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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.
In a survey of 85 people this is found that 31 want to drink milk 43 like coffee and 39 wish tea. As well 13 want both milk and tea, 15 like milk & coffee, 20 like tea and coffee
These can be expressed in terms of two fundamental operations of addition and multiplication. If a, b and c are any three real numbers, then; 1.
Frederick bought six books which cost d dollars each. What is the total cost of the books? Frederick would multiply the number of books, 6, through how much each one costs, d.
Realtors estimate that 23% of homes purchased in 2004 were considered investment properties. If a sample of 800 homes sold in 2004 is obtained what is the probability that at most
1 ream uses 6% of a tree, Estimate the reams of paper used in one month in an office(may be your father, mother or neighbour), hence find the number of trees that need to be cut fo
1) Let the Sample Space S = {1, 2, 3, 4, 5, 6, 7, 8}. Suppose each outcome is equally likely. Compute the probability of event E = "an even number is selected". P(E) = 2) A s
Obligatory application/interpretation problem : Next, we need to do our obligatory application/interpretation problem so we don't forget about them. Example : Assume that the
describe the end behavior of the following function using Limit notation f(x)= 2x-1/x-1
Describe and sketch the surfaces z + |y| = 1 and (x 2) 2 y + z 2 = 0.
Children of the same age can be at different operational stages, and children of different ages, can be at the same developmental stage." Do you agree with this statement? If so, g
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