3 sort the following running times so that f n is

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Unformatted text preview: 3. Sort the following running times so that f ( n ) is left from g ( n ) if f ( n ) is O ( g ( n )), indicating whether any two are Θ to each other. No proof required. n √ n,n log n, 1 . 5 n ,n 2 , 2 n 2 , log( n 2 ) n 4. Given that f 1 ( n ) is O ( f 2 ( n )), f 2 ( n ) is O ( f 3 ( n )), g 1 ( n ) is O ( g 2 ( n )), and g 2 ( n ) is O ( g 3 ( n )), state whether the following statements are true or false. If true, no proof is required. If false, give a counterexample. (a) f 1 ( n ) * g 1 ( n ) is O ( f 2 ( n ) * g 2 ( n )) (b) f 1 ( n ) is O ( f 3 ( n )) (c) f 3 ( n ) + g 3 ( n ) cannot be O ( f 1 ( n ) + g 1 ( n )) (d) log ( f 1 ( n )) is O ( log ( f 2 ( n ))) (e) f 1 ( n ) /g 1 ( n ) is O ( f 2 ( n ) /g 2 ( n )) 5. (30 pts) For each statement below, decide whether it is true or false. In each case attach a very brief explanation of your answer. (a) A binary heap of height 2 has at least 4 nodes and at most 7 nodes, true or false? (b) Suppose that the worst-case running time of method qq is Θ( n log n ) and the worst case running time of method uu is Θ( n 2 ). Then there is no input for which uu runs faster than qq , true or false? (c) It is possible to reverse the content of a stack using at most three other auxilliary stacks, true or false? 4 (d) A priority queue is a queue in which elements are enqueued in order of their priority, true or false? (e) Suppose we decide to change our model of computation (step-counting) by counting instructions that involve the creation of an array new int[n] as taking 10 n steps. In this new model, a program can have a different asymptotic complexity than in the original model, true or false? (f) In a binary min-heap with at least three elements the largest element is always in the right subtree of the root: true or false? (g) Suppose that f ( n ) is a function defined only for integers n ≥ 1 and that f ( n ) is O (1). Then, there exists a constant c such that f ( n ) ≤ c for all n ≥ 1, true or false? 6. A binary heap contains the keys 1 , 2 ,..., 10 , 11. If we traverse it in postorder we list the keys as follows: 10 , 8 , 7 , 5 , 4 , 3 , 2 , 11 , 9 , 6 , 1. Draw the heap. 7. After a removeMin operation that returns the key a , a binary min-heap looks like this (the keys are ordered alphabetically): b / \ / \ / \ c f / \ / \ / \ / \ g d k i / \ / j h e Draw two different trees that could have been what the heap looked like before the removeMin operation. 8. Consider a binary heap of height h and let L be the number of nodes at depth h in this tree. Assuming that the tree has 1000 nodes, calculate h and L . 9. (15 points) In this problem you NOT allowed to use any of the theorems about Big-Oh stated in the lecture slides, the textbook, or the lab writeups. Your proof should rely only on the definition of Big-Oh....
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