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Once again, here s our pair of equations: 10x + 17y = 8 and 11x + 41y = 7 We must get both of these into SI form, with x all by itself on the left sides of the equals signs Step-by-step, the first equation morphs like this: 10x + 17y = 8 10x = 17y + 8

314 The chapter suggests an alternative algorithm for linearization (topological sorting), which repeatedly removes source nodes from the graph (page 93) Show that this algorithm can be implemented in linear time 315 The police department in the city of Computopia has made all streets one-way The mayor contends that there is still a way to drive legally from any intersection in the city to any other intersection, but the opposition is not convinced A computer program is needed to determine whether the mayor is right However, the city elections are coming up soon, and there is just enough time to run a linear-time algorithm (a) Formulate this problem graph-theoretically, and explain why it can indeed be solved in linear time (b) Suppose it now turns out that the mayor s original claim is false She next claims something weaker: if you start driving from town hall, navigating one-way streets, then no matter where you reach, there is always a way to drive legally back to the town hall Formulate this weaker property as a graph-theoretic problem, and carefully show how it too can be checked in linear time 316 Suppose a CS curriculum consists of n courses, all of them mandatory The prerequisite graph G has a node for each course, and an edge from course v to course w if and only if v is a prerequisite for w Find an algorithm that works directly with this graph representation, and computes the minimum number of semesters necessary to complete the curriculum (assume that a student can take any number of courses in one semester) The running time of your algorithm should be linear

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10x = 17y 8 x = (17/10)y 8/10 The second equation morphs as follows: 11x + 41y = 7 11x = 41y + 7 11x = 41y 7 x = (41/11)y 7/11 Now we mix the right-hand sides, getting a first-degree equation in y: (17/10)y 8/10 = (41/11)y 7/11

You re traveling to another dimension, a dimension not only of sight and sound, but of mind a journey into a wondrous land whose boundaries are that of imagination The signpost up ahead: Fibonacci All teasing aside, a number of extremely serious traders are devoted to the Fibonacci sequence, that mysterious indicator with a really cool name Short-term traders primarily use it to identify key support and resistance areas The support and resistance levels are based on the retracement of a previous move, either up or down You ll have to read a book or take classes to fully understand Fibonacci, but I ll do my best to summarize it A brilliant thirteenthcentury Italian mathematician, Leonardo Fibonacci, helped to promote a sequence of numbers that is the sum of the previous two

Let s get a common denominator here We can multiply the numerators and denominators on the left side of the equals sign by 11, and multiply the numerators and denominators on the right side of the equals sign by 10 That gives us (187/110)y 88/110 = (410/110)y 70/110 Multiplying the whole equation through by 110 to get rid of the fractions, we obtain 187y 88 = 410y 70 Adding 88 to each side produces 187y = 410y + 18 Subtracting 410y from each side, we have 223y = 18 We can divide through by 223 to get y = 18/( 223) = 18/223 This agrees with the result we obtained by double elimination We can plug this into either of the SI equations we derived above, solving for x The first equation will do Step-by-step: x = (17/10)y 8/10 = (17/10)( 18/223) 8/10 = 306/2,230 8/10 = 306/2,230 1,784/2,230 = ( 306 1,784)/2,230 = 2,090/2,230 = 209/223

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It agrees again! This practically guarantees that our answers are correct We ve found x and y by two different routes, and they ve come out the same both times If we made an error somewhere, the answers would almost certainly disagree

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