[Undecidability and Reductions] OVERLAPPING = {{M, D) : | M is a Turing Machine and D is a DFA such that L(M)n L(D) # Ø }. That is, there exists at least one string w E L(D) such that M halts on w and accepts it. 1. Show that OVERLAPPING is undecidable using reductions. You can reduce ATM or ETM or any of the undecidable languages that was discussed in class, homework, or practice problems.

An NP-complete problem is a fascinating kind of problem because till now no one has discovered the polynomial-time algorithm to solve it and also no one has proved that no polynomial-time algorithm can exist for any NP-complete problem. It is an open research problem since it was first posed in 1971 to prove P#NP. The NxN Queens problem can be summarized as follows: putting N chess queens on an N×N chessboard such that none of them is able to attack any other queen using the standard chess queen's moves (row-column- diagonal). Thus, a solution requires that no two queens share the same row, column, or diagonal. Solutions exist only for N = 1 or N 2 4. Use the given function below to test whether a queen is attacked by another or not. You are not allowed to use any other code to check if a queen is safe. Implement a backtracking solution for the algorithm in Java that finds all possible solutions for N queens and measure the execution time it takes for N=4 to 12 and compare them…

Recall a set A is countable if |N| ≥ |A|. Recall that Cantor’s theorem shows that P(N) = {X ⊆ N}is uncountable. Prove this.Show that F = {X ⊆ N | |X| < ∞} is countable, with an onto functiong : N → F that is computable.   Write a program/function (e.g., in Python) that runs forever and prints every element of F,e.g., if run it would print something like{0}{1}{0,2}{3}{0,2}{0,5,7}{1}{0,1}{}{2}...(Note that you must prove that the program works, i.e., it prints every element of F, and onlyelements of F.)

Exercise 12.9 Given a weighted (connected) undirected graph G = (V, E, w), with m =n+ 0(1) edges (i.e., the number of edges is at most the number of vertices plus some constant), give an O(n)-time algorithm to compute the minimum spanning tree of G. Prove the correctness and the running time of your algorithm. (Hint: Think of how one can eliminate an edge from being in the MST.)

9. Limitation Question. Using the idea of Pumping Lemma, please explain, why we can't build a Finite Automaton to generate all and only strings of the form 1³n0n where n>=0.

You are given an adjacency list representation of a weighted directed graph G = (V, E) with n vertices, m edges, and no negative cycles. Furthermore, you are given two designated vertices s, t ∈ V and a subset P ⊆ E of the edges are labeled as “premium” edges. Describe a O(mn)-time algorithm to find the shortest path from s to t that uses at most one premium edge (along with any number of non-premium edges).

Let G be an undirected edge-weighted graph with V vertices and O(V2) edges. Describe an efficient algorithm to compute a minimum spanning tree that runs in O(V2) time. You are only allowed to use data structures such as arrays and priority queues. Do not use complex data structures such as Fibonacci heaps. Give a formal proof of correctness for your algorithm. An answer without any formal proof will be considered incomplete. Analyze the running time of the algorithm.