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- Correct answer will be upvoted else downvoted. Computer science. You are given a number n and an exhibit b1,b2,… ,bn+2, acquired by the accompanying calculation: some exhibit a1,a2,… ,a was speculated; cluster a was composed to exhibit b, for example bi=ai (1≤i≤n); The (n+1)- th component of the cluster b is the amount of the numbers in the exhibit a, for example bn+1=a1+a2+… +an; The (n+2)- th component of the cluster b was thought of some number x (1≤x≤109), for example bn+2=x; The cluster b was rearranged. For instance, the cluster b=[2,3,7,12,2] it very well may be acquired in the accompanying ways: a=[2,2,3] and x=12; a=[3,2,7] and x=2. For the given cluster b, find any exhibit a that might have been speculated at first. Input The main line contains a solitary integer t (1≤t≤104). Then, at that point, t experiments follow. The primary line of each experiment contains a solitary integer n (1≤n≤2⋅105). The second column of each experiment…8. Answer the following questions for a random graph generated by the Erdos-Renyi random graph model ER(n, p). You do not need to prove your answers are correct but still need to show your work. Your answer should be in terms of n and p. 8(a) What is the expected number of triangles? 8(b) What is the probability that d(u, v) > 2 for two particular vertices u v? (Hint: First, find the probability that d(u, v) > 1. Then consider, for all other vertices x, the probability that uxv is NOT a path in the random graph. Multiply all these probabilities together.) Note that for this question, no graph is given. However, we know that the expected number of edges for ER(n, p) random graph model is , and that the expected degrees of the vertex for the random graph model ER(n, p) is (n-1) *pHow can the Wumpus world rule that all cells surrounding edge cell (2,1) will have a stench if the Wumpus is in (2.1) be represented in Propositional Logic? Let Wij and Sij represent a Wumpus and stench in (i.) respectively. O W21- (S11 v S31 v 522) O S21 (W11A W31 A W22) O WI1A W31A W22) S21 O w21- (S11 A S31 A S22)
- Given the following example of UAG graphs: 5 7 7 1 2 7. 12 2 2 6 12 6, 10 21 3 5 3 3 21 14 15 8 15 8. 4 Graph A Graph B Graph C (degree(v) <= 1) a)- Give implementation to find the shortest path b)- Does your implementation take all considerations and could accept any kind of inputs? Explain c)- Justify your choice by providing the approximating time/space complexity of each type graph.Exercise 3 Consider calculation of PageRank for a directed graph. Consider two jumping probabilities a and B such that a > B. For every node u, let Pa(u) be the PageRank of u calculated using a and let Pg (u) be the PageRank of u calculated using B. For each of the following statements, indicate whether the statement is true or false and provide a brief explanation for your answer. 1. The inequality Pa (u) > Pg (u) must hold for all nodes u. 2. The inequality Pa(u) > Pg (u) must hold for at least one node u.a) Let M ({go, qı, q2, q3, q4, qs}, {a, b, c}, qo, fs, {q1,q3, qs}) be the Deterministic Finite Automaton (DFA) with state transition, f, is defined as in Table 2. Table 2 fs State a b 91 90 91 91 92 91 92 92 93 94 93 93 93 93 94 94 95 94 95 95 95 95 i. Draw the transition diagram for the machine, M. ii. Determine the final state for the input string bacc. Is the input string aabcba accepted by the DFA? Show the sequence of transition of each state for the input string. iv. State one input string that start and end with b and accepted by the DFA.
- Suppose there is undirected graph F with nonnegative edge weights we ≥ 0. You have also calculated the minimum spanning tree of F and also the shortest paths to all nodes from a particular node p ∈ V . Now, suppose that each edge weight is increased by 1, so the new weights are we′ = we + 1. (a) Will there be a change of the minimum spanning tree? Provide an example where it does or prove that it cannot change. (b) (3 points) Will the shortest paths from p change? Provide an example where it does or prove that it cannot change.Give a constructive proof to show that a DFA Mk = (Qk, Σk, δk, q0, Fk) exists that recognizes each Lk = {w: w ends in k 1’s} (for all k > 0) over Σ = {0, 1.= gcd(2a, a - b) = gcd(a + b, 2b). 9. Prove that gcd(a + b, a - b) Show your work! Justify each step.
- We showed the reduction from 3-SAT to INDEPENDENT SET in class. Given the following graph G, the correct statements are: -D U -A -B -C G corresponds to a 3-SAT formula (A V BV C) A (D VEVA) A (AVDV C). OG corresponds to a 3-SAT formula (A A BAC) V (DAEAA) V (AADA C). OG corresponds to a 3-SAT formula (A V¬BV C) A (-DVEVA) A (¬AV DV¬C). OG corresponds to a 3-SAT formula (AA¬BAC) V (DAEAA) V (¬AADA ¬C).I have to prove that: "If G1 and G2 are consistent connected graphs with at least one vertex in common, then G1UG2 (G1 union G2) is connected. What I know so far is that I either need to show that both vertices are in the same vertex set or the two vertices are in different sets. However, I'm not sure if this is the correct approach. Please help with this proof.in theory graph Define and give examples to all : what is the total number of maximal independent dominating sets ? and find the total number of maximal independent dominating Cn sets to Pand