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How do you solve a staircase problem?
One can find a lot of similar DP problems asked during the coding interview.Solution Idea
How many ways are there to climb a staircase with n stairs if you can walk up the stairs 1 or 2 at a time?
There are 274 ways to climb the stairs.
How many ways can we climb a staircase with 8 steps if we can take either 1 or 2 steps at a time?
Since, either he can climb one step or two steps only, we have to partition 8 steps as a combination of 1 and 2. This can be done in 8 C 0 = 1 ways. This can be done in 7 C 1 = 7 ways. This can be done in 6 C 2 = 15 ways.13-Sept-2021
How many ways can you climb 12 steps?
The number of ways to get to the 12th step is 233. The reason for this pattern is very simple. For the first step Liam has two choices: i) Jump down 1 step.
How do you code stairs?
Summary of Code Requirements for Residential Stairs Minimum 36 inch clear width for stairway. Maximum 4-1/2 inch handrail projection into stairway width, on either side. Minimum 6′-8″ headroom height clearance for stairway. Maximum 12′-7″ vertical height (rise) for a flight of stairs.
How do you print on stairs?
In the first row, we need to print height-1 spaces followed by one # symbol. So, the number of spaces we need to print in each row is equal to the (row index number)+(height-1). And the number of # symbols which follow these spaces is equal to (row index number)+1. This is how we print the staircase pattern in C++.
How does the staircase illusion work?
The staircase could then be ascended or descended forever without ever reaching the end. This illusion is caused by perspective distortion. The brain focuses on separate sections of the image and perceives that the stairs are joined together, despite the fact that it would be impossible in a 3D object.02-Jun-2017
How do you solve a Hackerrank staircase?
How many ways can you climb a staircase with 10 steps if you can take either 1 or 2 steps at a time hint group the different ways according to the number of double steps?
So there are 89 ways of climbing a flight of ten steps using steps of one and two!17-Dec-2014
A child is running up a staircase with n steps and can hop either 1 step, 2 steps, or 3 steps at a time. Implement a method to count how many possible ways the child can run up the stairs.
Examples:Input : 4 Output : 7 Explanation: Below are the seven ways 1 step + 1 step + 1 step + 1 step 1 step + 2 step + 1 step 2 step + 1 step + 1 step 1 step + 1 step + 2 step 2 step + 2 step 3 step + 1 step 1 step + 3 step Input : 3 Output : 4 Explanation: Below are the four ways 1 step + 1 step + 1 step 1 step + 2 step 2 step + 1 step 3 step
There are two methods to solve this problem:
Method 1: Recursive. There are n stairs, and a person is allowed to jump next stair, skip one stair or skip two stairs. So there are n stairs. So if a person is standing at i-th stair, the person can move to i+1, i+2, i+3-th stair. A recursive function can be formed where at current index i the function is recursively called for i+1, i+2 and i+3 th stair.
There is another way of forming the recursive function. To reach a stair i, a person has to jump either from i-1, i-2 or i-3 th stair or i is the starting stair.
Note: The Time Complexity of the program can be optimized using Dynamic Programming.
Method 2: Dynamic Programming.
Matrix Exponentiation is mathematical ways to solve DP problem in better time complexity. Matrix Exponentiation Technique has Transformation matrix of Size K X K and Functional Vector (K X 1) .By taking n-1th power of Transformation matrix and Multiplying It With functional vector Give Resultant Vector say it Res of Size K X 1. First Element of Res will be Answer for given n value. This Approach Will Take O(K^3logn) Time Complexity Which Is Complexity of Finding (n-1) power of Transformation Matrix.
K = No of Terms in which F(n) depend ,from Recurrence Relation We can Say That F(n) depend On F(n-1) and F(n-2). => K =3
F1 = Vector (1D array) that contain F(n) value of First K terms. Since K=3 =>F1 will have F(n) value of first 2 terms. F1=[1,2,4]
T = Transformation Matrix that is a 2D matrix of Size K X K and Consist Of All 1 After Diagonal And Rest All Zero except last row. Last Row Will have coefficient Of all K terms in which F(n) depends In Reverse Order. => T =[ [0 1 0] ,[0 0 1], [1 1 1] ].
Algorithms:1)Take Input N 2)If N < K then Return Precalculated Answer //Base Condition 3)construct F1 Vector and T (Transformation Matrix) 4)Take N-1th power of T by using Optimal Power(T,N) Methods and assign it in T 5)return (TXF)
for Optimal Power(T, N) Methods Refer Following Article: https://www.geeksforgeeks.org/write-a-c-program-to-calculate-powxn/amp/
Method 4: Using four variables
The idea is based on the Fibonacci series but here with 3 sums. we will hold the values of the first three stairs in 3 variables and will use the fourth variable to find the number of ways.
Time Complexity: O(n)
Method 5: DP using memoization(Top down approach)
We can avoid the repeated work done in method 1(recursion) by storing the number of ways calculated so far.
We just need to store all the values in an array.
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