**π‘ Problem Formulation:** Finding the minimum adjustment cost of an array involves altering each element so that the absolute difference between any two consecutive elements is less than or equal to a target value. The objective is to determine the minimum total cost of adjusting the array’s values. For instance, given an array `[1, 4, 2, 3]`

and target `1`

, the desired output is the minimum cost to make the differences between any two consecutive numbers not more than 1.

## Method 1: Dynamic Programming

This method uses dynamic programming to find the minimum adjustment cost efficiently. By maintaining a matrix that keeps track of costs for reaching each element, we can systematically build the minimum cost from the bottom up. The function `find_min_cost()`

will be defined to encapsulate this functionality.

Here’s an example:

def heuristic_optimize(arr, target): import numpy as np adjusted_arr = np.array(arr, dtype=np.int) changed = True while changed: changed = False for i in range(1, len(arr)): if abs(adjusted_arr[i] - adjusted_arr[i-1]) > target: adjustment = (adjusted_arr[i-1] + adjusted_arr[i+1 if i+1 < len(arr) else i]) // 2 adjusted_arr[i] = adjustment changed = True return sum(abs(arr[i] - adjusted_arr[i]) for i in range(len(arr))) # Example usage: print(heuristic_optimize([1, 4, 2, 3], 1))

Output: `2`

(although results may vary because it’s a heuristic approach)

The `heuristic_optimize()`

function applies a heuristic that adjusts each array element to the average of its neighbors if the target constraint is not met. The process is repeated until no further changes occur. The cost is then computed as the sum of absolute adjustments from the original array.

## Summary/Discussion

**Method 1: Dynamic Programming.**Highly efficient for large arrays. Returns the exact minimum cost. Can become memory-intensive.**Method 2: Recursion with Memoization.**More intuitive than dynamic programming. Still efficient and less memory-intensive. Slower for large arrays.**Method 3: Brute Force.**Simple and straightforward. Impractical for large-scale problems due to exponential time complexity.**Bonus Method 4: Heuristic Optimization.**Provides a quick and often good-enough solution. Not guaranteed to find the minimum cost. Useful when approximations are acceptable.

from itertools import product def brute_force_cost(arr, target): values_range = range(min(arr), max(arr)+1) all_combinations = product(values_range, repeat=len(arr)) min_cost = float('inf') for combo in all_combinations: if all(abs(combo[i] - combo[i+1]) <= target for i in range(len(combo)-1)): cost = sum(abs(arr[i] - combo[i]) for i in range(len(arr))) min_cost = min(min_cost, cost) return min_cost # Example usage: print(brute_force_cost([1, 4, 2, 3], 1))

Output: `2`

The `brute_force_cost()`

function computes the minimum adjustment cost using brute force. It generates all possible combinations of values in the array’s range and filters out combinations that do not meet the target constraint. For each valid combination, it computes the cost and keeps track of the minimum cost found.

## Bonus One-Liner Method 4: Heuristic Optimization

If performance is less critical, a heuristic optimization approach can be used, which makes gradual adjustments to the array based on a cost-effective heuristic until no further improvements can be made. This method won’t guarantee the minimum cost but can yield a sufficiently good solution.

Here’s an example:

def heuristic_optimize(arr, target): import numpy as np adjusted_arr = np.array(arr, dtype=np.int) changed = True while changed: changed = False for i in range(1, len(arr)): if abs(adjusted_arr[i] - adjusted_arr[i-1]) > target: adjustment = (adjusted_arr[i-1] + adjusted_arr[i+1 if i+1 < len(arr) else i]) // 2 adjusted_arr[i] = adjustment changed = True return sum(abs(arr[i] - adjusted_arr[i]) for i in range(len(arr))) # Example usage: print(heuristic_optimize([1, 4, 2, 3], 1))

Output: `2`

(although results may vary because it’s a heuristic approach)

The `heuristic_optimize()`

function applies a heuristic that adjusts each array element to the average of its neighbors if the target constraint is not met. The process is repeated until no further changes occur. The cost is then computed as the sum of absolute adjustments from the original array.

## Summary/Discussion

**Method 1: Dynamic Programming.**Highly efficient for large arrays. Returns the exact minimum cost. Can become memory-intensive.**Method 2: Recursion with Memoization.**More intuitive than dynamic programming. Still efficient and less memory-intensive. Slower for large arrays.**Method 3: Brute Force.**Simple and straightforward. Impractical for large-scale problems due to exponential time complexity.**Bonus Method 4: Heuristic Optimization.**Provides a quick and often good-enough solution. Not guaranteed to find the minimum cost. Useful when approximations are acceptable.

def compute_cost(arr, target, i, prev, memo): if i == len(arr): return 0 if (i, prev) in memo: return memo[(i, prev)] min_cost = float('inf') for j in range(max(0, prev - target), min(100, prev + target) + 1): cost = abs(arr[i] - j) + compute_cost(arr, target, i + 1, j, memo) min_cost = min(min_cost, cost) memo[(i, prev)] = min_cost return min_cost # Example usage: # Initial call to the function with prev as the first element # because it doesn't have a previous element print(compute_cost([1, 4, 2, 3], 1, 0, 0, {}))

Output: `2`

In the `compute_cost()`

function, the recursion navigates through the array, adjusting the current value within the target range and calculates the cost, storing intermediary results in the memoization dictionary `memo`

. The recursion base case is when we reach the end of the array. We return the minimum cost accumulated from recursive calls.

## Method 3: Brute Force

The brute force approach tries every possible combination of adjustments and computes the costs, picking the minimum overall cost. Although not efficient for large arrays, it is straightforward and can be suitable for small-scale problems.

Here’s an example:

from itertools import product def brute_force_cost(arr, target): values_range = range(min(arr), max(arr)+1) all_combinations = product(values_range, repeat=len(arr)) min_cost = float('inf') for combo in all_combinations: if all(abs(combo[i] - combo[i+1]) <= target for i in range(len(combo)-1)): cost = sum(abs(arr[i] - combo[i]) for i in range(len(arr))) min_cost = min(min_cost, cost) return min_cost # Example usage: print(brute_force_cost([1, 4, 2, 3], 1))

Output: `2`

The `brute_force_cost()`

function computes the minimum adjustment cost using brute force. It generates all possible combinations of values in the array’s range and filters out combinations that do not meet the target constraint. For each valid combination, it computes the cost and keeps track of the minimum cost found.

## Bonus One-Liner Method 4: Heuristic Optimization

If performance is less critical, a heuristic optimization approach can be used, which makes gradual adjustments to the array based on a cost-effective heuristic until no further improvements can be made. This method won’t guarantee the minimum cost but can yield a sufficiently good solution.

Here’s an example:

def heuristic_optimize(arr, target): import numpy as np adjusted_arr = np.array(arr, dtype=np.int) changed = True while changed: changed = False for i in range(1, len(arr)): if abs(adjusted_arr[i] - adjusted_arr[i-1]) > target: adjustment = (adjusted_arr[i-1] + adjusted_arr[i+1 if i+1 < len(arr) else i]) // 2 adjusted_arr[i] = adjustment changed = True return sum(abs(arr[i] - adjusted_arr[i]) for i in range(len(arr))) # Example usage: print(heuristic_optimize([1, 4, 2, 3], 1))

Output: `2`

(although results may vary because it’s a heuristic approach)

The `heuristic_optimize()`

function applies a heuristic that adjusts each array element to the average of its neighbors if the target constraint is not met. The process is repeated until no further changes occur. The cost is then computed as the sum of absolute adjustments from the original array.

## Summary/Discussion

**Method 1: Dynamic Programming.**Highly efficient for large arrays. Returns the exact minimum cost. Can become memory-intensive.**Method 2: Recursion with Memoization.**More intuitive than dynamic programming. Still efficient and less memory-intensive. Slower for large arrays.**Method 3: Brute Force.**Simple and straightforward. Impractical for large-scale problems due to exponential time complexity.**Bonus Method 4: Heuristic Optimization.**Provides a quick and often good-enough solution. Not guaranteed to find the minimum cost. Useful when approximations are acceptable.

def find_min_cost(arr, target): max_num = max(arr) dp = [[float('inf')] * (max_num+1) for _ in arr] for i in range(max_num+1): dp[0][i] = abs(arr[0] - i) for i in range(1, len(arr)): for j in range(max_num+1): for k in range(max(0, j-target), min(max_num, j+target)+1): dp[i][j] = min(dp[i][j], dp[i-1][k] + abs(arr[i] - j)) return min(dp[-1]) # Example usage: print(find_min_cost([1, 4, 2, 3], 1))

Output: `2`

The code defines the `find_min_cost()`

function that uses dynamic programming to compute the adjustment cost of the array. Each row of the `dp`

matrix represents different element adjustments, and the columns represent the potential new values. We iterate through each element and its potential new values, updating the matrix with the minimum cost found. Finally, we return the smallest cost from the last row of the matrix.

## Method 2: Recursion with Memoization

Using recursion with memoization, we solve smaller subproblems and store their results to avoid repeated computation. The recursive function `compute_cost()`

explores all possibilities for adjusting each element and cached results are stored in a dictionary `memo`

.

Here’s an example:

def compute_cost(arr, target, i, prev, memo): if i == len(arr): return 0 if (i, prev) in memo: return memo[(i, prev)] min_cost = float('inf') for j in range(max(0, prev - target), min(100, prev + target) + 1): cost = abs(arr[i] - j) + compute_cost(arr, target, i + 1, j, memo) min_cost = min(min_cost, cost) memo[(i, prev)] = min_cost return min_cost # Example usage: # Initial call to the function with prev as the first element # because it doesn't have a previous element print(compute_cost([1, 4, 2, 3], 1, 0, 0, {}))

Output: `2`

In the `compute_cost()`

function, the recursion navigates through the array, adjusting the current value within the target range and calculates the cost, storing intermediary results in the memoization dictionary `memo`

. The recursion base case is when we reach the end of the array. We return the minimum cost accumulated from recursive calls.

## Method 3: Brute Force

The brute force approach tries every possible combination of adjustments and computes the costs, picking the minimum overall cost. Although not efficient for large arrays, it is straightforward and can be suitable for small-scale problems.

Here’s an example:

from itertools import product def brute_force_cost(arr, target): values_range = range(min(arr), max(arr)+1) all_combinations = product(values_range, repeat=len(arr)) min_cost = float('inf') for combo in all_combinations: if all(abs(combo[i] - combo[i+1]) <= target for i in range(len(combo)-1)): cost = sum(abs(arr[i] - combo[i]) for i in range(len(arr))) min_cost = min(min_cost, cost) return min_cost # Example usage: print(brute_force_cost([1, 4, 2, 3], 1))

Output: `2`

The `brute_force_cost()`

function computes the minimum adjustment cost using brute force. It generates all possible combinations of values in the array’s range and filters out combinations that do not meet the target constraint. For each valid combination, it computes the cost and keeps track of the minimum cost found.

## Bonus One-Liner Method 4: Heuristic Optimization

If performance is less critical, a heuristic optimization approach can be used, which makes gradual adjustments to the array based on a cost-effective heuristic until no further improvements can be made. This method won’t guarantee the minimum cost but can yield a sufficiently good solution.

Here’s an example:

Output: `2`

(although results may vary because it’s a heuristic approach)

`heuristic_optimize()`

function applies a heuristic that adjusts each array element to the average of its neighbors if the target constraint is not met. The process is repeated until no further changes occur. The cost is then computed as the sum of absolute adjustments from the original array.

## Summary/Discussion

**Method 1: Dynamic Programming.**Highly efficient for large arrays. Returns the exact minimum cost. Can become memory-intensive.**Method 2: Recursion with Memoization.**More intuitive than dynamic programming. Still efficient and less memory-intensive. Slower for large arrays.**Method 3: Brute Force.**Simple and straightforward. Impractical for large-scale problems due to exponential time complexity.**Bonus Method 4: Heuristic Optimization.**Provides a quick and often good-enough solution. Not guaranteed to find the minimum cost. Useful when approximations are acceptable.

Output: `2`

`brute_force_cost()`

function computes the minimum adjustment cost using brute force. It generates all possible combinations of values in the array’s range and filters out combinations that do not meet the target constraint. For each valid combination, it computes the cost and keeps track of the minimum cost found.

## Bonus One-Liner Method 4: Heuristic Optimization

Here’s an example:

Output: `2`

(although results may vary because it’s a heuristic approach)

`heuristic_optimize()`

function applies a heuristic that adjusts each array element to the average of its neighbors if the target constraint is not met. The process is repeated until no further changes occur. The cost is then computed as the sum of absolute adjustments from the original array.

## Summary/Discussion

**Method 1: Dynamic Programming.**Highly efficient for large arrays. Returns the exact minimum cost. Can become memory-intensive.**Method 2: Recursion with Memoization.**More intuitive than dynamic programming. Still efficient and less memory-intensive. Slower for large arrays.**Method 3: Brute Force.**Simple and straightforward. Impractical for large-scale problems due to exponential time complexity.**Bonus Method 4: Heuristic Optimization.**Provides a quick and often good-enough solution. Not guaranteed to find the minimum cost. Useful when approximations are acceptable.

def find_min_cost(arr, target): max_num = max(arr) dp = [[float('inf')] * (max_num+1) for _ in arr] for i in range(max_num+1): dp[0][i] = abs(arr[0] - i) for i in range(1, len(arr)): for j in range(max_num+1): for k in range(max(0, j-target), min(max_num, j+target)+1): dp[i][j] = min(dp[i][j], dp[i-1][k] + abs(arr[i] - j)) return min(dp[-1]) # Example usage: print(find_min_cost([1, 4, 2, 3], 1))

Output: `2`

The code defines the `find_min_cost()`

function that uses dynamic programming to compute the adjustment cost of the array. Each row of the `dp`

matrix represents different element adjustments, and the columns represent the potential new values. We iterate through each element and its potential new values, updating the matrix with the minimum cost found. Finally, we return the smallest cost from the last row of the matrix.

## Method 2: Recursion with Memoization

Using recursion with memoization, we solve smaller subproblems and store their results to avoid repeated computation. The recursive function `compute_cost()`

explores all possibilities for adjusting each element and cached results are stored in a dictionary `memo`

.

Here’s an example:

def compute_cost(arr, target, i, prev, memo): if i == len(arr): return 0 if (i, prev) in memo: return memo[(i, prev)] min_cost = float('inf') for j in range(max(0, prev - target), min(100, prev + target) + 1): cost = abs(arr[i] - j) + compute_cost(arr, target, i + 1, j, memo) min_cost = min(min_cost, cost) memo[(i, prev)] = min_cost return min_cost # Example usage: # Initial call to the function with prev as the first element # because it doesn't have a previous element print(compute_cost([1, 4, 2, 3], 1, 0, 0, {}))

Output: `2`

In the `compute_cost()`

function, the recursion navigates through the array, adjusting the current value within the target range and calculates the cost, storing intermediary results in the memoization dictionary `memo`

. The recursion base case is when we reach the end of the array. We return the minimum cost accumulated from recursive calls.

## Method 3: Brute Force

The brute force approach tries every possible combination of adjustments and computes the costs, picking the minimum overall cost. Although not efficient for large arrays, it is straightforward and can be suitable for small-scale problems.

Here’s an example:

Output: `2`

`brute_force_cost()`

function computes the minimum adjustment cost using brute force. It generates all possible combinations of values in the array’s range and filters out combinations that do not meet the target constraint. For each valid combination, it computes the cost and keeps track of the minimum cost found.

## Bonus One-Liner Method 4: Heuristic Optimization

Here’s an example:

Output: `2`

(although results may vary because it’s a heuristic approach)

`heuristic_optimize()`

function applies a heuristic that adjusts each array element to the average of its neighbors if the target constraint is not met. The process is repeated until no further changes occur. The cost is then computed as the sum of absolute adjustments from the original array.

## Summary/Discussion

**Method 1: Dynamic Programming.**Highly efficient for large arrays. Returns the exact minimum cost. Can become memory-intensive.**Method 2: Recursion with Memoization.**More intuitive than dynamic programming. Still efficient and less memory-intensive. Slower for large arrays.**Method 3: Brute Force.**Simple and straightforward. Impractical for large-scale problems due to exponential time complexity.**Bonus Method 4: Heuristic Optimization.**Provides a quick and often good-enough solution. Not guaranteed to find the minimum cost. Useful when approximations are acceptable.

Output: `2`

`compute_cost()`

function, the recursion navigates through the array, adjusting the current value within the target range and calculates the cost, storing intermediary results in the memoization dictionary `memo`

. The recursion base case is when we reach the end of the array. We return the minimum cost accumulated from recursive calls.

## Method 3: Brute Force

Here’s an example:

Output: `2`

`brute_force_cost()`

function computes the minimum adjustment cost using brute force. It generates all possible combinations of values in the array’s range and filters out combinations that do not meet the target constraint. For each valid combination, it computes the cost and keeps track of the minimum cost found.

## Bonus One-Liner Method 4: Heuristic Optimization

Here’s an example:

Output: `2`

(although results may vary because it’s a heuristic approach)

`heuristic_optimize()`

function applies a heuristic that adjusts each array element to the average of its neighbors if the target constraint is not met. The process is repeated until no further changes occur. The cost is then computed as the sum of absolute adjustments from the original array.

## Summary/Discussion

**Method 1: Dynamic Programming.**Highly efficient for large arrays. Returns the exact minimum cost. Can become memory-intensive.**Method 2: Recursion with Memoization.**More intuitive than dynamic programming. Still efficient and less memory-intensive. Slower for large arrays.**Method 3: Brute Force.**Simple and straightforward. Impractical for large-scale problems due to exponential time complexity.**Bonus Method 4: Heuristic Optimization.**Provides a quick and often good-enough solution. Not guaranteed to find the minimum cost. Useful when approximations are acceptable.

def find_min_cost(arr, target): max_num = max(arr) dp = [[float('inf')] * (max_num+1) for _ in arr] for i in range(max_num+1): dp[0][i] = abs(arr[0] - i) for i in range(1, len(arr)): for j in range(max_num+1): for k in range(max(0, j-target), min(max_num, j+target)+1): dp[i][j] = min(dp[i][j], dp[i-1][k] + abs(arr[i] - j)) return min(dp[-1]) # Example usage: print(find_min_cost([1, 4, 2, 3], 1))

Output: `2`

The code defines the `find_min_cost()`

function that uses dynamic programming to compute the adjustment cost of the array. Each row of the `dp`

matrix represents different element adjustments, and the columns represent the potential new values. We iterate through each element and its potential new values, updating the matrix with the minimum cost found. Finally, we return the smallest cost from the last row of the matrix.

## Method 2: Recursion with Memoization

Using recursion with memoization, we solve smaller subproblems and store their results to avoid repeated computation. The recursive function `compute_cost()`

explores all possibilities for adjusting each element and cached results are stored in a dictionary `memo`

.

Here’s an example:

Output: `2`

`compute_cost()`

function, the recursion navigates through the array, adjusting the current value within the target range and calculates the cost, storing intermediary results in the memoization dictionary `memo`

. The recursion base case is when we reach the end of the array. We return the minimum cost accumulated from recursive calls.

## Method 3: Brute Force

Here’s an example:

Output: `2`

`brute_force_cost()`

function computes the minimum adjustment cost using brute force. It generates all possible combinations of values in the array’s range and filters out combinations that do not meet the target constraint. For each valid combination, it computes the cost and keeps track of the minimum cost found.

## Bonus One-Liner Method 4: Heuristic Optimization

Here’s an example:

Output: `2`

(although results may vary because it’s a heuristic approach)

`heuristic_optimize()`

function applies a heuristic that adjusts each array element to the average of its neighbors if the target constraint is not met. The process is repeated until no further changes occur. The cost is then computed as the sum of absolute adjustments from the original array.

## Summary/Discussion

**Method 1: Dynamic Programming.**Highly efficient for large arrays. Returns the exact minimum cost. Can become memory-intensive.**Method 2: Recursion with Memoization.**More intuitive than dynamic programming. Still efficient and less memory-intensive. Slower for large arrays.**Method 3: Brute Force.**Simple and straightforward. Impractical for large-scale problems due to exponential time complexity.**Bonus Method 4: Heuristic Optimization.**Provides a quick and often good-enough solution. Not guaranteed to find the minimum cost. Useful when approximations are acceptable.

Output: `2`

`brute_force_cost()`

function computes the minimum adjustment cost using brute force. It generates all possible combinations of values in the array’s range and filters out combinations that do not meet the target constraint. For each valid combination, it computes the cost and keeps track of the minimum cost found.

## Bonus One-Liner Method 4: Heuristic Optimization

Here’s an example:

Output: `2`

(although results may vary because it’s a heuristic approach)

`heuristic_optimize()`

function applies a heuristic that adjusts each array element to the average of its neighbors if the target constraint is not met. The process is repeated until no further changes occur. The cost is then computed as the sum of absolute adjustments from the original array.

## Summary/Discussion

**Method 1: Dynamic Programming.**Highly efficient for large arrays. Returns the exact minimum cost. Can become memory-intensive.**Method 2: Recursion with Memoization.**More intuitive than dynamic programming. Still efficient and less memory-intensive. Slower for large arrays.**Method 3: Brute Force.**Simple and straightforward. Impractical for large-scale problems due to exponential time complexity.**Bonus Method 4: Heuristic Optimization.**Provides a quick and often good-enough solution. Not guaranteed to find the minimum cost. Useful when approximations are acceptable.

Output: `2`

`compute_cost()`

function, the recursion navigates through the array, adjusting the current value within the target range and calculates the cost, storing intermediary results in the memoization dictionary `memo`

. The recursion base case is when we reach the end of the array. We return the minimum cost accumulated from recursive calls.

## Method 3: Brute Force

Here’s an example:

Output: `2`

`brute_force_cost()`

function computes the minimum adjustment cost using brute force. It generates all possible combinations of values in the array’s range and filters out combinations that do not meet the target constraint. For each valid combination, it computes the cost and keeps track of the minimum cost found.

## Bonus One-Liner Method 4: Heuristic Optimization

Here’s an example:

Output: `2`

(although results may vary because it’s a heuristic approach)

`heuristic_optimize()`

function applies a heuristic that adjusts each array element to the average of its neighbors if the target constraint is not met. The process is repeated until no further changes occur. The cost is then computed as the sum of absolute adjustments from the original array.

## Summary/Discussion

**Method 1: Dynamic Programming.**Highly efficient for large arrays. Returns the exact minimum cost. Can become memory-intensive.**Method 2: Recursion with Memoization.**More intuitive than dynamic programming. Still efficient and less memory-intensive. Slower for large arrays.**Method 3: Brute Force.**Simple and straightforward. Impractical for large-scale problems due to exponential time complexity.**Bonus Method 4: Heuristic Optimization.**Provides a quick and often good-enough solution. Not guaranteed to find the minimum cost. Useful when approximations are acceptable.

Output: `2`

`find_min_cost()`

function that uses dynamic programming to compute the adjustment cost of the array. Each row of the `dp`

matrix represents different element adjustments, and the columns represent the potential new values. We iterate through each element and its potential new values, updating the matrix with the minimum cost found. Finally, we return the smallest cost from the last row of the matrix.

## Method 2: Recursion with Memoization

`compute_cost()`

explores all possibilities for adjusting each element and cached results are stored in a dictionary `memo`

.

Here’s an example:

Output: `2`

`compute_cost()`

function, the recursion navigates through the array, adjusting the current value within the target range and calculates the cost, storing intermediary results in the memoization dictionary `memo`

. The recursion base case is when we reach the end of the array. We return the minimum cost accumulated from recursive calls.

## Method 3: Brute Force

Here’s an example:

Output: `2`

`brute_force_cost()`

function computes the minimum adjustment cost using brute force. It generates all possible combinations of values in the array’s range and filters out combinations that do not meet the target constraint. For each valid combination, it computes the cost and keeps track of the minimum cost found.

## Bonus One-Liner Method 4: Heuristic Optimization

Here’s an example:

Output: `2`

(although results may vary because it’s a heuristic approach)

`heuristic_optimize()`

function applies a heuristic that adjusts each array element to the average of its neighbors if the target constraint is not met. The process is repeated until no further changes occur. The cost is then computed as the sum of absolute adjustments from the original array.

## Summary/Discussion

**Method 1: Dynamic Programming.**Highly efficient for large arrays. Returns the exact minimum cost. Can become memory-intensive.**Method 2: Recursion with Memoization.**More intuitive than dynamic programming. Still efficient and less memory-intensive. Slower for large arrays.**Method 3: Brute Force.**Simple and straightforward. Impractical for large-scale problems due to exponential time complexity.**Bonus Method 4: Heuristic Optimization.**Provides a quick and often good-enough solution. Not guaranteed to find the minimum cost. Useful when approximations are acceptable.