Given n integers {a1, az, ..., an}, let S = a1 + a2 + + an. Suppose that S is even. Design an algorithm that finds a subset of these n integers whose sum is S/2.

We can sort n integers between 0 and 2n in O(n) time. True False Given a sorted array of n integers, we can find the median of them in O(1) time. True False If ƒ = 0(g), then g = 0(f). False True

Java Programming Consider a set of n intervals I1,...,In, each given as an integer tuple (si,fi) with si<fi, such that the ith interval starts at si and ends at fi. We want to determine the maximum number of nonoverlapping intervals. More formally, we want to determine the size of a largest subset S⊆{1,...,n} such that for i, j∈S, with i≠j we have fj≤si or fi≤sj. Note that the intervals [1,2] and [2,3] are not considered to be overlapping. Input The first line of input consists of an integer n∈{1,...,105}, the number of intervals. Each of the following n lines consists of two space-separated integers ss and ff, indicating an interval starting at s and ending at f, with 0≤s<f≤109. requirements: CPU Time limit: 4 seconds Memory limit: 1024 MB Output Output a single integer, the largest number of nonoverlapping intervals.

= Trace the Winnow algorithm with 3: 1 for the following input. Suppose the domain is vectors of length n = 6 over {V1, V2, V3, V4, V5, V6} and the actual labels are with respect to formula v1/v3/v6; for example, the true label of (1, 0, 1, 0, 1, 1) is '1' and that of 1, 1, 1, 1, 1,0 is '0'. The input vectors are: x1 = (1, 0, 1, 1, 0, 1) x2 = (1,0,1,1,0,0) x3 = x4= X5 x6 x7 x8 = = = = x9 = (0, 1, 0, 1, 0, 1) 1, 1, 1, 0, 0, 1) (1, (0, 1, 1, 1, 1,0) (1, 1, 1, 1, 1, 1) (0, 1, 0, 1, 1, 1) (1, 1, 0, 1, 1, 1) (1, 0, 1, 0, 0, 1) You need to follow the same assumptions as in the example in Slide 4 of Module 11. Show your work and specify whether it is possible that the algorithm makes more mistakes after processing the above vectors; justify your answer.

write a computer program that produces the desired output from the given input. Input: Elements in a finite set SOutput: Elements in ℘(S)Algorithm: Use recursion.

use java programming lanuage if needed to explain

1. Write an algorithm to determine whether a given element x belongs to a set S := {s1, . . . , sn}.

For part A, how what would the pseudo-code visually be when trying to visualize an efficient algorithm? As for Part B, what would determine the space and complexity?

Let S be a set of n distinct positive integers, where n is odd. The median of S is the (n+1)/2th smallest number of this set - i.e., the number in the middle of the set once the set is sorted. For example, if S = [70, 10, 20, 60, 30], then the sorted set is [10, 20, 30, 60, 70], from which we see that the median is 30. There is an obvious linearithmic algorithm to determine the median of S. First you sort the set, which takes O(n log n) time. Then you output the value of S[(n+1)/2}, which takes O(1) time. But is this the best we can do? Here is a bold claim: there is an O(n) algorithm to determine the median of set S. Determine whether the above claim is TRUE or FALSE.