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midtermsol - FUNDAMENTAL ALGORITHMS MIDTERM SOLUTIONS You...

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FUNDAMENTAL ALGORITHMS MIDTERM SOLUTIONS You must omit at least 10 points. Mark on your booklet which part or parts of problems you are omitting. Maximum score: 140. What is slight can still be great, if it is written in a natural, flowing and easy style - and at the same time bears the mark of sound composition. – letter to Wolfgang Mozart from his father, 1778 1. (35) These questions concern a max-heap A with auxilliary structures length[A] , Left[i] , Right[i] and Parent[i] . (a) (5) Give the max-heap property. (That is, what is the relationship between the values A [ i ] that one needs for a heap.) Solution: A[Parent[i]] A[i] (b) (5) Suppose length[A] =100 and the values A [ i ] are distinct. What are the possible i for which A [ i ] is the second largest value? (You must give all possible i .) Solution: The biggest is the root and the second biggest is one of its children, so i = 2 , 3 are the possibilities. (c) (5) Suppose length[A] =100 and the values A [ i ] are distinct. What are the possible i for which A [ i ] is the smallest value? (You must give all possible i .) Solution: It must be a leaf so 51 i 100. (d) (10) Describe the algorithm MAX-HEAP-INSERT(A,key) which adds to heap A a new entry with value key . (Either a description in words or a pseudocode program would be fine.) Solution: First increment heapsize and place key in the last po- sition. Set x = heapsize. Now while x negationslash = 1 and A [ x ] is less than A [ parent [ x ]], interchange those two and reset x parent [ x ]. (e) (10) How long (in worst case) does the above algorithm take as a function of n = length[A] ? Give a reason for your answer. Solution: You climb up the tree to the top so it takes Θ(lg n ) steps. 2. (35) These questions concern QUICKSORT . We will take PARTITION(A,p,r) as given . (This procedure rearranges A [ p ··· r ], placing all entries less
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than A [ r ] to the left and all entries greater than A [ r ] to the right. You are not being asked to write it!!)
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