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CSC375 Homework3 George Corser 2009-01-26

# CSC375 Homework3 George Corser 2009-01-26 - CSC 375...

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CSC 375 Homework3 George Corser 2009 January 26 3.1 All five expressions in Figure 3.1 return zero at n=0. We don’t allow negative efficiency, i.e. negative values of n. So 2n 2 is most efficient between 0 and 1; 5n log n is most efficient between 1 and about 1.3; 2 n is most efficient between about 1.3 and 5.9; and 10n is most efficient above that. These expressions intersect twice, once where 5n log n = 2 n , and again where 2 n =10n. We can solve for more precise values at the intersections of the expressions as explained below. For 5n log n = 2 n we can write log 2 n 5n = 2 n which is the same as n 5n = 2 2 n , or n 5n = 4 n . Reducing the equation by taking the logarithm of both sides, we write log n 5n = 5n log n = n log 4, which means that 5 log n = log 4, log n 5 = log 4, or n 5 = 4. That means n= 5 4 ≈ 1.3195, or more accurately that the most efficient algorithm is 5n log n in the range 0 < n ≤ 5 4 . From 2 n = 10n, using the Lambert W function (explained at en.wikipedia.org/wiki/Lambert_W ) we can reduce as follows: 2 n = 10n; 1 = 10n/ 2 n ; 1 = 10ne -n( ln 2)

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CSC375 Homework3 George Corser 2009-01-26 - CSC 375...

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