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 33.3, Problem 6E
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To show the process of implementing the incremental method for computing the convex hull of n points so that it runs in
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Let P be a set of n points in the plane. The points of P are given one point at a time. After receiving eachpoint, we compute the convex hull of the points seen so far.
(a) As a naive approach, we could run Graham’s scan once after receiving each point, with a total runningtime of O(n2log n). Write down the psuedo-code for this algorithm
We have learned the algorithm that solves the Closest pair problem in 2D in Θ(n log n) time. (Closest pair problem in 2D: Given n points in the 2D plane, find a pair with smallest Euclidean distance between them.)
Give an algorithm that solves the Closest pair problem in 3D in Θ(n log n) time. (Closest pair problem in 3D: Given n points in the 3D space, find a pair with smallest Euclidean distance between them.)
Prove or disprove that for any x ∈ N, x(x+1)/2 ∈ N (where N = {0, 1, 2, 3, ….}
Chapter 33 Solutions
Introduction to Algorithms
Ch. 33.1 - Prob. 1ECh. 33.1 - Prob. 2ECh. 33.1 - Prob. 3ECh. 33.1 - Prob. 4ECh. 33.1 - Prob. 5ECh. 33.1 - Prob. 6ECh. 33.1 - Prob. 7ECh. 33.1 - Prob. 8ECh. 33.2 - Prob. 1ECh. 33.2 - Prob. 2E
Ch. 33.2 - Prob. 3ECh. 33.2 - Prob. 4ECh. 33.2 - Prob. 5ECh. 33.2 - Prob. 6ECh. 33.2 - Prob. 7ECh. 33.2 - Prob. 8ECh. 33.2 - Prob. 9ECh. 33.3 - Prob. 1ECh. 33.3 - Prob. 2ECh. 33.3 - Prob. 3ECh. 33.3 - Prob. 4ECh. 33.3 - Prob. 5ECh. 33.3 - Prob. 6ECh. 33.4 - Prob. 1ECh. 33.4 - Prob. 2ECh. 33.4 - Prob. 3ECh. 33.4 - Prob. 4ECh. 33.4 - Prob. 5ECh. 33.4 - Prob. 6ECh. 33 - Prob. 1PCh. 33 - Prob. 2PCh. 33 - Prob. 3PCh. 33 - Prob. 4PCh. 33 - Prob. 5P
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- Let Z be the set of all integers. An integer a has f as a factor if a = fj for some integer j. An integer is even if it has 2 as a factor. An integer a is odd if it is not even. Prove by contradiction that an odd number cannot have an even number as a factor.arrow_forwardMake K-map as followsarrow_forwardLet Z be the set of all integers. An integer a has f as a factor if a = integer j. An integer is even if it has 2 as a factor. An integer a is odd if it is not even. Prove by contradiction that an odd number cannot have an even number as a factor.arrow_forward
- Please explain Give an algorithm that solves the Closest pair problem in 3D in Θ(n log n) time. (Closest pair problem in 3D: Given n points in the 3D space, find a pair with the smallest Euclidean distance between them.)arrow_forward1: Given a fixed integer B (B ≥ 2), we demonstrate that any integer N (N ≥ 0) can bewritten in a unique way in the form of the sum of p+1 terms as follows:N = a0 + a1×B + a2×B2 + … + ap×Bpwhere all ai, for 0 ≤ i ≤ p, are integer such that 0 ≤ ai ≤ B-1.The notation apap-1…a0 is called the representation of N in base B. Notice that a0 is theremainder of the Euclidean division of N by B. If Q is the quotient, a1 is the remainder of theEuclidean division of Q by B, etc.1. Write an algorithm that generates the representation of N in base B. 22. Compute the time complexity of your algorithm.arrow_forwardAbout a set X of numbers we say that it is almost sum-free if the sum of two different elements of X never belongs to X. For instance, the set {1, 2, 4} is almost sum-free. Almost-Schur number A(k) is the largest integer n for which the interval {1, . . . , n} can be partitioned into k almost sum-free sets. Use clingo to find the exact values of A(1), A(2), A(3) and try to find the largest lower bound for A(4), i.e., the largest number l such that A(4) ≥ l. Hint: you do not need to find all partitions to find the values of A(k). PLEASE USE CLINGO.arrow_forward
- * The iteration of Newton-Raphson method stop if | Xn+1-Xn] & O | Xn+1-Xnl>aOarrow_forwardDo 2-iterations of the Secant method for finding a root of the equation f(x)=cos(x)-x*e^x=O start with the interval [1,2]. The root is ... * O 0.85778 O 0.8320 None of them 0.83853 O 0.855555arrow_forwardGiven f(n) ∈ Θ(n), prove that f(n) ∈ O(n²).arrow_forward
- Let T be a sorted array of n elements. An element x is said to be a majority element in T if the number of elements i, with T[i] = x, is greater than n/2. Give an algorithm (code or pseudo-code) that can decide whether T includes a majority element (it cannot have more than one), and if so, find it. Your algorithm must run in linear time.arrow_forwardLets say you need to arrange seating in a club. There is a finite amount of seating, that is close to VIP seating,L. Therefore, there is a fixed amount of people you can seat near VIP. The goal is to choose a set of L seats so that the max distance between VIP seating and the partyer is minimized. Write a poly-time approximation algorithm for this problem, prove it has a specific approximation ratio.arrow_forwardWrite a recurrence for this algorithm and solve it to obtain a tight upper bound on the worst case runtime of this algorithm. You can use any method you like for solving this recurrence.arrow_forward
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