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Unformatted text preview: Solutions to Homework Four CS 191 : Discrete Structures I Mar. 6 , Winter 2007 Due Date: Mar. 20, 2007, Tuesday, before the class meeting. Section 4.2, p155 Exercise (15) Write an algorithm whose input is a sequence s 1 , , s n sorted in nondecreasing order and a value x . (Assume that all the values are real numbers.) The algorithm inserts x into the sequence so that the resulting sequence is sorted in nondecreqsing order. Example: If the sequence is 2, 6, 12, 14, and x = 5, the resulting sequence is 2, 5, 6, 12, 14. Answer : insert ( s , n , x ){ i :=1 while ( i n ) and x > s i do i := i + 1 // x should be inserted in front of the i th location. We need to right shift s i , , s n one place. j := n while ( j i ) { do { s i +1 := s i j := j 1 } //Insert x s i := x } Exercise (18) Describe the worst case input for algorithm 4.2.1. Answer : The worst case occurs when the for loop and the while loop run as long as possible. This situation is achieved when t consists of n 0s and p consists of m1 0s followed by one 1. 1 Section 4.3, p168 Select a theta notation from Table 4.3.3 for each expression in exercises 112. Exercise (6) 4 6 6 + + n n Answer: 4 6 6 + + n n = ( n 6 ) Exercise (9) (6 n + 4)(1+lg n ) Answer : (6 n + 4)(1+lg n ) = ( n lg n ) Exercise (11) n n n n n 2 2 ) 1 )( lg ( + + + Answer : n n n n n 2 2 ) 1 )( lg ( + + + = ( n ) Exercise (12) 2+4+8+16++ 2 n Answer: 2+4+8+16++ 2 n = 2 n +1 2 = (2 n ) In Exercises 1315, select a theta notation for f(n) + g(n). 2 Exercise (15) f ( n ) = ( n 3/2 ), g ( n ) = ( n 5/2 ), Answer : f(n) + g(n) = ( n 3/2 ) + ( n 5/2 ) = ( n 5/2 ), In exercises 1626, select a theta notation from among (1), (lg n ), ( n ), ( n lg n ), ( n 2 ), ( n 3 ),...
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