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 17.2, Problem 1E
Program Plan Intro
To show that the cost of n stack operations including copying the stack is
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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.
Consider the following operations performed on a stack of size 5. Push(1);Pop();Push(2);Push(3);Pop();Push(4);Pop();Push(5); After the completion of all operations, the number of elements in the stack is:
Consider that you have a stack S and a queue Q. Draw S and Q after executing the following operations (Op).
You have to indicate for each operation, the stack or the queue content in addition to the index value of the top (stack) or the front and the rear (queue)
Stack S; Queue Q;
int a=3, b=7
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- 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? [15+5] POP(), PUSH(25), PUSH(20), POP(), POP(), PUSH(45), PUSH(15), POP(), PUSH(30), PUSH(17), PUSH(33), PUSH(24), PUSH(54), POP(), PUSH(99)arrow_forwardS1 and S2 are two sorted stacks of n and m numbers sorted in decreasing order, with their top items pointing to the smallest in their lists, respectively. Create a stack MERGE that merges the items in stacks S1 and S2, so that at the conclusion of the merge, all of the elements in S1 and S2 are available in MERGE in decreasing order, with the largest element at the top.Keep in mind that the number of components in stack MERGE is (n + m).arrow_forwardmultiple 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)arrow_forward
- Suppose an initially empty stack, S, has performed a total of 75 push() operations, 5 peek() operations and 10 pop operations, 4 of which returned null to indicate an empty stack. What is the current size of the stack, S? (Explain your answer/Show your workings)arrow_forwardConsider 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?arrow_forwardThe two sorted stacks S1 and S2 each contain n and m numbers arranged in decreasing order, with the top elements of each stack pointing to the list item that is smallest. Make a stack MERGE that combines the items in stacks S1 and S2 so that all of the elements in S1 and S2 are accessible in MERGE in descending order, with the largest element as its top element, at the conclusion of the merge.Keep in mind that stack MERGE would have (n + m) elements.arrow_forward
- Q: 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_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_forwardThe advantages and disadvantages of this strategy are discussed, along with examples from the real world when an array (linear) form of a stack could be appropriate.arrow_forward
- Analyze the following series of insertion (I) and deletion (D) operations is provided for debugging a module, coded in C language, being applied on STACK and circular Queue. The given series is: I D I I I D D I I I I D I D I D D D I I Compute the position of TOP or any warnings message in case of STACK (Size = 5) Compute the positions of FRONT and REAR or any warning message in case of Circular Queue after each operation (Size = 5)arrow_forwardConsider a “superstack” data structure which supports four operations: create, push, pop, andsuperpop. The four operations are implemented using an underlying standard stack S as shownbelow.def create():S = Stack.create()def push(x):S.push(x)def pop():return S.pop()def superpop(k,A): // k is an integer, A is an array with size >= ki = 0while i < kA[i] = S.pop()i = i + 1Show that each of these operations uses a constant amortized number of stack operations. In yoursolution:• Define your potential function Φ.• State, for each operation, its actual time, the change in potential, and the amortized time.3. Suppose we add a superpush operation to the superstack from Problem 2. The superpush operationis defined as follows:def superpush(k,A): // k is an integer, A is an array with size >= ki = 0while i < kS.push(A[i])i = i + 1Is it still true that each of the superstack operations uses a constant amortized number of stackoperations? Answer YES or NO.• If your answer is YES, give an…arrow_forwardi need the answer in MASM Write a procedure, EvaluatePoly, to evaluate a polynomial: P(x) = Coeff(n)*x^n + Coeff(n-1)*x^(n-1) + ... + Coeff(1)*x + Coeff(0) at a given value of x. You are to assume that the calling routine first pushes the address of the coefficient array onto the stack, then pushes the degree of the polynomial onto the stack, and then finally passes the value of x onto the stack. The value of P(x) is to be returned in the ax register. The algorithm you should use to evaluate the polynomial should be modeled after the following higher level language code: n = degree;value = Coeff[n];if ( n == 0 )return;elsefor ( i = n - 1; i >= 0; i-- )value = value*x + Coeff[i];EvaluatePoly PROCpush ebpmov ebp,esp; Write code hereEvaluatePoly ENDP View keyboard shortcutsarrow_forward
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