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1. a) Given a digraph G = (V,E), prove that if we add a constant k to the length of every arc coming out from the root node r, the shortest path tree remains the same. Do this by using potentials:
i) Show there is a potential y* for the new costs for which the paths in the tree to each node v have cost y*v, and
ii) explain why this proves it. What is the relationship between the shortest path distances of the modified problem and those of the original problem?
b) Can adding a constant k to the length of every arc coming out from a non-root node produce a change in the shortest path tree? Justify your answer.
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What is our aim when teaching children multiplication? Firstly they should be able to judge which situations they need to multiply in, and the numbers that are to be multiplied sec
Question: Let f be a quartic polynomial (ie. a polynomial of degree 4). Suppose that f has zeros at -2; 1; 3; 4 and that f(0) = 4. Sketch a graph of f. If f(x) is
If y 1 (t) and y 2 (t) are two solutions to a linear, homogeneous differential equation thus it is y (t ) = c 1 y 1 (t ) + c 2 y 2 (t ) ........................(3) Remem
What fraction of the full price will you pay for 2 shirts? 3 4 11 2 $45.001 2 .
1/2+3/4
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Question: Classify the following differential equations as linear/nonlinear. Also, what is the order of the following differential equations? Xy'-2y =x Xy'' -2y' =xsin(y)
1. (‡) Prove asymptotic bounds for the following recursion relations. Tighter bounds will receive more marks. You may use the Master Theorem if it applies. 1. C(n) = 3C(n/2) + n
Consider the system of linear equations X + ay = 1 2x + 8y = b Where a and b are real numbers. (a) Write out the augmented matrix for this system of linear equations.
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