Prove that Every marking algorithm is strictly k-competitive.
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Q: demonstrate that all marking algorithms are strictly k-competitive
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Prove that Every marking
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- 5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…Long chain of friends: You are given a list of people, and statements of the form “x knows y”. You are asked to find, is there a chain of k distinct people, such as x1 knows x2, x2 knows x3, and xk-1 knows xk. Prove that this problem is NP-complete by using one of the known NP-complete problems (CLIQUE, 3-SAT, Hamiltonian Path, Hamiltonian Cycle, Independent Set, etc.)Prove that the following problem, given a set S of integers and a number t, is of the NP class. Is there a subset of S whose elements add up to t?Note: An issue with data structures and algorithms
- Reword the statement below as a theorem about graphs and then prove it. Assume that if A is a friend of B, then B is a friend of A and that for all A, A is not a friend of A. • In any group of n >= 2 people, there are two people with the same number of friends in the group.Prove by Induction that for all integers n ≥ 1, n < n2 + 1 .Yes this problem is silly, but still do it by induction! Prove by Induction that for all integers n ≥ 3, 2n < n2 .Appendix A 10-Fold Cross Validation for Parameter Selection Cross Validation is the standard method for evaluation in empirical machine learning. It can also be used for parameter selection if we make sure to use the training set only. To select parameter A of algorithm A(X) over an enumerated range d E [A1,..., A] using dataset D, we do the following: 1. Split the data D into 10 disjoint folds. 2. For each value of A e (A1,..., Ar]: (a) For i = 1 to 10 Train A(A) on all folds but ith fold Test on ith fold and record the error on fold i (b) Compute the average performance of A on the 10 folds. 3. Pick the value of A with the best average performance Now, in the above, D only includes the training data and the parameter A is chosen without the knowledge of the test data. We then re-train on the entire train set D using the chosen A and evaluate the result on the test set.
- Prove by induction that there exists a knight’s walk of an n-by-n chessboard for any n ≥ 4. (It turns out that knight’s tours exist for all even n ≥ 6, but you don’t need to prove this fact.)Note: For all integers k,n it is true that kn, k+n, and k-n are integers. An integer k is even if and only if there exists an integer r such that k=2r. An integer k is odd if and only if there exists an integer r such that k=2r+1. For every integer k it is true that if k is even then k is not odd. For every integer k it is true that if k is odd then k is not even. For every integer k it is true that if k is not even then k is odd. For every integer k it is true that if k is not odd then k is even. If P then Q means the same thing as P → Q (P implies Q). 3. Consider the argument form pvr .. p→r Is this argument form valid? Prove that your answer is correct. 4. Prove that for every integer d, if d³ is odd then d is odd.You are organizing a programming competition, where contestants implement Dijkstra's algorithm. Given adirected graph G = (V, E) with integer-weight edges and a starting vertex s ∈ V , their programs are supposedto output triplets (v, v.d, v.π) for each vertex v ∈ V . Design an O(V +E) time algorithm that takes as inputthe original graph G in both adjacency matrix (G.M) and adjacency list (G.Adj) representations, startingvertex s, and the output of a contestant's program (given as an array A of triplets), and returns whetherA is the correct output for G. Write down the pseudocode for your algorithm, explain why it correctlyveries the output, and analyze your algorithm's running time. You may assume that all edge weights of the input graph provided to the contestantsare nonnegative and A (the output of their programs) is in the valid format, i.e., you don't need to verifythat A is actually an array of triplets, with v and v.π being valid vertices and v.d being an integer.Can you…
- LOWER BOUNDS BY THE METHOD OF ADVERSARIES. We have n numbers that take only at most k ≥ 2 different values (as in exercise 2 above). We are interested in a lower bound for an algorithm that groups these numbers by value when we can only use an identity oracle, i.e., an oracle that decides whether two numbers x and y are equal or not. Convince yourself that there is an obvious O(kn) algorithm for finding the groups. Using the method of adversaries, show that (kn) is a lower bound.An efficient algorithm has been designed to perform comparison sort of nnumbers. It is claimed that the number comparisons will always be ≤2n in the worst case scenario for n≤8. Do you agree with this claim or not? Justify will proper explanationThere are n men and women, each of which has ranked all n members of the other sex in order of preference. Consider the following algorithm for matching them up: Start with an arbitrary matching and as long as there is a man and woman who prefer each other to whoever they are currently matched with, switch them. Will this algorithm always result in a stable matching? O true O false