From 50efd35e266d2288b9f29c9b64243fd5cd3848bf Mon Sep 17 00:00:00 2001 From: Sachin Malhotra Date: Fri, 8 Feb 2019 15:10:50 -0800 Subject: [PATCH] Added article, images, and solutions for Climbing Stairs --- Dynamic-Programming/Climbing-Stairs/README.md | 123 ++++++++++++++++++ .../dynamic_programming_solution.py | 9 ++ .../Climbing-Stairs/recursive_solution.py | 23 ++++ Images/Climbing-Stairs/main.png | Bin 0 -> 281585 bytes Images/Climbing-Stairs/overflow.gif | Bin 0 -> 1574160 bytes Images/Climbing-Stairs/recursion.png | Bin 0 -> 97162 bytes Images/Climbing-Stairs/recursion_tree.png | Bin 0 -> 295848 bytes Images/Climbing-Stairs/what.gif | Bin 0 -> 299288 bytes 8 files changed, 155 insertions(+) create mode 100644 Dynamic-Programming/Climbing-Stairs/README.md create mode 100644 Dynamic-Programming/Climbing-Stairs/dynamic_programming_solution.py create mode 100644 Dynamic-Programming/Climbing-Stairs/recursive_solution.py create mode 100644 Images/Climbing-Stairs/main.png create mode 100644 Images/Climbing-Stairs/overflow.gif create mode 100644 Images/Climbing-Stairs/recursion.png create mode 100644 Images/Climbing-Stairs/recursion_tree.png create mode 100644 Images/Climbing-Stairs/what.gif diff --git a/Dynamic-Programming/Climbing-Stairs/README.md b/Dynamic-Programming/Climbing-Stairs/README.md new file mode 100644 index 0000000..e500a8b --- /dev/null +++ b/Dynamic-Programming/Climbing-Stairs/README.md @@ -0,0 +1,123 @@ +

+ +

+ +--- +### Solution 1: Memoization + +#### Motivation + +Some problems have a certain defined structure to them which begs to be solved using recursion. Let's see what the problem asks us to do in a little more detail before moving onto the algorithm. + +So, at each point in time we are allowed two different choices as to the kind of move we can make. We can either climb one step or two steps at a time. Eventually, we have to reach a given number of steps. Let's look at all the ways of reaching `N = 4`. + +``` +1, 1, 1, 1 +1, 1, 2 +2, 2 +2, 1, 1 +1, 2, 1 +``` + +There are 5 different ways of climbing `4` steps given that we can either climb `1` step or `2` steps at a time. Looking at the solutions above, we can't really see a well defined pattern. + +> What do we do when we have a given task/goal to achieve and we have multiple choices but we don't know which one to take exactly? + +

+ +

+ +Yes, that's right. We try all all the possible options using recursion. So, that's the motivation for our first approach (and the second one as well!). + +#### Algorithm + +1. Let's assume we have a function called `findAll` which takes in a value of `N` and this is the recursive function that we will work with. +2. For every recursive function, we ought to define a base case where the recursion would terminate. What would happen if we don't provide a base case? + +

+ +

+ + > You'll get a stack-overflow exception because your recursion stack would soon run out of memory. + +3. The base case for our recursion would be when `N` becomes `0`. In these cases, we don't need to recurse any further and we simply return `1` as the result for this particular recursive call. The reason we don't recurse any further is because we don't have any more steps left to climb 😅😅. +4. Within the recursion, we simply make two recursive calls: `findAll(N - 1)` and `findAll(N - 2)`, add the results returned by these two calls and return them. The catch is that we only make the second recursive call i.e. `findAll(N - 2)` if we have at least 2 steps left to climb. Otherwise, we only stick to the `findAll(N - 1)` call. +5. As the name of the approach says, we need memoization here. If we don't use that, our simple recursive solution would give a time limit exceeded error on the OJ. +6. Adding memoization is pretty simple to a recursive solution especially to something as simple as our current problem. We simply cache the results calculated for a given `N` in a cache called `memo`. In a recursive call, if we find that we have already calculated the result for `N`, we simply return `memo[N]`. + +Have a look at the recursion tree below to realize the importance of memoization in this problem. We can clearly see in the figure below that there are overlapping sub-problems in our recursion tree and we can save on a lot of computation by using caching. + +

+ +

+ +#### Complexity Analysis + +* Time Complexity: `O(N)` since we calculate the result for a given value of `N` only once (due to caching) and for a given input, we have to go calculate all the results from `N` down to `0` i.e. our base case. +* Space Complexity: `O(N)` since our recursion stack occupies space equivalent to the height of the recursion tree. The height would be the path `N, N - 1, N - 2, N - 3 ... 0`. + +--- +### Solution 2: Dynamic Programming + +#### Motivation + +The motivation for this approach is the same as the previous one. The only downfall of the previous approach is that it uses a lot of space. If you look at the recurrence relation: + +``` +findAll(N) = findAll(N - 1) + findAll(N - 2) +``` + +you can see that we only need the solutions for the previous two values to compute the result for the current subproblem. We don't really need to *store* all the `N` values in a cache to find the answer for `N`. Rather, we can adopt a more iterative approach to solve this problem. + +#### Algorithm + +1. Initialize two variables `a` and `b` with values `1` and `1` respectively. They represent the results for `N = 0` and `N = 1` respectively. +2. There onwards, we have a loop from `2 ... N` and at every step we perform the following computations: + ``` + result = a + b + a = b + b = result + ``` +3. We continue doing this and in the end, we return the `result`. + +In the above computations, `result` represents `findAll(N)` and `a` and `b` represent `findAll(N - 2)` and `findAll(N - 1)` respectively. + +#### Complexity Analysis + +* Time Complexity: `O(N)` since we loop from `2..N`. +* Space Complexity: `O(1)` since we only make use of 3 variables for keeping track of results. + +--- +### Solution 3: Mathematical + +#### Motivation + +Let's look at the results for the first `N = 1..10`. + +``` +n = 1, 1 +n = 2, 2 +n = 3, 3 +n = 4, 5 +n = 5, 8 +n = 6, 13 +n = 7, 21 +n = 8, 34 +n = 9, 55 +n = 10, 89 +``` + +> Isn't this the fibonacci series? + +

+ +

+ +Well, yes! It is indeed the [fibonacci series](https://www.mathsisfun.com/numbers/fibonacci-sequence.html). The problem simply asks us to find the `N^th` fibonacci number as efficiently as possible. We've already seen a recursive and an iterative approach to solve this problem. These solutions already perform well for relatively smaller values of `N`. But, for larger values of `N`, they won't work. + +Read [this](https://kukuruku.co/post/the-nth-fibonacci-number-in-olog-n/) article to look at an approach based on ***Matrix Multiplication*** for finding the `N^th` fibonacci number in logarithmic time. That is blazingly fast! + +#### Complexity Analysis + +* Time Complexity: `O(logN)` +* Space Complexity: `O(logN)` occupied by the recursion stack. The matrix used in the approach is of a fixed size `2*2` and doesn't vary with `N` and hence it's not counted in the space complexity. diff --git a/Dynamic-Programming/Climbing-Stairs/dynamic_programming_solution.py b/Dynamic-Programming/Climbing-Stairs/dynamic_programming_solution.py new file mode 100644 index 0000000..717b553 --- /dev/null +++ b/Dynamic-Programming/Climbing-Stairs/dynamic_programming_solution.py @@ -0,0 +1,9 @@ +class Solution: + def climbStairs(self, n: 'int') -> 'int': + a, b, result = 1, 1, 1 + for i in range(2, n + 1): + result = a + b + a, b = b, result + return result + + diff --git a/Dynamic-Programming/Climbing-Stairs/recursive_solution.py b/Dynamic-Programming/Climbing-Stairs/recursive_solution.py new file mode 100644 index 0000000..5966165 --- /dev/null +++ b/Dynamic-Programming/Climbing-Stairs/recursive_solution.py @@ -0,0 +1,23 @@ +class Solution: + def climbStairs(self, n: 'int') -> 'int': + memo = {} + def findAll(n : 'int'): + # Base case + if n == 0: + return 1 + + # If we have the cached result, return + if n in memo: + return memo[n] + + # This will always be called + ans = findAll(n - 1) + + # Only call this if n > 2 + if n >= 2: + ans += findAll(n - 2) + + # cache the results for future use + memo[n] = ans + return ans + return findAll(n) diff --git a/Images/Climbing-Stairs/main.png b/Images/Climbing-Stairs/main.png new file mode 100644 index 0000000000000000000000000000000000000000..2c37fd6df44e4ca1ee14b0c347f513cdd8955107 GIT binary patch literal 281585 zcmeFZbzD@@x;{>K3P?8+BCYhmPy(WiiV6qu3*?DGm;C zKMoEd9tjclo2O4~+;MOyG;Gz>9&4+qaXof*wz74w#KF;eo2XB0px1d5ZuFG90tfG@ zs&h8|+EZ1V54WF}(0&A}l1gw#j(yM|92Mm;Q`6U=e?`t^-)?fVeTMU0p5Twwca(`T zLYeuqemj1s^OYH8>%LP^=aio{PJh9CWV>A6O`O_ne{Ev}8Vmmiw?l?WX)=l2gG6@c zz+cR*931$9B)u`Y^>w&~S;ih8-7Xa|rW$D-$*#CSb8cOfc5N-q4l$0{{JUGHI5Z8J zGp;WY1~CGv1;S8L)jLUx4{wQnSaiQt^lXf;t(0qq6vq(O`r!>;<^aV}6rL0R(m)Mi zX?dgoJ#MCU_*jPU+%6YYgtxkr6P!&=!7TZT9kik>I}{-+m)^r&P@uuRWR~?Ty*1@3 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