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 14.1, Problem 7E
Program Plan Intro
To count the number of inversions in an array of size n for an order-statistic tree in
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Apply the Backtracking algorithm for the n-Queens problem to the problem instance in which n= 8, and show the actions step by step. Draw the pruned state space tree produced by this algorithm up to the point where the first solution is found.
Explain how to use an AVL tree or a red-black tree to sort n comparable elements in O(n log n) time in the worst case.
Answer the given question with a proper explanation and step-by-step solution.
Question 1
What is the worst time complexity for inserting an arbitrary item into a AVL tree?
0(n)
0 (log2 (n))
0(1)
0(n^2)
0 (n log2 (n))
Question 2
What is the worst time complexity for inserting an arbitrary item into a binary search tree?
0(n)
0 (log2(n))
0(1)
0 (n^2)
0 (n log2 (n))
Question 13
What is the time complexity for efficiently ordering elements (sorting) from a binary search tree?
0 (n)
0 (Log2 (n))
0(1)
0(n^2)
0 (n log2 (n))
Question 11
The average Big O runtime to find an arbitrary element in an arbitrary AVL tree is roughly equal to the element in which other data structure or structures?
Queue
Hash Table
Sorted Array
LinkedList
None of the above
Question 13
What is the time complexity for efficiently ordering elements (sorting) from a binary search tree?
0 (n)
0 (log2 (n))
0(1)
0 (n^2)
0 (n log2 (n))
Question 23
What is the time complexity of the most time efficient algorithm for…
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- Refine the mathematical model to better explain the experimental results in the table given in the text. Specifically, show that the average numberof compares for a successful search in a tree built from random keys approaches thelimit 2 ln N 2 – 3 1.39 lg N – 1.85 as N increases, where .57721... is Euler’sconstant. Hint : Referring to the quicksort analysis in Section 2.3, use the fact that theintegral of 1/x approaches ln N .arrow_forwardUse structural induction to show that n(T ) ≥ 2h(T ) + 1, where T is a full binary tree, n(T ) equals the number of vertices of T, and h(T) is the height of T.arrow_forwardIt was claimed that:(a, b) ≤ (c, d) ⇔ (a < c) ∨ (a = c ∧ b ≤ d) defines a well-ordering on N x N. Show that this is actually the case.arrow_forward
- How about O(log n) time complexity operations that a red-black tree might perform? Select one that best fits the situation: The correct answers are (a), (b), and (d), respectively.arrow_forwardIllustrate that via AVL single rotation, any binary search tree T1 can betransformed into another search tree T2 (with the same items) Give an algorithm to perform this transformation using O(N log N) rotation on averagearrow_forwardAm i correct to assume that the possible outcomes for the algorithm of topological Sorting are: 4,3,6,2,8,7,5,1 and 3,6,4,1,2,8,7,5 Are there other possible outcomes?arrow_forward
- Consider a sort of items, according to their keys, that inserts all the items one at a time into an initially empty regular binary search tree and then applies an in-order traversal to complete the sort. Assume that all items have distinct keys. Using big-Theta notation, what is the worst-case complexity of the sort? What is the average-case complexity of the sort? Now answer the same two questions if an AVL tree is used instead of a regular binary search tree.arrow_forwardDraw an ordered tree representing the formula 'q(X - Y * cos(Z), log(K), A / (B – C)'arrow_forwardFor any given AVL tree T of n elements, we need to design an algorithm that finds elements x in T such that a <= x <= b, where a and b are two input values. Assume that T has m elements x satisfying a <= x <= b. What is the best time complexity for such an algorithm? A. O(m) B. O(m log n) C.O(m + log n) D.O(log m + log n)arrow_forward
- Find the values of N for which using a binary search tree to build a symbol table of N random double keys becomes 10, 100, and 1,000 times faster than binary search. Predict the values with analysis and verify them experimentally.arrow_forwardGeneralize Huffman's algorithm to ternary codewords (i.e., codewords using the symbols 0, 1, and 2), and write down the corresponding pseudo-code using a priority queue.arrow_forwardWhat kinds of operations can a red-black tree carry out with a time complexity of O(log n), and what are those operations? Pick one from the following options: a) Insertion and deletion; b) The search for a predecessor; c) The search for a successor; d) All of the abovearrow_forward
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