A Practical Introduction to Data Structures and Algorithm Analysis

94 Chap. 3 Algorithm Analysis3.17 Given an array storing integers ordered by value, modify the binary searchroutine to return the position of the first integer with value K in the situationwhere K can appear multiple times in the array. Be sure that your algorithmis Θ(log n), that is, do not resort to sequential search once an occurrence ofK is found.3.18 Given an array storing integers ordered by value, modify the binary searchroutine to return the position of the integer with the greatest value less thanK when K itself does not appear in the array. Return ERROR if the leastvalue in the array is greater than K.3.19 Modify the binary search routine to support search in an array of infinitesize. In particular, you are given as input a sorted array and a key valueK to search for. Call n the position of the smallest value in the array thatis equal to or larger than X. Provide an algorithm that can determine n inO(log n) comparisons in the worst case. Explain why your algorithm meetsthe required time bound.3.20 It is possible to change the way that we pick the dividing point in a binarysearch, and still get a working search routine. However, where we pick thedividing point could affect the performance of the algorithm.(a) If we change the dividing point computation in function binary fromi = (l + r)/2 to i = (l + ((r − l)/3)), what will the worst-case runningtime be in asymptotic terms? If the difference is only a constanttime factor, how much slower or faster will the modified program becompared to the original version of binary?(b) If we change the dividing point computation in function binary fromi = (l + r)/2 to i = r − 2, what will the worst-case running time be inasymptotic terms? If the difference is only a constant time factor, howmuch slower or faster will the modified program be compared to theoriginal version of binary?3.21 Design an algorithm to assemble a jigsaw puzzle. Assume that each piecehas four sides, and that each piece’s final orientation is known (top, bottom,etc.). Assume that you have available a functionbool compare(Piece a, Piece b, Side ad)that can tell, in constant time, whether piece a connects to piece b on a’sside ad and b’s opposite side bd. The input to your algorithm should consistof an n × m array of random pieces, along with dimensions n and m. Thealgorithm should put the pieces in their correct positions in the array. Youralgorithm should be as efficient as possible in the asymptotic sense. Write