## Prefix Sum

Precompute cumulative totals once → any range sum in **O(1)**.

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## Recognition

- Many range-sum queries over a static array
- "Count subarrays with sum k" (negatives allowed — window won't work)
- "Product/sum of everything except me" → prefix from both directions

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## Template — range query (LC 303)

```js
// prefix[i] = sum of nums[0..i-1], prefix[0] = 0
const prefix = [0];
for (const x of nums) prefix.push(prefix[prefix.length - 1] + x);

// sum of nums[l..r] inclusive:
const rangeSum = (l, r) => prefix[r + 1] - prefix[l];
```

The leading 0 removes every edge case at `l = 0`.

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## Template — prefix + hashmap (LC 560)

```js
function subarraySum(nums, k) {
  const count = new Map([[0, 1]]); // empty prefix
  let sum = 0;
  let total = 0;
  for (const x of nums) {
    sum += x;
    total += count.get(sum - k) ?? 0; // earlier prefix to cut off
    count.set(sum, (count.get(sum) ?? 0) + 1);
  }
  return total;
}
```

sum(l..r) = prefix[r] − prefix[l−1] — so look up `sum - k`.

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## Pitfalls

- Forgetting the seed `(0, 1)` misses subarrays starting at index 0
- Count **before** inserting the current prefix (a subarray has length ≥ 1)
- Sliding window fails here when values can be negative

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## Recall

- Why does LC 560 need a hashmap instead of a window?
- LC 238 Product Except Self: how do prefix and suffix passes combine?
- LC 525 Contiguous Array: what do you turn 0s into, and why?

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## Drill

- 303 Range Sum Query · 1480 Running Sum
- 238 Product of Array Except Self
- 560 Subarray Sum Equals K
- 525 Contiguous Array · 974 Subarray Sums Divisible by K