A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 17.6 Projects 59117.14 The last paragraph of Section 17.2.3 discusses a strategy for developing asolution to a new problem by alternating between finding a polynomial timesolution and proving the problem N P-complete. Refine the “algorithm fordesigning algorithms” from Section 15.1 to incorporate identifying and dealingwith N P-complete problems.17.15 Prove that the set of real numbers is uncountable. Use a proof similar to theproof in Section 17.3.1 that the set of integer functions is uncountable.17.16 Prove, using a reduction argument such as given in Section 17.3.2, that theproblem of determining if an arbitrary program will print any output is unsolvable.17.17 Prove, using a reduction argument such as given in Section 17.3.2, that theproblem of determining if an arbitrary program executes a particular statementwithin that program is unsolvable.17.18 Prove, using a reduction argument such as given in Section 17.3.2, that theproblem of determining if two arbitrary programs halt on exactly the sameinputs is unsolvable.17.19 Prove, using a reduction argument such as given in Section 17.3.2, that theproblem of determining whether there is some input on which two arbitraryprograms will both halt is unsolvable.17.20 Prove, using a reduction argument such as given in Section 17.3.2, that theproblem of determining whether an arbitrary program halts on all inputs isunsolvable.17.21 Prove, using a reduction argument such as given in Section 17.3.2, that theproblem of determining whether an arbitrary program computes a specifiedfunction is unsolvable.17.22 Consider a program named COMP that takes two strings as input. It returnsTRUE if the strings are the same. It returns FALSE if the strings are different.Why doesn’t the argument that we used to prove that a program to solve thehalting problem does not exist work to prove that COMP does not exist?17.6 Projects17.1 Implement VERTEX COVER; that is, given graph G and integer k, answerthe question of whether or not there is a vertex cover of size k or less. Beginby using a brute-force algorithm that checks all possible sets of vertices ofsize k to find an acceptable vertex cover, and measure the running time on anumber of input graphs. Then try to reduce the running time through the useof any heuristics you can think of. Next, try to find approximate solutions to