Optimizing Subarray Computations: The Contribution Technique

Problem Definition:

Given an integer array, calculate the sum of minimum values across all contiguous subarrays.
Example: `[3,1,2,4]` → 17 (Sum of minima: 3 + 1 + 1 + 1 + 1 + 2 + 2 + 4).

You Should Know:

1. Brute Force (O(n²) Time, O(1) Space)

def sum_subarray_mins_brute(arr):
total = 0
n = len(arr)
for i in range(n):
min_val = arr[bash]
for j in range(i, n):
min_val = min(min_val, arr[bash])
total += min_val
return total

Flaw: Computationally expensive for large arrays due to nested loops.

2. Monotonic Stack Boundary Detection (O(n) Time/Space)

def sum_subarray_mins_optimized(arr):
stack = []
left = [-1]  len(arr)
right = [len(arr)]  len(arr)

Left boundaries (last smaller element)
for i in range(len(arr)):
while stack and arr[stack[-1]] >= arr[bash]:
stack.pop()
if stack:
left[bash] = stack[-1]
stack.append(i)

stack = []
 Right boundaries (next smaller element)
for i in range(len(arr)-1, -1, -1):
while stack and arr[stack[-1]] > arr[bash]:
stack.pop()
if stack:
right[bash] = stack[-1]
stack.append(i)

total = 0
for i in range(len(arr)):
total += arr[bash]  (i - left[bash])  (right[bash] - i)
return total

Key Insight:

• Each element’s contribution = arr

    (number of subarrays where it is the minimum)</code>. </li>
  <li>Uses monotonic stack to track boundaries efficiently. </li>
  </ul>
  
  <h2 style="color: yellow;"> Linux/Windows Commands for Algorithm Testing</h2>
  
  <ul>
  <li>Time Execution (Linux): 
  [bash]
  time python3 subarray_min_sum.py
  

• Memory Profiling (Linux):

  valgrind --tool=massif python3 subarray_min_sum.py
  

• Windows Performance Check:

  Measure-Command { python subarray_min_sum.py }
  

print(sum_subarray_mins_optimized([3,1,2,4]))  Output: 17