Data Structures and Algorithms in Java
6th Edition
ISBN: 9781119278023
Author: Michael T. Goodrich; Roberto Tamassia; Michael H. Goldwasser
Publisher: Wiley Global Education US
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Chapter 4, Problem 35C
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Possibility of three-way dis-jointness in
It is assumed that the “n” numbers can be sorted in
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Question 3)
Use the master theorem to give an asymptotic tight bound for the following recurrences.
Tell me the values of a, b, the case from the master theorem that applies (and why), and the
asymptotic tight bound.
3a) T(n) = : 2T (n/4) + n
3b) T(n) = 16T(n/4) + (√√n)³
Given an n-element sequence of integers, an algorithm executes an O(n)-time computation for each even number in the
sequence, and an O(logn)-time computation for each odd number in the sequence. What are the best-case and worst-case
running times of this algorithm? Why? Show with proper notations.
Second order Spada numbers are well established in the insurance industry. Formally they are defined by the recurrence: Si={Si−1 %( Si−2+1) :i≥ 2 , 1:i=0 ,7 :i=1} Give an O(n) time algorithm to compute the n-th Second order Spada number
Chapter 4 Solutions
Data Structures and Algorithms in Java
Ch. 4 - Prob. 1RCh. 4 - The number of operations executed by algorithms A...Ch. 4 - The number of operations executed by algorithms A...Ch. 4 - Prob. 4RCh. 4 - Prob. 5RCh. 4 - Prob. 6RCh. 4 - Prob. 7RCh. 4 - Prob. 8RCh. 4 - Prob. 9RCh. 4 - Prob. 10R
Ch. 4 - Prob. 11RCh. 4 - Prob. 12RCh. 4 - Prob. 13RCh. 4 - Prob. 14RCh. 4 - Prob. 15RCh. 4 - Prob. 16RCh. 4 - Prob. 17RCh. 4 - Prob. 18RCh. 4 - Prob. 19RCh. 4 - Prob. 20RCh. 4 - Prob. 21RCh. 4 - Prob. 22RCh. 4 - Show that 2n+1 is O(2n).Ch. 4 - Prob. 24RCh. 4 - Prob. 25RCh. 4 - Prob. 26RCh. 4 - Prob. 27RCh. 4 - Prob. 28RCh. 4 - Prob. 29RCh. 4 - Prob. 30RCh. 4 - Prob. 31RCh. 4 - Prob. 32RCh. 4 - Prob. 33RCh. 4 - Prob. 34RCh. 4 - Prob. 35CCh. 4 - Prob. 36CCh. 4 - Prob. 37CCh. 4 - Prob. 38CCh. 4 - Prob. 39CCh. 4 - Prob. 40CCh. 4 - Prob. 41CCh. 4 - Prob. 42CCh. 4 - Prob. 43CCh. 4 - Draw a visual justification of Proposition 4.3...Ch. 4 - Prob. 45CCh. 4 - Prob. 46CCh. 4 - Communication security is extremely important in...Ch. 4 - Al says he can prove that all sheep in a flock are...Ch. 4 - Consider the following justification that the...Ch. 4 - Consider the Fibonacci function, F(n) (see...Ch. 4 - Prob. 51CCh. 4 - Prob. 52CCh. 4 - Prob. 53CCh. 4 - Prob. 54CCh. 4 - An evil king has n bottles of wine, and a spy has...Ch. 4 - Prob. 56CCh. 4 - Prob. 57CCh. 4 - Prob. 58CCh. 4 - Prob. 59CCh. 4 - Prob. 60PCh. 4 - Prob. 61PCh. 4 - Perform an experimental analysis to test the...Ch. 4 - Prob. 63P
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- An array of n elements is almost sorted if and only if every element is at most k spots away from its actual location. Assuming that you can only perform pairwise comparisons, formally prove that any algorithm which can sort almost sorted arrays must have running time S2(n log k), You may assume that n is a multiple of k.arrow_forwardGiven a list of n positive integers, show that there must two of these integers whose difference is divisible by n-1arrow_forwardimplement Running time algorithm Careful(n) pre-cond: n is an integer. post-cond: Q(n) “Hi”s are printed for some odd function Qarrow_forward
- Determine φ (m), for m=12,15, 26, according to the definition: Check for each positive integer n smaller m whether gcd(n,m) = 1. (You do not have to apply Euclid’s algorithm.)arrow_forwardFor any input array of n 3-digit numbers, prove that Radix Sort is guaranteed to correctly sort these n numbers, with the algorithm running in O(n) time.arrow_forwardGiven two sorted arrays, each of length n, we can merge them into a sorted array in O(n) time. True False If we solve T(n) = 9T(n/3) + O(n), then we have T(n) = 0(n log n). True False The running time of the Strassen's algorithm for multiplying two n-by-n matrices is O(n®). True Falsearrow_forward
- Given a sorted array A of n distinct integers, some of which may be negative, give an O(log(n)) algorithm to find an index i such that 1 ≤ i ≤ n and A[i] = i provided such an index exists. If there are many such indices, the algorithm can return any one of them.arrow_forward4 points Given an n-element array X of integers, Algorithm A executes an O(n3.4)-time computation for each even positive number in X, an Oin2.3-time computation for each odd positive number in X, and an O(n2.5)-time computation for each negative number in X. What are the best-case and worst-case running times of Algorithm A? Justify your answer. For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac) BIUS Paragraph V Arial 10pt EVE 2 IXO QFS3Earrow_forwardShow by induction that T(n) ≤ 3n for all n ≥ 1arrow_forward
- Solve the following recurrences using iteration methods and Master's Theorem (if possible) a. T(n) = 2T (n/3) +3 b. T(n) = 3T (n/6) + narrow_forwardProve by Induction that for all integers n ≥ 1, n < n2 + 1 .Yes this problem is silly, but still do it by induction! Prove by Induction that for all integers n ≥ 3, 2n < n2 .arrow_forwardSolve the following recurrence using substitution method with the initial guess as O(n) and assume T(1)=1 as a base case. T(n) = 2T(n/3)+n This question is an algorithm analysis problem. Please solve it on paper.arrow_forward
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