first_is_consecutive: it uses a single stack as auxiliary storagesecond_is_consecutive: it uses a single queue as auxiliary storage"""import collections def first_is_consecutive(stack): storage_stack = [] for i in range(len(stack)): first_value = stack.pop() if len(stack) == 0: # Case odd number of values in stack return True second_value = stack.pop() if first_value - second_value != 1: # Not…arrow_forwardQ: Consider an empty stack STK of size 5 (array-based implementation). What will be the output after applying the following stack operations? Draw a diagram in support of your answer. How many elements are there in the stack at the end of the processing? POP(), PUSH(9), PUSH(11), PUSH(25), POP(), POP(), PUSH(42), POP(), PUSH(3), PUSH(7), PUSH(30), PUSH(15), PUSH(54),POP(), PUSH(50)arrow_forwardUse a stack to reverse the input integer values. Check the stack grows as more elements are pushed. Use with and without function pointer in each of the problem (Please give correct answer with explanation)arrow_forward
- A function f defined on stack of integer satisfies the following properties f(empty) = 0 and f (push (S, i)) = max (f(s), 0) + i for all stacks S and integer i. After pushing integers 2, -3, 2, -1, 2 in order from bottom to top. f(s) returns the top of the stack. What is f(s)?arrow_forwardThe pseudocode of Figure 6.16 illustrates the basic push() and pop() operations of an array-based stack. Assuming that this algorithm could be used in a concurrent environment, answer the following questions: a. What data have a race condition? b. How could the race condition be fixed? push(item) { if (top < SIZE) { stack (top] top++; } else = item; ERROR } pop () { if (!is.empty()) { top--; return stack[top]; else ERROR } is.empty() { if (top 0) return true; else return false; } Figure 6.16 Array-based stack for Exercise 6.12.arrow_forwardConsider the following infix expression.( 5 + 8 ) * 9 – 7 * 10 + 9.Apply infix-to-postfix conversion algorithm to generate the correspondingpostfix expression from the given infix expression. You have to show thecontent of stack at each stage of conversion.arrow_forward
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