Midterm_sample

# Midterm_sample - IE 170 Midterm Examination Practice...

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Dr. T.K. Ralphs 1. You are given information that the running time of algorithm A is O ( n lg n ) and that the running time of algorithm B is O ( n 3 ). What does this tell you about the relative performance of A and B? 2. Show that n lg n O ( n 3 / 2 ). Is it also true that n lg n o ( n 3 / 2 )? 3. For each pair of functions f and g , state whether f is o , O , Θ, Ω, ω of g (choose as many as apply). Justify your assertion. (a) f ( n ) = lg n , g ( n ) = ln n . (b) f ( n ) = n 1 /n , g ( n ) = 1. (c) f ( n ) = n lg n , g ( n ) = n . 4. Solve the following 3 recurrences. Assume that T (1) = 1. (a) T ( n ) = T ( n - 2) + n/ 2 (b) T ( n ) = 7 T ( n/ 3) + n 2 (c) T ( n ) = T ( n ) + 1 5. Observe that the while loop of lines 5-7 of the insertion sort procedure on page 17 of CLRS uses a linear search to scan through the sorted subarray A [1 . . . j - 1] to ﬁnd the insertion point. Since the subarray is already sorted, can we improve the worst-case running time by using binary search to ﬁnd the insertion point? If so, what would be the improved worst-case running time? 6. A generalized queue is a data structure that combines the functionality of a stack, in which the item most recently added to the list is the next to be removed, with that of a standard (FIFO) queue, in which the item least recently added to the list is the next to be removed. In a basic generalized queue interface, the following operations are supported. put : add an item to the queue. getMostRecent : return a copy of the most recently added item and then delete it from the queue. getLeastRecent : return a copy of the least recently added item and then delete it from the queue. empty : indicate whether the queue is empty or not. Note that we don’t have operations to look at items in the queue without deleting them. 1

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Midterm_sample - IE 170 Midterm Examination Practice...

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