Sum of Total Strength of Wizards

class Solution {
    public int computeStrength(int[] values) {
        int modulo = (int)1e9 + 7, len = values.length;
    
        Stack<Integer> indxStack = new Stack<>();
        int[] rightSmaller = new int[len];
        Arrays.fill(rightSmaller, len);
        for (int i = 0; i < len; ++i) {
            while (!indxStack.isEmpty() && values[indxStack.peek()] >= values[i]) {
                rightSmaller[indxStack.pop()] = i;
            }
            indxStack.add(i);
        }
    
        int[] leftSmaller = new int[len];
        Arrays.fill(leftSmaller, -1);
        indxStack.clear();
        for (int i = len - 1; i >= 0; --i) {
            while (!indxStack.isEmpty() && values[indxStack.peek()] > values[i])
                leftSmaller[indxStack.pop()] = i;
            indxStack.add(i);
        }
    
        long total = 0;
        long[] cumulativeSum = new long[len + 2];
        for (int i = 0; i < len; ++i)
            cumulativeSum[i + 2] = (cumulativeSum[i + 1] + values[i]) % modulo;
        for (int i = 1; i <= len; ++i)
            cumulativeSum[i + 1] = (cumulativeSum[i + 1] + cumulativeSum[i]) % modulo;
            
        for (int i = 0; i < len; ++i) {
            int left = leftSmaller[i], right = rightSmaller[i];
            int countLeft = i - left, countRight = right - i;
            long negativeSum = (cumulativeSum[i + 1] - cumulativeSum[i - countLeft + 1]) % modulo;
            long positiveSum = (cumulativeSum[i + countRight + 1] - cumulativeSum[i + 1]) % modulo;
    
            total = (total + (positiveSum * countLeft - negativeSum * countRight) % modulo * values[i] % modulo) % modulo;
        }
    
        return (int)(total + modulo) % modulo;
    }
}

This Java solution tackles the problem of computing the sum of the total strength of wizards, utilizing an algorithm that focuses on identifying the strength based on smaller elements to the left and right of each wizard's strength.

Follow these steps in your understanding of the provided solution:

1. Initialize rightSmaller and leftSmaller arrays to hold the indices of the next smaller element to the right and the last smaller element to the left for each element in the input values array.
2. Utilize a stack to help determine the nearest smaller values for each element with respect to both left and right sides.
3. Implement the logic to populate the rightSmaller array by examining elements from left to right. For leftSmaller, the array is examined from right to left.
4. Compute cumulative sums forward and then backward to facilitate efficient range sum queries, essential for calculating strengths.
5. For each index, compute the product of the index's value and the difference between sums of elements located between that index and its next and last smaller elements. The aim is to calculate the influence of each wizard's strength based on the strength of surrounding wizards.
6. Apply modulo operations frequently to ensure the results remain within bounds and avoid integer overflow.
7. Sum up all calculated strengths to get the final result.

The use of data structures like arrays for direct access and stacks for maintaining indices efficiently handling the problem within acceptable time complexities typical of such range query and modified stack-using problems. This systematic use of modular arithmetic alongside cumulative sums and stack-based index management helps efficiently solve a problem of determining collective strength influenced by neighbors.

class Calculator:
    def calculateStrength(self, svc: List[int]) -> int:
        modulo, length = 10**9 + 7, len(svc)
    
        # Right bounds for each element where the next smaller exists.
        right_bounds = [length] * length
        temp_stack = []
        for idx in range(length):
            while temp_stack and svc[temp_stack[-1]] >= svc[idx]:
                right_bounds[temp_stack.pop()] = idx
            temp_stack.append(idx)
    
        # Left bounds for each element where the previous smaller exists.
        left_bounds = [-1] * length
        temp_stack = []
        for idx in range(length - 1, -1, -1):
            while temp_stack and svc[temp_stack[-1]] > svc[idx]:
                left_bounds[temp_stack.pop()] = idx
            temp_stack.append(idx)
    
        # Precomputing prefix sums of sums for rapid calculations.
        sum_of_sums = list(accumulate(accumulate(svc, initial=0), initial=0))
        result = 0
    
        # Calculating the contribution of each element.
        for i in range(length):
            l_idx = left_bounds[i]
            r_idx = right_bounds[i]
            l_count = i - l_idx
            r_count = r_idx - i
            negative_sum = (sum_of_sums[i + 1] - sum_of_sums[i - l_count + 1]) % modulo
            positive_sum = (sum_of_sums[i + r_count + 1] - sum_of_sums[i + 1]) % modulo
            result += svc[i] * (positive_sum * l_count - negative_sum * r_count)
            result %= modulo
    
        return result

In this Python solution, the goal is to efficiently calculate the total strength of wizards where the strength is defined by specific boundaries in the svc list, which contains the strength values. The approach uses a stack to determine the next and previous smaller elements for each list element, which defines the valid bounds for subarray calculations affecting a wizard's strength.

Here's an outline of each major part of the solution:

• Initialization: Constants and structures such as right_bounds, left_bounds, and mod are initialized. These variables are used for storing bounds and the modulo for results, respectively.

• Stack Operations:

  • Determine right_bounds for each element by maintaining a stack that keeps track of elements until a smaller one is found to the right.
  • Similarly, determine left_bounds by iterating from the end to the start of the list and using a stack to track the smaller elements on the left.

• Prefix Sums Calculation:

  • Compute prefix sums and prefix sums of sums, which helps in quick range sum calculations.

• Calculate Total Strength:

  • For each wizard, apply the bounds defined by left_bounds and right_bounds to compute its contributions based on its influence range, both to the left and right. These contributions are weighted by the wizard's strength itself and differences in precomputed prefix sums.
  • Each contribution is accumulated into the result, applying modulo operations to prevent overflow.

This approach ensures that every element's bounds are processed exactly once, and range sum queries required for strength calculation are optimized via prefix sums, making the entire operation efficient even for relatively large inputs.