Count ways to build street under given constraints

4

There is a street of length n and as we know it has two sides. Therefore a total of 2 * n spots are available. In each of these spots either a house or an office can be built with following 2 restrictions:
1. No two offices on the same side of the street can be adjacent.
2. No two offices on different sides of the street can be exactly opposite to each other i.e. they can’t overlook each other.
There are no restrictions on building houses and each spot must either have a house or office.
Given length of the street n, find total number of ways to build the street.

Source: Intuit Interview
Examples:

Input : 2
Output : 7
Please see below diagram for explanation.

Input : 3
Output : 17

Following image depicts the 7 possible ways for building the street with N = 2

Ways for building street with length 1
with 2 houses: (H H) = {1}
with 1 office and 1 house: (O H) (H O) = {2}
(O O) is not allowed according to the problem statement.
Total = 1  + 2 = 3
For length = 2,
with 2 houses: (H H) can be added to all
the cases of length 1:

(H H)    (H H)    (H H)
(H H)    (O H)    (H O) = {3}

with 1 office and 1 house:
(O H) and (H O) can both be added to all 
the cases of length 1 with last row (H H)

(O H)    (H O)
(H H)    (H H) = {2}

when last row of a case of length 1 
contains 1 office, it can only be 
extended in one way to build an office
in row 2. 
(H O)    (O H)    
(O H)    (H O) = {2}

(O H)    (H O) (O O)    
(O H)    (H O) (O H) etc are not allowed.

Total = 3 + 2 + 2 = 7

Since the problem can be solved by finding solution for smaller subproblems and then extending the same logic, it can be solved using dynamic programming. We move in steps of one unit length. For each row we have two options:
Build houses in both the spots
Build one house and one office

The first one can be done without any constraints. There is one way of building houses in both the spots at length i. So total ways using this choice = total ways for length i – 1.

For the second choice, if row (i-1) had houses in both spots we have two ways of building a office i.e. (H O) and (O H)
if row(i-1) had an office in one of its two spots we only have one way to build an office in row i.If prev row had (O H) curr row would have (H O) and similarly for prev row = (H O) curr row = (O H).
From the above logic, total ways with this choice = 2 * (choice1(i-1)) + choice2(i-1)

We will build a 2D dp for this.
dp[0][i] indicates choice1 and dp[1][i] indicates choice2 for row i.
Below is the c++ implementation for above logic

// C++ program to count ways to build street 
// under given constraints
#include <bits/stdc++.h>
using namespace std;

// function to count ways of building
// a street of n rows
long countWays(int n)
{
    long dp[2][n + 1];

    // base case
    dp[0][1] = 1;
    dp[1][1] = 2;

    for (int i = 2; i <= n; i++) {

        // ways of building houses in both
        // the spots of ith row
        dp[0][i] = dp[0][i - 1] + dp[1][i - 1];

        // ways of building an office in one of
        // the two spots of ith row
        dp[1][i] = dp[0][i - 1] * 2 + dp[1][i - 1];
    }

    // total ways for n rows
    return dp[0][n] + dp[1][n];
}

// driver program for checking above function
int main()
{

    int n = 5;
    cout << "Total no of ways with n = " << n
         << " are: " << countWays(n) << endl;
}

Output:

Total no of ways with n = 5 are: 99

Time Complexity: O(N)
Auxiliary Space : O(N)

This article is contributed by Aditi Sharma. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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