1. Why not solve the knapsack problem in the same way as the matrix-chain and optimum binary search tree problems: by minimizing, for k from I to M, the sum of the best value achievable for a knapsack of size k and the best value achievable for a knapsack of size M-k?

2. Extend the program for the shortest-paths problem to include a procedure paths that fills an array path with the shortest path from i to j. This procedure should take time proportional to the length of the path each time it is called, using an auxiliary data structure built up by a modified version of the program given in Chapter 32.

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1. Write an OpenGL program that renders a cube with six faces of different colors. Form the cube from six quadrilaterals, making sure that the front faces are facing outwards. If you already know how to perform rotations, let your program include the ability to spin the cube around.(Ref er to Chapter II and see the Wrap Torus program for code that does this.)

If you rendered the cube using triangles instead, how many triangles would be needed?

2. Repeat question 1. but render the cube using two quad strips, each containing three quadrilaterals.

3. Repeat question 1. but render the cube using two triangle fans.

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1. Write a program to find the longest simple path from x to y in a given weighted graph.

2. Could there be an algorithm which solves an NP-complete problem in an average time of N log N , if P ¢ NP? Explain your answer.

3. Give a nondeterministic polynomial-time algorithm for solving the partition  problem.

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1. Write a program to solve the knapsack problem of Chapter 42 when the sizes can be real numbers.

2. Write a program to count the number of spanning trees of a set of N points in the plane with no intersecting edges.

3.

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• 1- You have been tasked by your audit manager to develop an audit plan of a major bank. Propose the key elements of your audit plan and the end result you expect from implementing the audit. Justify the key elements you chose for the plan.
• 2- From the e-Activity, analyze the systems the company used and propose a computer analysis plan that would have detected the fraud. Support your analysis with examples.
• I only need 1 or 2 paragraph each. 200 words each.

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