A general definition of the big-oh notation for algorithm analysis

Algorithm analysis O-notation Prevailing definition Implied properties Correctness analysis Complexity analysis Algorithm analysis 2 / 101 Algorithm analysis O-notation Prevailing definition Implied properties Correctness analysis Complexity analysis Algorithm analysis Algorithm An algorithm is a finite sequence of instructions for transforming data to another form, a process which can be followed with pen and paper. 3 / 101 Algorithm analysis O-notation Prevailing definition Implied properties Correctness analysis Complexity analysis Algorithm analysis Algorithm An algorithm is a finite sequence of instructions for transforming data to another form, a process which can be followed with pen and paper. Example An algorithm could provide a way to sort a sequence of integers in increasing order. Algorithm analysis O-notation Prevailing definition Implied properties Correctness analysis Complexity analysis Algorithm analysis Algorithm An algorithm is a finite sequence of instructions for transforming data to another form, a process which can be followed with pen and paper. Example An algorithm could provide a way to sort a sequence of integers in increasing order. Algorithm analysis Algorithm analysis studies the correctness and complexity of a given algorithm. Algorithm analysis O-notation Prevailing definition Implied properties Correctness analysis Complexity analysis Correctness analysis 6 / 101 Algorithm analysis O-notation Prevailing definition Implied properties Correctness analysis Complexity analysis Correctness analysis Correctness analysis Does the algorithm do what it is claimed to do? 7 / 101 Algorithm analysis O-notation Prevailing definition Implied properties Correctness analysis Complexity analysis Correctness analysis Correctness analysis Does the algorithm do what it is claimed to do? Example of correctness Prove that a given algorithm sorts any sequence of integers in increasing order, and does so in finite time for a finite sequence. 8 / 101 Algorithm analysis O-notation Prevailing definition Implied properties Correctness analysis Complexity analysis Complexity analysis 9 / 101 Algorithm analysis O-notation Prevailing definition Implied properties Correctness analysis Complexity analysis Complexity analysis Complexity analysis How much time / memory does it take to follow the algorithm on a given input in the worst case / best case / average case (etc.)? 10 / 101 Algorithm analysis O-notation Prevailing definition Implied properties Correctness analysis Complexity analysis Complexity analysis Complexity analysis How much time / memory does it take to follow the algorithm on a given input in the worst case / best case / average case (etc.)? Example of complexity Prove that a given algorithm never uses more than n(n − 1)/2 number of order-comparisons to sort any …