TADM2E 2.1

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2-1 ans

This loop can be expressed as the sum:

<math> \sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\sum_{k=1}^{j}1 </math>

Reducing this, sum by sum from the rhs:

<math> \begin{align} &\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\sum_{k=1}^{j}1 =\\ &\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}j =\\ &\sum_{i=1}^{n-1}\left(\sum_{j=1}^{n}j - \sum_{j=1}^{i}j\right) =\\ &\sum_{i=1}^{n-1}\left(\frac{n(n+1)}{2} - \frac{i(i+1)}{2}\right) =\\ &\frac{1}{2}\sum_{i=1}^{n-1}n^2+n-i^2-i =\\ &\frac{1}{2}\left((n-1)n^2 + (n-1)n - \left(\frac{n(n+1)(2n+1)}{6} - n^2\right) -

   \left(\frac{n(n+1)}{2} - n\right)\right) =\\

&f(n) = \frac{n(n(n+1))}{2} - \frac{n(n+1)(2n+1)}{12} - \frac{n(n+1)}{4} \end{align} </math>

n=1 gives zero; and order is O((n^3)/3)


Alternative Derivation

<math> \begin{align} &mystery(n)=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\sum_{k=1}^{j} 1\\ &=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}j\\ \end{align} </math>

Taking the following well-known formula for the sum of the integers <math>1</math> to <math>n</math>:-

<math> \sum_{x=1}^{n}x = \frac{1}{2}n(n + 1) </math>

And noting that:-

<math> \sum_{j=i+1}^{n}j = \sum_{j=1}^{n}j - \sum_{j=1}^{i}j </math>

We continue our derivation thus:-

<math> \begin{align} &=\sum_{i=1}^{n-1}(\frac{1}{2}n(n+1) - \frac{1}{2}i(i+1))\\ &=\frac{1}{2}n(n-1)(n+1) - \frac{1}{2}\sum_{i=1}^{n-1}(i^2 + i)\\ \end{align} </math>

Given our formula for the sum of integers, substituting <math>n-1</math> for <math>n</math> gives us:-

<math> \sum_{x=1}^{n-1}x=\frac{1}{2}(n-1)((n-1)+1)=\frac{1}{2}n(n-1) </math>

Thus continuing with our derivation:-

<math> \begin{align} &=\frac{1}{2}n(n-1)(n+1)- \frac{1}{2}\frac{1}{2}n(n-1) - \frac{1}{2}\sum_{i=1}^{n-1}i^2\\ &=\frac{1}{2}n(n-1)(n + 1 - \frac{1}{2}) - \frac{1}{2}\sum_{i=1}^{n-1}i^2\\ &=\frac{1}{2}n(n-1)(n + \frac{1}{2}) - \frac{1}{2}\sum_{i=1}^{n-1}i^2\\ \end{align} </math>

Let us consider the well-known formula for the sum of the squares of the integers between <math>1</math> and <math>n</math>:-

<math> \sum_{x=1}^{n}x^2=\frac{1}{6}n(n+1)(2n+1) </math>

As before, let us substitute <math>n-1</math> for <math>n</math>:-

<math> \sum_{x=1}^{n-1}x^2=\frac{1}{6}(n-1)((n-1)+1)(2(n-1)+1)=\frac{1}{6}n(n-1)(2n-1) </math>

Substituting this into our derivation gives us:-

<math> \begin{align} &=\frac{1}{2}n(n-1)(n + \frac{1}{2}) - \frac{1}{2}\frac{1}{6}n(n-1)(2n - 1)\\ &=\frac{1}{2}n(n-1)(n + \frac{1}{2} - \frac{1}{6}(2n-1))\\ &=\frac{1}{2}n(n-1)(n + \frac{1}{2} - \frac{1}{3}n + \frac{1}{6})\\ &=\frac{1}{2}n(n-1)(\frac{2}{3}n + \frac{2}{3})\\ &=\frac{1}{3}n(n-1)(n+1) \end{align} </math>

Thus the solution is <math>mystery(n)=\frac{1}{3}n(n-1)(n+1)</math>. This is the equivalent to the solution given by the previous derivation.

The Big-Oh complexity of this function (ignoring constants) is O(n^3), as the RAM model dictates that each iteration of the increment of <math>r</math> is a time-step. We can ignore the first and last line's contribution to running time for the purposes of Big-Oh, as they do not contribute to the growth of the time taken with input size.


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