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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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)
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)
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.
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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- Given a typical Stack s with elements [T,W,X,Y,Z] (where Top/left of Stack at T) anda typical Queue q with elements [E,D,C,B,A] (where Front/left of Queue at E andRear/right of Queue at A), determine and list elements of the updated Stack and those of theupdated Queue in their similar forms after the following execution:arrow_forwardIt is known that the sequence of the output of a string of integers is from left to right, respectively: 35, 78, 42, 59, 11, 93, 27, 66, 85, and it is known that the current results of the Linked Stack are as shown in the attached figure, it is possible What is the correct sequence of Stack operations (push and pop)? (multiple choice questions)arrow_forwardImplement a stack S using two queues Q1 and Q2 so that operations PUSH (S,X) and POP(S, Y), where S is supposedly a stack and X is the element to be insertedinto S and Y the element deleted from S, are worked upon by the queues that operatetogether as a stack. Assume that the ADT operations of ENQUEUE, DEQUEUE andEMPTYQUEUE are only available for the queues.Demonstrate the working of your method on a list {a, b, c}, which is to beoperated upon as a stack by the queues Q1 and Q2.arrow_forward
- Given a max-priority queue A = <8, 5, 7, 3, 2, 4, 1>, consider calling Max-Heap-Insert(A, 6). Note: Array index starts from 1. (1) What will be the resulting max-priority queue A? Please write your answer as a sequence of numbers separated by ", ". For example: 1, 2, 3, 4, 5, 6, 7, 8arrow_forwardConsider 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=7arrow_forwardConsider 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?arrow_forward
- Consider the following implementation of a queue using 2 stacks. Let S1 andS2 be the two stacks and initially are both empty. enqueue(x): Push x into S1.dequeue(x): If S2 is not empty, then pop an item from S2 and return it. If S2 isempty, pop all items from S1 and push them into S2, then pop an item S2 and returnit. Show that the amortized cost for each queue operation is O(1).arrow_forward12.22 Demonstrate how, in an in-order traversal, the associated stack can beremoved and replaced with a single reference. (Hint: We only need to know thetop of the stack, and the elements below the stack top are determined by thestack top.)12.23 Which other traversals can be rewritten by replacing their Linear structure with a single reference? How does this change impact the complexity ofeach of the iterations?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_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_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_forwardShow how to replace the related stack with a single reference in an in-order traversal by removing the accompanying stack. (Hint: The items below the stack top are decided by the stack top; we just need to know the top of the stack.) 12.23 Which additional traversals can be reformulated by substituting a single reference for their Linear structure? How does this modification affect how complicated each iteration is?arrow_forward
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