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@ -3,11 +3,11 @@ know what the fibonacci sequence is and how to calculate it. |
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For those who don't know: [Fibonacci](https://en.wikipedia.org/wiki/Fibonacci) |
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is a sequence of numbers starting with 0,1 whose next number is the sum |
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of the two previous numbers. After having multiple of my CS classes |
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gave lectures and multiple homework on the Fibonacci sequence; I decided |
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that it would be a great idea to write a blog post going over |
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give lectures and homeworks on the Fibonacci sequence; I decided |
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to write a blog post going over |
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the 4 main ways of calculating the nth term of the Fibonacci sequence. |
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In addition to providing python code for calculating the nth perm of the sequence, a proof for their validity |
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and analysis of their time complexities both mathematically and empirically will |
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In addition to providing the python code for calculating the nth perm of the sequence, a proof for their validity |
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and an analysis of their time complexities both mathematically and empirically will |
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be examined. |
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# Slow Recursive Definition |
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@ -25,28 +25,26 @@ def fib(n): |
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##Time Complexity |
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Observing that each call has two recursive calls we can place an upper bound on this |
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function as $O(2^n)$. However, if we solve this recurrence we can compute the exact value |
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and place a tight bound for time complexity. |
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function as $O(2^n)$. However, if we solve this recurrence we can place a tight bound for time complexity. |
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We can write a recurrence for the number of times fib is called: |
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$$ |
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F(0) = 0\\ |
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F(1) = 1\\ |
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F(n) = F(n-1) + F(n-2)\\ |
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$$ |
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Next we replace each instance of F(n) with $a^n$ since we want to solve for the roots since that |
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will allow us to put a tight asymptotic limit on the growth. |
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Next, we replace F(n) with $a^n$ since we want to find rate of exponential growth. |
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$$ |
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a^n = a^{n-1} + a^{n-2}\\ |
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\frac{a^n}{a^{n-2}} = \frac{a^{n-1} + a^{n-2}}{a^{n-2}}\\ |
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a^2 = a + 1\\ |
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a = \frac{1 + sqrt(5)}{2}\\ |
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a = \frac{1 \pm sqrt(5)}{2}\\ |
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$$ |
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From this calculation we can conclude that F(n) $\in \Theta 1.681^n$ |
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From this calculation we can conclude that F(n) $\in \Theta 1.681^n$. We don't have to worry about |
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the negative root since it would not be asymptotically relevant by the definition of $\Theta$. |
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@ -57,14 +55,13 @@ Here is a graph of the actual performance that I observed from this recursive de |
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# Accumulation Solution |
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The problem with the previous recursive solution is that you had to recalculate certain |
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terms of fibonacci a ton of times. A summation variable would help us avoid this problem. |
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You could write this using a simple loop, however, it is still possible to do this with |
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recursion. |
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You could write this solution using a simple loop or dynamic programming |
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, however, I chose to use recursion to demonstrate that it's recursion which made the first |
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problem slow. |
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```Python |
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@ -80,9 +77,11 @@ def fibIterative(n): |
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return fibHelper(n, 0, 1) |
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``` |
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In this code example fibHelper is a method which accumulates the previous two terms. |
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In this code example, fibHelper is a method which accumulates the previous two terms. |
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The fibIterative is a wrapper method which sets the two initial terms equal to 0 and 1 |
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representing the fibonacci sequence. |
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representing the fibonacci sequence. At first it may not be obvious that fibIterative(n) |
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is equivalent to fib(n). To demonstrate that these two are in fact equivalent, I broke this |
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into two inductive proofs. |
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## Proof for Fib Helper |
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**Lemma:** For any n $\epsilon$ N if n $>$ 1 then |
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@ -90,7 +89,7 @@ representing the fibonacci sequence. |
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**Proof via Induction** |
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Base Case: n = 2: |
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**Base Case**: n = 2: |
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$$ |
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LHS = fibHelper(2, a, b)\\ |
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= fibHelper(1, b, a + b) = a + b\\ |
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@ -98,7 +97,7 @@ $$ |
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= a + b\\ |
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$$ |
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Inductive Step: |
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**Inductive Step:** |
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Assume proposition is true for all n and show n+1 follows. |
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@ -118,19 +117,19 @@ $\Box$ |
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**Proof via Strong Induction** |
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Base Case: n = 0: |
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**Base Case**: n = 0: |
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$$ |
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fibIterative(0, 0, 0) = 0\\ |
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= fib(0) |
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$$ |
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Base Case: n = 1: |
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**Base Case**: n = 1: |
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$$ |
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fibIterative(1, 0, 0) = 1\\ |
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= fib(1) |
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$$ |
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Inductive Step: |
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**Inductive Step:** |
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Assume proposition is true for all n and show n+1 follows. |
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@ -145,9 +144,9 @@ $\Box$ |
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## Time Complexity |
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Suppose that we wish to solve for time complexity in terms of the number of additions needed to be |
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computed. By observing the algorithm for fibHelper we can see that we perform one addition every time |
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which we have a recursive call. We can now form a recurrence for time complexity and solve for it. |
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Suppose that we wish to solve for the time complexity in terms of the number of additions needed to be |
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computed. Based on fibHelper we can see that it performs one addition every recursive call. |
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We can now form a recurrence for time complexity. |
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$$ |
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T(0) = 0\\ |
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@ -169,8 +168,8 @@ Fibonacci. |
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# Matrix Solution |
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We can actually get better than linear time for performance while calculating |
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the Fibonacci sequence recursively using this fact: |
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We can actually get better than linear for performance for Fibonacci while still using |
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recursion. However, to do so we need to know this fact: |
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$$ |
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\begin{bmatrix} |
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@ -183,10 +182,11 @@ F_n & F{n-1} |
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\end{bmatrix}^n |
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$$ |
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Without any other tricks, raising a matrix to a power n times would not get |
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Without any tricks, raising a matrix to a power n times would not get |
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us better than linear performance. However, if we use the [Exponentiation by Squaring](https://en.wikipedia.org/wiki/Exponentiation_by_squaring) |
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method, we can expect to see logarithmic time. Since two spots in the matrix are always equal, |
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I represented the matrix as an array with only three elements. |
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I represented the matrix as an array with only three elements to reduce the space and |
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computations required. |
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```Python |
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@ -216,9 +216,9 @@ def fibPower(n): |
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## Time Complexity |
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For this algorythem lets solve for the time complexity as the number of additions and multiplications. |
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For this algorithm, lets solve for the time complexity as the number of additions and multiplications required. |
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Since we are always multiplying two 2x2 matrices, that is constant time. |
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Since we are always multiplying two 2x2 matrices, that operation is constant time. |
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$$ |
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T_{multiply} = 9 |
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@ -246,11 +246,14 @@ $$ |
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T_{fibPower}(n) = T_{power}(n)\\ |
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$$ |
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Now we can state that $fibPower(n) \in \Theta(log(n))$. |
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## Inductive Proof for Matrix Method |
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I would like to now prove that this matrix identity is valid since it is not at first obvious. |
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**Lemma:** For any n $\epsilon$ N if n $>$ 0 then |
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$$ |
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\begin{bmatrix} |
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@ -314,20 +317,20 @@ $\Box$ |
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As expected by our mathmatical calcuations, the algorthem appears to be running in |
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As expected by our mathematical calculations, the algorithm appears to be running in |
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logarithmic time. |
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## Measured Performance With Large Numbers |
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When calculating the fibonacci term for extremely large numbers dispite having a polynomial |
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time complexity, the space required to compute Fibonacci grows exponentially. Since our |
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When calculating the fibonacci term for extremely large numbers despite having a polynomial |
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time complexity, the space required to compute each Fibonacci term grows exponentially. Since our |
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performance is only pseudo-polynomial we see a degrade in our performance when calculating |
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large terms of the fibonacci sequence. |
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The one amazing thing to point out here is that despite calculating the 10,000 term of Fibonacci, |
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this algorithm is nearly 400 times faster than the recursive algorithm when it was calculating |
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this algorithm is nearly 400 times faster than the recursive algorithm when calculating |
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the 30th term of Fibonacci. |
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@ -346,10 +349,10 @@ def fibClosedFormula(n): |
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return (p-v)/math.sqrt(5) |
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``` |
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## Derivation of Formula |
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## Derivation of Binet's Formula |
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Similar to when we were calculating for the time complexity, we want to start by finding the |
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two roots of the equation. |
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Similar to when we were calculating the time complexity of the basic recursive definition |
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, we want to start by finding the two roots of the equation in terms of exponents. |
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$$ |
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a^n = a^{n-1} + a^{n-2}\\ |
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@ -385,7 +388,6 @@ $$ |
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## Time Complexity |
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Since we managed to find the closed form of the fibonacci sequence we can expect to see constant performance. |
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## Measured Performance |
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Jeffery Russell
6 years ago
GitHub