In one pass and space O(n log n), the above algorithm constructs a (1/6)-approximate weighted matching M.give proof

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) =

4.2. Consider the following code. class A { void m (A x) { System.out.println("AA"); } void m(B x) {System.out.println("AB"); } void m(C x) { System.out.println("AC"); } } class B extends A { } void m (A x) { System.out.println("BA"); void m (B x) {System.out.println("BB"); void m(C x){ System.out.println("BC"); } } } class C extends B { void m (A x) {System.out.println("CA"); } void m(B x) {System.out.println("CB"); } void m(C x) { System.out.println("CC"); } } A a1 = new A(); A a2 = new B(); A a3 = new C(); B b1 = new B(); B b2=new C(); C c1 = new C(); Fill in each box with the output of the corresponding method invocation. For example: - fill in the 4th row and 2nd column with the output of b1.m(a2). - fill in the 2nd row and 4th column with the output of a2.m(b1) Hint: this problem goes fast once you see the pattern. al a2 a3 bl b2 cl al a2 a3 b1 b2 cl

Find a tight bound solution for the following recurrence: T(n) = 4 T(n/2) + c n (c is a positive constant) That is, find a function g(n) such that T(n) ∊ Θ(g(n)). For convenience, you may assume that n is a power of 2, i.e., n=2k for some positive integer k. Justify your answer. [Note: Read question 1-(2) first before writing your answer] Prove your answer in 1-(1) either using the iteration method or using the substitution method.

A recursive algorithm is applied to some data A = (a1,..., am) where m≥ 2. The running time T is characterised using the following recurrence equations: T(2) = c when the size of A is 2 T(m) = T(m-1) + 2c otherwise Determine the running time complexity of this algorithm.

You are given a weighted, undirected graph G = (V, E) which is guaranteed to be connected. Design an algorithm which runs in O(V E + V 2 log V ) time and determines which of the edges appear in all minimum spanning trees of G. Do not write the code, give steps and methods. Explain the steps of algorithm, and the logic behind these steps in plain English.Please give total time complexity. test whether a particular edge appears in all MSTs