Show that f(n) is Θ(g(n)) for the correct function g(n). Prove your result using the definitions from class, including an explicit value for k justifying your statement is true for all n ≥ k. f(n) = n 1.5 log(2n) + n 2 log(n 2 ) + √ n

Consider a sequence of integers defined by the following recurrence: f(0) = 0, f(1) = 1, and f(i) = f(i – 1) + f([i/2]) for i > 2. We would like to compute f(n) using DP. If we use bottom-up approach, what is the running time? Explain how you obtained your answer. (Assume every basic operation such as plus takes O(1) time as usual.) Give a pseudo-code for a top-down approach with memoization.

You are given a N*N maze with a rat placed at maze[0][0]. Find whether any path exist that rat can follow to reach its destination i.e. maze[N-1][N-1]. Rat can move in any direc­tion ( left, right, up and down).Value of every cell in the maze can either be 0 or 1. Cells with value 0 are blocked means rat can­not enter into those cells and those with value 1 are open.Input FormatLine 1: Integer NNext N Lines: Each line will contain ith row elements (separated by space)Output Format :The output line contains true if any path exists for the rat to reach its destination otherwise print false.Sample Input 1 :31 0 11 0 11 1 1Sample Output 1 :trueSample Input 2 :31 0 11 0 10 1 1Sample Output 2 : false Solution: //// public class Solution {     public static boolean ratInAMaze(int maze[][]){         int n = maze.length;        int path[][] = new int[n][n];         return solveMaze(maze, 0, 0, path);    }       public static boolean solveMaze(int[][] maze, int i, int j, int[][] path) {//…

Asymptotic notation Prove each of the following. You can use either the definitions of the form "exists c such that for all no..." or the definitions using limits. In both cases, show your work. (a) 3n² logn+ 5n²/ log log n e e(n² log³n). Note: log* n means (log n)*. (b) 8n345n E w(2") (c) 8n³45n € 20(n). (d) n? + 10n! is in o(2"). Hint: use Stirling's approximation formula.