Consider the function f : N × N → N given by f(m, n) = 2m-1(2n − 1), (m, n) ∈ N × N Show that f is bijective

Let m be a matrix with n rows and n columns whose entries are either 1 or 0. recall that the element of m on row i and column j is denoted by mij . the diagonal entries of m are {mii} for 1 ≤ i ≤ n. We call M permutable if it is possible to swap some of the rows and some of columns so that all diagonal entries are 1. Design a polynomial time algorithm that decides whether a binary matrix M is permutable or not. Note that we can swap any two rows or two columns. Also, the order in which these swaps are done is not important.

Show that:   logn = O(n)

Consider an n by n matrix, where each of the n2 entries is a positive integer. If the entries in this matrix are unsorted, then determining whether a target number t appears in the matrix can only be done by searching through each of the n2 entries. Thus, any search algorithm has a running time of O(n²). However, suppose you know that this n by n matrix satisfies the following properties: • Integers in each row increase from left to right. • Integers in each column increase from top to bottom. An example of such a matrix is presented below, for n=5. 4 7 11 15 2 5 8 12 19 3 6 9 16 22 10 13 14 17 24 1 18 21 23 | 26 | 30 Here is a bold claim: if the n by n matrix satisfies these two properties, then there exists an O(n) algorithm to determine whether a target number t appears in this matrix. Determine whether this statement is TRUE or FALSE. If the statement is TRUE, describe your algorithm and explain why your algorithm runs in O(n) time. If the statement is FALSE, clearly explain why no…

Use & for •, v for V, > for Ɔ, and = for = What can you derive by applying De Morgan's (DM) to this formula: ~(N • E) Ɔ G (N v E) > G O (-N v -E) > G O (N & E) v G O (N & E) v G O (N v E) > G O (N & E) > G

Let T(n) be defined by the T(n) = 4T(n/2) + 2n, T(1) = c Compute T(4) by iterating forwards and express your answer in terms of c. T(4) = divide-and-conquer recurrence

Use the Transform-and-Conquer algorithm design technique with Instance Simplification variant to design an O(nlogn) algorithm for the problem below. Show the pseudocode. Given a set S of n integers and another integer x, determine whether or not there exist two elements in S whose sum is exactly x.

Let T(n) be defined by the first-order linear recurrence T(n) = 2T(n-1) +8 Suppose it is given that T(2) = c. Compute T(0) by iterating backwards and express your answer in terms of c. T(0) =

Let ENFA = {N | N is an NFA and L(N) = Ø}. Prove that ENFA is in P.

give set of R(x): x < x^2 where the domain is Z