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- Let A = {1, 2, 3, 4, 5} and B = {2, 3, 4, 5, 6, 7} and C = {a, b, c, d, e} 15. Give an example of f: A -> B that is not 1-1.Consider the following network: a.) Find the shortest path from node 1 to node 8 using Dijkstra’s algorithm. Write the length of the shortest path as well as the edges that compose it. b.) Write an integer programming formulation to find the second-shortest path from 1 to 8 in this network. (Hint: if the shortest path is not available, the second-shortest path becomes the shortest path.)How exactly would I be able to paraphrase this pseudo-code so that it can make a bit more sense? 1.Initialtzation: generate a random solution X ← RandomSolution() 2. Solve the traffic assignment problem at equilibrium for each link: (flow, cost) ← NonLinearComplementary(X) 3. Evaluate the objective value for the current solution: f (X) ← EvaluateObj Val (f low, cost) 4. T ← Tmax 5 while (T > Tmin) do 6 I ← 1 7 while (t < ltermax)do 8 X' ← FindNeighborSolution(X) 9 (flow,cost) ← NonLinearComplementary(X') 10 f(X') ← EvaluateObf Val (f low, Cost) 11 Δ ←(f(x) - f(x')) 12 if(Δ<0)then 13 BoltzmannValue ← exp(-Δ/T) 14 if (BoltzmannValue < Random (0,1)) then 15 Accept the new solution: X ← X' 16 end If 17 Else I8 Accept the new solution: X ← X' 19 end if 20 i ← i+1 21 end while 22 T ←T ∞ 23 end while ________
- 6. Let M = (Q,Sigma,s, q0, F) be a dfa and define cfg g= (v,sigma, R,S) as follows: 1. V=Q; 2. For each q in Q and a in sigma, define rule q->aq' where q' = s(q,a); 3. S = q0 Prove L(M) = L(G)Consider the classical hub location formulations we studied in the class. In these formulations, we assumed that every origin-destination pair uses at most two hubs. This assumption will result in suboptimal solutions if the travel costs are calculated considering the rectilinear distances instead of the Euclidean distances. True or falseConsider the elliptic curve group based on the equation y^2 ≡ x^3 + ax + b mod p where a=1190, b=213, and p=2011. We will use these values as the parameters for a session of Elliptic Curve Diffie-Hellman Key Exchange. We will use P=(2,51) as a subgroup generator. You may want to use mathematical software to help with the computations, such as the Sage Cell Server (SCS). On the SCS you can construct this group as: G=EllipticCurve(GF(2011),[1190,213]) Here is a working example. (Note that the output on SCS is in the form of homogeneous coordinates. If you do not care about the details simply ignore the 3rd coordinate of output.) Alice selects the private key 34 and Bob selects the private key 22. What is A, the public key of Alice? What is B, the public key of Bob? After exchanging public keys, Alice and Bob both derive the same secret elliptic curve point ???. The shared secret will be the x-coordinate of ???. What is it?
- The Manhattan Tourist Problem (1) Given setting in Figure 6.4, calculate exact number of different paths by dynamic programming. (2) For a general setting, i.e., from source(0,0) to sink (n, m), how many different paths by dynamic programming?1 4 3 3 4 1 0 4 3 N 5 1Please answer the following question in detail and explain all the proofs and assumptions for all parts. The question has three parts, (a), (b) and (c). Iterative lengthening search is an iterative analogue of uniform-cost search. The basic idea is to use increasing limits on path cost. If a node is generated whose path cost exceeds the current limit, it is immediately discarded. For each new iteration, the limit is set to the lowest path cost of any node discarded in the previous iteration. (a) Show that this algorithm is optimal for general path costs. You may assume that all costs are integers (this is not a loss of generality if the search space is finite). You may wish to consider the minimal path cost C; what happens when we set the path cost to be some limit l < C? (b) Consider a uniform tree with branching factor b, solution depth d, and unit step costs (each action costs one unit). How many iterations will iterative lengthening require? (c) (7 points) Now consider the…Construct a LCS table for Input1=’Student name’ and Input2=’Hometown’. Also find the Length of LCS and Actual LCS. Construct LCS for both scenarios given below. a)Take Input1—Vertical and Input2—Horizontal b) Take Input2—Vertical and Input1—Horizontal
- 5 Consider the degree-4 LFSR given by x to the power of 4 space plus space x squared space plus space x space plus space 1. Assume that the LFSR is initialized with the string left parenthesis s subscript 0 s subscript 1 s subscript 2 s subscript 3 right parenthesis equals left parenthesis 0110 right parenthesis. How long is the cycle generated?Please answer the following question in depth with full detail. Suppose that we are given an admissible heuristic function h. Consider the following function: 1-h'(n) = h(n) if n is the initial state s. 2-h'(n) = max{h(n),h'(n')−c(n',n)} where n' is the predecessor node of n. where c(n',n) min_a c(n',a,n). Prove that h' is consistent.Please find an optimal parenthesization of a matrix chain product whose sequence of dimensions is <10, 5, 10, 4, 8>. Please show your works. Also, explain the computational complexity and real cost of calculating the matrix chain product.