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COP3530Fall1999Diagnostic

# COP3530Fall1999Diagnostic - COP 3530 1 Fall 1999 Diagnostic...

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COP 3530 Fall 1999 Diagnostic Test Name:_____________________ 1. The algorithmic (programming) techniques of recursion and iteration can be related to the mathematical proof technique of induction in a manner that allows inductive proofs of correctness and run-time complexity . Show this relationship by proving that the first of the following two code segments correctly computes N 2 , for N 0, and that the second has run time complexity 2 N –1, N 1. In this latter case, we base complexity on the number of recursive calls made. function sq(N : integer ); begin if N<=0 then sq := 0 else sq := 2*N – 1 + sq(N-1) end ; { sq } HINT : Prove S(N): sq(N) = N 2 , N 0 procedure Move (n:integer; X, Y, Z:char); begin if n = 1 then writeln('Move ', X, ' to ', Y) else begin Move (n-1, X, Z, Y); writeln('Move ', X, ' to ', Y); Move (n-1, Z, Y, X) end end ; { Move } HINT : Prove S(N): T(N) = 2 N –1, N 1, where T(1) = 1; T(N) = 2 * T(N–1) + 1, N>1.

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2. A dictionary is an ADT that responds to the messages insert (newWord), lookup (oldWord) and delete (oldWord). There are many competing abstract implementations for a dictionary, three of which are a sorted linear list (not a linked list), a balanced binary search tree and a trie . Focusing on the lookup only, I have given informal algorithms and analyses of the costs of looking up a word. Discuss the pros and cons of each abstract implementation. Be sure to specify the complexity of the other operations (insert and delete) for each implementation.
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COP3530Fall1999Diagnostic - COP 3530 1 Fall 1999 Diagnostic...

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