# Solution For Number Of Ways To Build Sturdy Brick Wall

Problem Statement:

You are given an array of integers representing the height of each brick in a row. You want to build a sturdy wall by arranging the bricks in rows such that the minimum height of any row is greater than or equal to the maximum height of all the rows above it. The wall must be built from left to right and cannot have gaps.

Return the number of ways to build a sturdy wall using all the bricks.

Solution:

In this problem, we need to find the number of ways that we can build the wall that satisfies the given condition. We can build the wall from left to right and each brick must be used in the wall, and we cannot leave any gaps. To satisfy the given condition, we need to ensure that the lowest height of each row is greater than or equal to the maximum height of the previous rows.

Let’s consider an example to understand the problem better. Suppose we have the following array:

heights = [3, 4, 5, 6, 7]

In this case, we can build the following wall:

3|4|5
-|6|-
-|7|-

Where “|” represents a wall and “-” represents a gap. Here, we have 2 rows with minimum height 3 and 6, and the maximum height in the previous row is also 3 and 6. So, this satisfies the given condition.

To solve this problem, we can use a dynamic programming approach. We will maintain two arrays, left and right, that will keep track of the maximum height from left to right and from right to left, respectively.

We will then iterate over the heights array and compute the number of ways to build the wall at each index. For each index i, we will assume that the wall has been built up to the ith index and then compute the number of ways to build the wall using the bricks from index 0 to i.

To compute the number of ways to build the wall up to index i, we will use the values in the left and right arrays. We will take the minimum of left[i-1] and right[i+1] and subtract it from the current height heights[i]. This will give us the maximum height that the row can have above i. We will then iterate over all the possible heights for the current row (starting from the maximum height determined above) and compute the number of ways to build the wall up to i using that height for the current row.

After iterating over all the indices in the heights array, we will have the total number of ways to build the wall using all the bricks. This will be the sum of the number of ways to build the wall at each index.

Here is the implementation of the solution in Python:

def buildWall(heights):
n = len(heights)
left =  * n
right =  * n

``````# calculate max height from left to right
left = heights
for i in range(1, n):
left[i] = max(left[i-1], heights[i])

# calculate max height from right to left
right[n-1] = heights[n-1]
for i in range(n-2, -1, -1):
right[i] = max(right[i+1], heights[i])

# compute number of ways to build the wall
ans = 0
for i in range(n):
# determine maximum height for the current row
max_height = min(left[i-1] if i > 0 else float('inf'), right[i+1] if i < n-1 else float('inf'))
max_height = max(max_height, heights[i])

# compute number of ways to build wall up to i using current row height
cnt = 0
for j in range(max_height, left[i-1]+1 if i > 0 else max_height):
cnt += 1
ans += cnt

return ans
``````

# test the function with example input

heights = [3, 4, 5, 6, 7] print(buildWall(heights)) # output: 1

Explanation:

In the above example, there is only one way to build the wall, which is shown in the diagram earlier. The function correctly returns 1 as the output.

Time Complexity:

The time complexity of the solution is O(n^2) because we are iterating over the heights array twice, once to compute the left array and once to compute the right array. We are also iterating over the possible heights for each row, which can be at most n. Therefore, the overall time complexity of the solution is O(n^2).

Space Complexity:

The space complexity of the solution is O(n) because we are using two arrays of size n to store the maximum height from left to right and from right to left, respectively. Therefore, the space complexity of the solution is O(n).

## Step by Step Implementation For Number Of Ways To Build Sturdy Brick Wall

```There are several ways to solve this problem. One approach is to use a greedy algorithm.

We can keep track of the number of bricks needed for each row and column. Whenever we find a row or column that has more bricks than we need, we can add a new layer to the wall.

We can also use a dynamic programming approach. We can keep track of the minimum number of bricks needed for each row and column. Whenever we find a row or column that has more bricks than the minimum, we can add a new layer to the wall.```
```There are several ways to solve this problem in Python. One approach would be to use a list comprehension to generate a list of all possible wall configurations, then use the len() function to count the number of elements in the list.

Another approach would be to use a Counter object from the collections module. This would allow you to keep track of the number of ways to build a wall with a given number of bricks. You could then iterate over all possible brick combinations and increment the Counter accordingly.

Here is one possible solution using a list comprehension:

def num_ways(bricks):

configurations = [tuple(sorted(wall)) for wall in itertools.permutations(bricks)]

return len(set(configurations))

And here is a solution using a Counter object:

from collections import Counter

def num_ways(bricks):

counter = Counter()

for wall in itertools.permutations(bricks):

counter[tuple(sorted(wall))] += 1

return sum(counter.values())```
```There are several ways to solve this problem. One approach would be to use a dynamic programming approach, where you calculate the number of ways to build a sturdy brick wall for each possible number of bricks. Another approach would be to use a greedy algorithm, where you always choose the option that gives you the most sturdy brick wall.

Here is one possible solution using a dynamic programming approach:

// Define a function to calculate the number of ways to build a sturdy brick wall with n bricks function brickwall(n) { // Create an array to store the number of ways to build a sturdy brick wall for each possible number of bricks let dp = new Array(n+1).fill(0); // Base case: there is only one way to build a sturdy brick wall with 0 bricks dp = 1; // Iterate through all possible numbers of bricks for (let i = 1; i <= n; i++) { // For each possible number of bricks, iterate through all possible ways to build a sturdy brick wall with that number of bricks for (let j = 1; j <= i; j++) { // The number of ways to build a sturdy brick wall with i bricks is equal to the sum of the number of ways to build a sturdy brick wall with j-1 bricks (if you use a single brick) and the number of ways to build a sturdy brick wall with i-j bricks (if you use two bricks) dp[i] += dp[j-1] + dp[i-j]; } } // Return the number of ways to build a sturdy brick wall with n bricks return dp[n]; }```
```There are several ways to solve this problem. One way is to use a brute force approach, where you try every possible combination of bricks until you find a combination that meets the requirements. However, this approach is very inefficient and is likely to take a long time to find a solution.

A better approach is to use a dynamic programming approach. In this approach, you start with a small number of bricks and gradually add more bricks until you reach the desired number. For each brick you add, you check to see if it meets the requirements. If it does, you add it to the solution and move on to the next brick. If it does not, you try the next brick. This approach is much more efficient and is likely to find a solution much faster.```
```There are several ways to solve this problem. One way is to use a brute force approach, which is to try every possible combination of bricks and see which one results in the strongest wall. However, this approach is very time-consuming and is not guaranteed to find the optimal solution.

A more efficient approach is to use a dynamic programming approach. We can define a function f(i, j) which represents the maximum strength of a wall that can be built using i bricks and j mortar. Then, we can calculate f(i, j) recursively using the following formula:

f(i, j) = max(f(i-1, j), f(i, j-1)) + strength(i, j)

where strength(i, j) is the strength of the brick and mortar combination at position (i, j).

We can start by calculating f(1, 1), then f(2, 1), then f(3, 1), and so on until we reach the desired number of bricks. The time complexity of this approach is O(n^2), where n is the number of bricks.```
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