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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.
Even and Odd Functions : This is the final topic that we have to discuss in this chapter. Firstly, an even function is any function which satisfies,
While we first looked at mechanical vibrations we looked at a particular mass hanging on a spring with the possibility of both a damper or/and external force acting upon the mass.
Don't count the number of divisions. Do not use asymptotic notation, instead provide exact answers. (i) What is the maximum number of multiplications required to solve a system
Assume that the amount of money in a bank account after t years is specified by, Find out the minimum & maximum amount of money in the account throughout the first 10 years
how is it done
A function is an equation for which any x which can be plugged into the equation will yield accurately one y out of the equation. There it is. i.e. the definition of functions w
-cot^2 90^0 + 4 sin 270^0 - 3 tan 180^0
solve the following simultaneous equations x+y=a+b ; a/x_b/y
Create a detailed diagram to describe the equation of an ellipse in terms of it’s eccentricity and indicate how the foci and major and minor semi-axes are involved. Y
Evaluate following limits. Solution In this part what we have to note (using Fact 2 above) is that in the limit the exponent of the exponential does this, Henc
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