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 8.2, Problem 4E
Program Plan Intro
To describe an
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Let n be a positive integer, and consider the following algorithm segment.
for i := 1 to n
for j := 1 to i
[Statements in body of inner loop.
None contain branching statements
that lead outside the loop.]
next j
next i
How many times will the inner loop be iterated when the algorithm is implemented and run?
Find the correct asymptotic complexity of an algorithm with runtime T(n) where T(x) = O(n) + T(3 * x /4) + T(x / 4)
Can't really understand the available solution hence, i am asking again
Consider the following recursive algorithm, where // denotes integer division: 3//2 = 1, 5//2 = 2, etc.
F(n):if n <= 1: returnF(n//2)for i from 0 to n for j from 0 to n//2 print(i+j)
Let function T(n) denote the running time of this algorithm. Derive T(n) and prove its worst case timecomplexity
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- Consider the following recursive algorithm, where // denotes integer division: 3//2 = 1, 5//2 = 2, etc. H(n): if n <= 1: return H(n//2) for i from 0 to n print(min(3, i)) Let function T(n) denote the running time of this algorithm. Derive T(n) and prove its worst case time complexityarrow_forwardPlease solve sections, Find the asymptotic (large-Θ) limits for the running times of the algorithms whose running time is given iteratively. 1. T (n) = 4T (n/4) + 5n2. T (n) = 4T (n/5) + 5n3. T (n) = 5T (n/4) + 4n4. T (n) = T (n/2) + 2T (n/5) + T (n/10) + 4narrow_forwardGive an O(n2)-time algorithm to find the longest monotonically increasing subsequence of a sequence of n numbers.arrow_forward
- Suppose we are to sort a set of 'n' integers using an algorithm with time complexity O (log2 n) that takes 0.5 ms (milliseconds) in the worst case when n = 20,000. How large an element can the algorithm then handle in 1 ms? (Note that only numeric answers can be provided.)arrow_forwardProve that the running time of an algorithm is ‚theta(g(n)) if and only if its worst-case running time is O(g(n)) and its best-case running time is Omega(g(n)).arrow_forwardConsider a divide-and-conquer algorithm that calculates the sum of all elements in a set of n numbers by dividing the set into two sets of n/2 numbers each, finding the sum of each of the two subsets recursively, and then adding the result. What is the recurrence relation for the number of operations required for this algorithm? Answer is f(n) = 2 f(n/2) + 1. Please show why this is the case.arrow_forward
- Let the sequence (n) be recursively defined by x1 = √2 and Xn+1 = √√2+xn, n≥ 1. Show that (n) converges and evaluate its limit.arrow_forwardCall a sequence X[1 · · n] of numbers oscillating if X[i] < X[i + 1] for all even i, and X[i] > X[i + 1] for all odd i. Describe an efficient algorithm to compute the length of the longest oscillating subsequence of an arbitrary array A of integers.arrow_forwardGive an example of an algorithm that is O(1), an algorithm that is O(n) and an algorithm that is O(n2). Discuss the difference between them.arrow_forward
- When the order of increase of an algorithm's running time is N log N, the doubling test leads to the hypothesis that the running time is a N for a constant a. Isn't that an issue?arrow_forwardConsider the problem of counting, in a given text, the number of substrings that start with an A and end with a B. For example, there are four such substrings in CABAAXBYA.a. Design a brute-force algorithm for this problem and determine its efficiency class.b. Design a more efficient algorithm for this problem with complexity O (n)arrow_forwardFind the correct asymptotic complexity of an algorithm with runtime T(n) where T(x) = O(n) + T(3 * x /4) + T(x / 4)arrow_forward
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