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zl8?WCE}Z|DaDF(!=3NYdiS=UeJz6YW@d`N{_RC6l65$WkT0Y!lOdcul9{;#}2s2w- z$n}63f#KtB$Hn_wwHyd0CR3&%`L;rGEQ)^X=`cNT#U7Qy-I;hSQ|BMmWxCenqb01> zW@apq7TNTmR8>D$Rm1iXGz7NPrkgC))y*SX3a~ZVH!l+u4GI+p0wQ61NYDaL5_cP4 zKiSaO+}s%-g>jkO1&YmEC=!S@>18_~#4pnIAbf$dq(&o~9)(52RV3 literal 0 HcmV?d00001 diff --git a/blogContent/posts/programming/everything-fibonacci.md b/blogContent/posts/programming/everything-fibonacci.md index 3ecb4a9..a34f14d 100644 --- a/blogContent/posts/programming/everything-fibonacci.md +++ b/blogContent/posts/programming/everything-fibonacci.md @@ -3,14 +3,16 @@ know what the fibonacci sequence is and how to calculate it. For those who don't know: [Fibonacci](https://en.wikipedia.org/wiki/Fibonacci) is a sequence of numbers starting with 0,1 whose next number is the sum of the two previous numbers. After having multiple of my CS classes -give lectures and multiple homeworks on the Fibonacci sequence; I decided -that it would be a good idea to write a blog post going over -the 4 main ways of calculating the nth term of the Fibonacci sequence -and proving their time complexities both mathematically and empirically. +gave lectures and multiple homework on the Fibonacci sequence; I decided +that it would be a great idea to write a blog post going over +the 4 main ways of calculating the nth term of the Fibonacci sequence. +In addition to providing python code for calculating the nth perm of the sequence, a proof for their validity +and analysis of their time complexities both mathematically and empirically will +be examined. # Slow Recursive Definition -By the definition of the Fibonacci sequence, it is natural to write it as +By the definition of the Fibonacci sequence, it is the most natural to write it as a recursive definition. ```Python @@ -20,29 +22,38 @@ def fib(n): return fib(n-1) + fib(n-2) ``` -#### Time Complexity +##Time Complexity Observing that each call has two recursive calls we can place an upper bound on this -function as O(2^n). However, if we solve this recurrence we can compute the exact value +function as $O(2^n)$. However, if we solve this recurrence we can compute the exact value and place a tight bound for time complexity. We can write a recurrence for the number of times fib is called: -```angular2html -T(1) = 1 -T(n) = T(n-1) + T(n-2) -a^n = a^{n-1} + a^{n-2} -a^2 = a + 1 -a = \frac{1 + sqrt(5)}{2} -T(n) = \frac{1 + sqrt(5)}{2}^n + \frac{1 1 sqrt(5}{2}^n +$$ + F(0) = 0\\ + F(1) = 1\\ + F(n) = F(n-1) + F(n-2)\\ +$$ -O(1.618^n) -``` +Next we replace each instance of F(n) with $a^n$ since we want to solve for the roots since that +will allow us to put a tight asymptotic limit on the growth. + +$$ + a^n = a^{n-1} + a^{n-2}\\ + \frac{a^n}{a^{n-2}} = \frac{a^{n-1} + a^{n-2}}{a^{n-2}}\\ + a^2 = a + 1\\ + a = \frac{1 + sqrt(5)}{2}\\ +$$ + +From this calculation we can conclude that F(n) $\in \Theta 1.681^n$ -#### Measured Performance -Here is a graph of the actual performance that I observed for this algorithm. +## Measured Performance + +Here is a graph of the actual performance that I observed from this recursive definition of Fibonacci. + ![Recursive Definition](media/fibonacci/RecursiveDefinition.png) @@ -73,20 +84,84 @@ In this code example fibHelper is a method which accumulates the previous two te The fibIterative is a wrapper method which sets the two initial terms equal to 0 and 1 representing the fibonacci sequence. -Proof in latex that fibHelper +## Proof for Fib Helper +**Lemma:** For any n $\epsilon$ N if n $>$ 1 then + $fibHelper(n, a, b) = fibHelper(n - 1, a, b) + fibHelper(n - 2, a, b)$. + +**Proof via Induction** + +Base Case: n = 2: +$$ + LHS = fibHelper(2, a, b)\\ + = fibHelper(1, b, a + b) = a + b\\ + RHS = fibHelper(2 -1, a, b) + fibHelper(2-2, a, b)\\ + = a + b\\ +$$ + +Inductive Step: + +Assume proposition is true for all n and show n+1 follows. + +$$ + RHS=fibHelper(n+1;a,b)\\ + = fibHelper(n;b,a+b)\\ + =fibHelper(n-1;b,a+b) + fibHelper(n-2;b,a+b)\\ + =fibHelper(n;a,b) + fibHelper(n-1;a,b)\\ + =LHS\\ +$$ + +$\Box$ + +## Proof That fibIterative = Fib -proof in latex that fib iterative = fib +**Lemma:** For any n $\in$ N, $fib(n)$ = $fibIterative(n, 0, 1)$ + +**Proof via Strong Induction** +Base Case: n = 0: +$$ + fibIterative(0, 0, 0) = 0\\ + = fib(0) +$$ -#### Time Complexity +Base Case: n = 1: +$$ + fibIterative(1, 0, 0) = 1\\ + = fib(1) +$$ -proof in latex for time complexity +Inductive Step: -#### Measured Performance +Assume proposition is true for all n and show n+1 follows. + +$$ + fib(n+1) = fib(n) + fib(n-1)\\ + = fibHelper(n, 0, 1) + fibHelper(n+1, 0 ,1) \quad\text{I.H}\\ + = fibHelper(n+1, 0, 1) \quad\text{from result in previous proof}\\ +$$ + +$\Box$ + + +## Time Complexity + +Suppose that we wish to solve for time complexity in terms of the number of additions needed to be +computed. By observing the algorithm for fibHelper we can see that we perform one addition every time +which we have a recursive call. We can now form a recurrence for time complexity and solve for it. + +$$ + T(0) = 0\\ + T(1) = 0\\ + T(n) = 1 + T(n-1)\\ + T(n) = n-1\\ +$$ + +From this recurrence we can say that fibHelper $\in \Theta(n)$. + +## Measured Performance Notice how much faster this solution is compared to the original recursive solution for -Fibonacci. Also, I only measured going out to 500 because beyond that I hit the maximum -number of recursive calls for my installation of Python. +Fibonacci. ![Iterative Performance](media/fibonacci/Iterative.png) @@ -95,13 +170,23 @@ number of recursive calls for my installation of Python. # Matrix Solution We can actually get better than linear time for performance while calculating -the Fibonacci sequence recursively. - -![Inductive Proof of Fibonacci Matrix](media/fibonacci/FibonacciMatrix.png) +the Fibonacci sequence recursively using this fact: + +$$ +\begin{bmatrix} +1 & 1\\ +1 & 0 +\end{bmatrix}^n = +\begin{bmatrix} +F_{n+1} & F_n\\ +F_n & F{n-1} +\end{bmatrix}^n +$$ Without any other tricks, raising a matrix to a power n times would not get us better than linear performance. However, if we use the [Exponentiation by Squaring](https://en.wikipedia.org/wiki/Exponentiation_by_squaring) -method, we can expect to see logarithmic time. +method, we can expect to see logarithmic time. Since two spots in the matrix are always equal, +I represented the matrix as an array with only three elements. ```Python @@ -128,17 +213,112 @@ def fibPower(n): return power(l, n)[1] ``` -#### Time Complexity - -latex proof for 9lg(n) performance +## Time Complexity + +For this algorythem lets solve for the time complexity as the number of additions and multiplications. + +Since we are always multiplying two 2x2 matrices, that is constant time. + +$$ + T_{multiply} = 9 +$$ + +Solving for the time complexity of fib power is slightly more complicated. +$$ + T_{power}(1) = 0\\ + T_{power}(n) = T(\left\lfloor\dfrac{n}{2}\right\rfloor) + T_{multiply}\\ + = T(\left\lfloor\dfrac{n}{2}\right\rfloor) + 9\\ + = T(\left\lfloor\dfrac{n}{2*2}\right\rfloor) + 9 + 9\\ + = T(\left\lfloor\dfrac{n}{2*2*2}\right\rfloor) + 9+ 9 + 9\\ + T_{power}(n) = T(\left\lfloor\dfrac{n}{2^k}\right\rfloor) + 9k\\ +$$ + +let $k=k_0$ such that $\left\lfloor\dfrac{n}{2^{k_0}}\right\rfloor = 1$ + +$$ + \left\lfloor\dfrac{n}{2^{k_0}}\right\rfloor = 1 \rightarrow 1 \leq \frac{n}{2^{k_0}} < 2\\ + \rightarrow 2^{k_0} \leq n < 2^{k_0 +1}\\ + \rightarrow k_0 \leq lg(n) < k_0+1\\ + \rightarrow k_0 = \left\lfloor lg(n)\right\rfloor\\ + T_{power}(n) = T(1) + 9*\left\lfloor lg(n)\right\rfloor\\ + T_{power}(n) = 9*\left\lfloor\ lg(n)\right\rfloor\\ + T_{fibPower}(n) = T_{power}(n)\\ +$$ + + + + +## Inductive Proof for Matrix Method + +**Lemma:** For any n $\epsilon$ N if n $>$ 0 then + $$ + \begin{bmatrix} + 1 & 1\\ + 1 & 0 + \end{bmatrix}^n = + \begin{bmatrix} + F_{n+1} & F_n\\ + F_n & F{n-1} + \end{bmatrix}^n + $$ + +Let + +$$ +A= +\begin{bmatrix} +1 & 1\\ +1 & 0 +\end{bmatrix}^n +$$ + +**Base Case:** n = 1 +$$ +A^1= +\begin{bmatrix} +1 & 1\\ +1 & 0 +\end{bmatrix}^n = +\begin{bmatrix} +F_{2} & F_2\\ +F_2 & F_{0} +\end{bmatrix}^n +$$ + +**Inductive Step:** Assume proposition is true for n, show n+1 follows +$$ +A^{n+1}= +\begin{bmatrix} +1 & 1\\ +1 & 0 +\end{bmatrix} +\begin{bmatrix} +F_{n+1} & F_n\\ +F_n & F{n-1} +\end{bmatrix}^n\\ += \begin{bmatrix} +F_{n+1} + F_n & F_n + F_{n-1}\\ +F_{n+1} & F_{n} +\end{bmatrix}\\ += \begin{bmatrix} +F_{n+2} & F_{n+1}\\ +F_{n+1} & F_{n} +\end{bmatrix}\\ +$$ + +$\Box$ + + +## Measured Performance -#### Measured Performance ![FibPower Performance](media/fibonacci/FibPower.png) + As expected by our mathmatical calcuations, the algorthem appears to be running in logarithmic time. -#### Measured Performance With Large Numbers +## Measured Performance With Large Numbers + ![FibPower Performance](media/fibonacci/FibPowerBigPicture.png) When calculating the fibonacci term for extremely large numbers dispite having a polynomial @@ -153,6 +333,61 @@ the 30th term of Fibonacci. # Closed Form Definition -It is actually possible to calculate Fibonacci in constant time using a closed form definition. +It is actually possible to calculate Fibonacci in constant time using Binet's Formula. + +$$ + F_n = \frac{(\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n}{\sqrt{5}} +$$ + +```Python +def fibClosedFormula(n): + p = ((1+ math.sqrt(5))/2)**n + v = ((1-math.sqrt(5))/2)**n + return (p-v)/math.sqrt(5) +``` + +## Derivation of Formula + +Similar to when we were calculating for the time complexity, we want to start by finding the +two roots of the equation. + +$$ + a^n = a^{n-1} + a^{n-2}\\ + \frac{a^n}{a^{n-2}} = \frac{a^{n-1} + a^{n-2}}{a^{n-2}}\\ + a^2 = a + 1\\ + 0 = a^2 - a - 1\\ + a = \frac{1 \pm sqrt(5)}{2}\\ +$$ + +Since there are two roots to the equation, the solution of $F_n$ is going to be +a linear combination of the two roots. + + +$$ + F_n = c_1(\frac{1 + \sqrt{5}}{2})^n + c_2(\frac{1 - \sqrt{5}}{2})^n +$$ + +Fact: $F_1$ = 1 + +$$ + F_1 = 1\\ + = c_1(\frac{1 + \sqrt{5}}{2}) + c_2(\frac{1 - \sqrt{5}}{2})\\ + = \frac{c_1}{2} + \frac{c_2}{2} + \frac{c_1\sqrt{5}}{2} - \frac{c_2\sqrt{5}}{2}\\ +$$ + +Let $c_1 = \frac{1}{\sqrt{5}}$, +Let $c_2 = \frac{-1}{\sqrt{5}}$ + +$$ + F_n = \frac{1}{\sqrt(5)}((\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n)\\ + = \frac{(\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n}{\sqrt{5}} +$$ + +## Time Complexity + + +Since we managed to find the closed form of the fibonacci sequence we can expect to see constant performance. + +## Measured Performance -latex proof of closed form definition \ No newline at end of file +![FibPower Performance](media/fibonacci/ConstantTimeComplexity.png) \ No newline at end of file diff --git a/blogContent/posts/programming/media/fibonacci/FibonacciMatrix.png b/blogContent/posts/programming/media/fibonacci/FibonacciMatrix.png deleted file mode 100644 index f0d6014f58a69f76509d6610345f3fae79f87027..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 62460 zcmeFZ2T)X7yDiFgZ)J;u+W<&b0m&dJAQBCLP0l$eNQOqq84Q4esrXdQ@2~n_g?pHRl(`H@-2~CnW`GQW9zsA|fJE z*z;#9L`3KLh=~3a`1=C*r0uGQ3;6Yy{S%nl-{8mN?^hqe|EU}#wH#D!O&y$H+L;iU 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