For \( n \) distinct elements \( a_1, \ldots, a_n \) with positive weights \( w_1, \ldots, w_n \) such that \( \sum_{i=1}^{n} w_i = 1 \), design a linear-time algorithm returning the element \( a_k \), which satisfies the following:\[\sum_{a_i < a_k} w_i \leq 1/2 \quad \text{and} \quad \sum_{a_i > a_k} w_i \leq 1/2.\]Assume that such an \( a_k \) always exists. You should also justify the time complexity.

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Answered: n For n distinct elements a₁,..., an…

Computer Networking: A Top-Down Approach (7th Edition)

7th Edition

ISBN:9780133594140

Author:James Kurose, Keith Ross

Publisher:James Kurose, Keith Ross

Chapter1: Computer Networks And The Internet

Section: Chapter Questions

Problem R1RQ: What is the difference between a host and an end system? List several different types of end...

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Given an array of intervals I[1, . . . , n] where I[i] = [si, fi], find the minimumnumber of intervals you need to remove to make the rest of the intervals non-overlapping. Precisely define the subproblem.Provide the recurrence equation.Describe the algorithm in pseudocode to compute the optimal value.Describe the algorithm in pseudocode to print out an optimal solution.
Please written by computer source Let x be a binary string. The minimal description of x, written d(x), is the shortest string ⟨M, w⟩ where TM M on input w halts with x on its tape. If several such strings exist, select the lexicographically first among them. The descriptive complexity, or Kolmogorov complexity, of x, written K(x), is K(x) = |d(x)|. Show that the function K(x) is not a computable function. HINTS: If K is a computable function, there is some TM which computes it. That TM can be used to find strings of large complexity. Try to design a program which outputs “complex” strings but which contradicts their supposed complexity, and even contradicting the supposed complexity of a single string suffices.
Write Algorithm for Non-commutative Gaussian Elimination Input: a set W of elements of G; Output: a cgs h = [hi, h2 ..... hr] ofH = ; procedure reduce_and_insert( g : element; var h : sequence of elements ); (* h = [h 1, h2 ..... hr] has the propertythat li (hi) < li(hj) for i < j. The element g is echelonized against h and its reduced form (possibly) inserted in the sequence. *)
1. Let S be a set of n arbitrary but distinct numbers. (a) Give a deterministic algorithm to output the largest n³/4 of the numbers in S in sorted order. Your algorithm must use O(n) comparisons. (b) Show that any deterministic comparison-based algorithm that correctly outputs the median on inputs of n distinct elements must use at least n 1 comparisons. To be completely precise, the median is defined to be the [n/2]th smallest element.
Let x[n] = {1,2,3}. Let y[n] = {1, 2, 3, 4, 5). Circularly convolve x and y where N = 6. Develop the flow-graph of N = 2 x 2 point Decimation-in-time FFT algorithm.
Consider a randomized algorithm SCRAMBLE-SEARCH that works by first randomly permuting the input array and then running the deterministic linear search given above on the resulting permuted array. Letting k be the number of indices i such that A[i] =x, give the worst-case and expected running times of SCRAMBLE-SEARCH for the cases 1. Generalize your solution to handle the case in which k > in which k O and k = 1.
Problem 2: Finding an element.(i) Let n be an even positive integer. Let A be an array containing n not necessarydifferent integers. Elements in A are not necessary sorted. Describe an algorithmwhich in time O(n) tests whether there exists an element x in A that appears in Aat least n/2 times.(ii) Let S be a set containing n integers. An element x ∈ S is an approximate median ofS, if|{y ∈ S | y < x}| ≥ n/4and |{y ∈ S | y > x}| ≥ n/4Given an array A containing n different integers, not necessary sorted, describe analgorithm which in time O(n) finds all approximate medians of the set {A[i] | i =1, 2, . . . , n}.

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