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assign1 - University of Waterloo CS240 Fall 2010 Assignment...

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University of Waterloo CS240, Fall 2010 Assignment 1 Due Date: Wednesday, October 6, at 5:00pm Please read http://www.student.cs.uwaterloo.ca/ ~ cs240/f10/guidelines.pdf for guidelines on submission. In particular, don’t forget to make a cover page ( https://www. student.cs.uwaterloo.ca/ ~ isg/makeCover?course=cs240 ) and attach it to the front of any written assignment. There are 85 marks available. The assignment will be marked out of 80. Problem 1 [5+5+5=15 marks] Provide a complete proof of the following statements from first principles (i.e., using the original definitions of order notation). a) 12 n 3 + 123 n 2 + 1234 n + 12345 O ( n 3 ) b) n 3 + 321 n 2 + 54321 Ω( n 2 ) c) 4 n + 12 o ( n 2 ) Problem 2 [6 marks] Let f ( n ) and g ( n ) be two positive functions. Prove or disprove the following statement: One of the following relations always holds: f ( n ) o ( g ( n )) f ( n ) Θ( g ( n )) f ( n ) ω ( g ( n )) Problem 3 [6 marks] Prove from definition that min( f, g ) Θ( fg/ ( f + g )). Assume that f and g are positive functions.
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