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aa:notations [2016/11/02 21:44] pdmatei created |
aa:notations [2019/11/04 13:52] (current) pdmatei |
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| For now, let us focus solely on the ''merge'' procedure presented in the previous lecture. There are many technical issues that prevent us from immediately computing an execution time: | For now, let us focus solely on the ''merge'' procedure presented in the previous lecture. There are many technical issues that prevent us from immediately computing an execution time: | ||
| * some instructions have complexity which **depends on the machine and implementation**: | * some instructions have complexity which **depends on the machine and implementation**: | ||
| - | * ''malloc'' may depend on the compiler implementation | + | * ''malloc'' may depend on the standard C library implementation |
| - | * we do not know the how many execution steps control instructions (''if'', ''while'') take | + | * we do not know how many execution steps control instructions (''if'', ''while'') take |
| * it is **difficult to combine** instruction times into a single time function | * it is **difficult to combine** instruction times into a single time function | ||
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| $end | $end | ||
| - | Thus, $math[g \in o(f(n))] should be read: //$math[g] grows assymptotically strictly less than $math[f]//. We have $math[o(f(n)) \cap \omega(f(n)) = \emptyset], $math[O(f(n)) \cap \Omega(f(n)) = \Theta(f(n))] and $math[\omega(f(n)) \cup \Theta(f(n)) = \Omega(f(n))]. | + | Thus, $math[g \in o(f(n))] should be read: //$math[g] grows asymptotically strictly less than $math[f]//. We have $math[o(f(n)) \cap \omega(f(n)) = \emptyset], $math[O(f(n)) \cap \Omega(f(n)) = \Theta(f(n))]. |
| ==== - Exercise ==== | ==== - Exercise ==== | ||
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| $math[left = right] | $math[left = right] | ||
| - | * each occurrence of an asymptotic notation in the //left// should be replaced by an **universally quantified** function belonging to the corresponding class. | + | * each occurrence of an asymptotic notation in the //left// should be replaced by an **unuversally quantified** function belonging to the corresponding class. |
| * each occurrence of an asymptotic notation in //right// should be replaced by an **existentially quantified** function from the corresponding class. | * each occurrence of an asymptotic notation in //right// should be replaced by an **existentially quantified** function from the corresponding class. | ||