A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 17.5 Exercises 58917.3 Use a reduction to prove that multiplying two upper triangular n × n matricesis just as expensive (asymptotically) as multiplying two arbitrary n × nmatrices.17.4 (a) Explain why computing the factorial of n by multiplying all valuesfrom 1 to n together is an exponential time algorithm.(b) Explain why computing an approximation to the factorial of n by makinguse of Stirling’s formula (see Section 2.2) is a polynomial timealgorithm.17.5 Consider this algorithm for solving the K-CLIQUE problem. First, generateall subsets of the vertices containing exactly k vertices. There are O(n k ) suchsubsets altogether. Then, check whether any subgraphs induced by thesesubsets is complete. If this algorithm ran in polynomial time, what wouldbe its significance? Why is this not a polynomial-time algorithm for the K-CLIQUE problem?17.6 Write the 3 SAT expression obtained from the reduction of SAT to 3 SATdescribed in Section 17.2.1 for the expression(a + b + c + d) · (d) · (b + c) · (a + b) · (a + c) · (b).Is this expression satisfiable?17.7 Draw the graph obtained by the reduction of SAT to the K-CLIQUE problemgiven in Section 17.2.1 for the expression(a + b + c) · (a + b + c) · (a + b + c) · (a + b + c).Is this expression satisfiable?17.8 A Hamiltonian cycle in graph G is a cycle that visits every vertex in thegraph exactly once before returning to the start vertex. The problem HAMIL-TONIAN CYCLE asks whether graph G does in fact contain a Hamiltoniancycle. Assuming that HAMILTONIAN CYCLE is N P-complete, prove thatthe decision-problem form of TRAVELING SALESMAN is N P-complete.17.9 Use the assumption that VERTEX COVER is N P-complete to prove that K-CLIQUE is also N P-complete by finding a polynomial time reduction fromVERTEX COVER to K-CLIQUE.17.10 We define the problem INDEPENDENT SET as follows.