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72 lines
1.6 KiB
Markdown
72 lines
1.6 KiB
Markdown
## Prefix Sum
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Precompute cumulative totals once → any range sum in **O(1)**.
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---
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## Recognition
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- Many range-sum queries over a static array
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- "Count subarrays with sum k" (negatives allowed — window won't work)
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- "Product/sum of everything except me" → prefix from both directions
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---
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## Template — range query (LC 303)
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```js
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// prefix[i] = sum of nums[0..i-1], prefix[0] = 0
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const prefix = [0];
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for (const x of nums) prefix.push(prefix[prefix.length - 1] + x);
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// sum of nums[l..r] inclusive:
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const rangeSum = (l, r) => prefix[r + 1] - prefix[l];
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```
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The leading 0 removes every edge case at `l = 0`.
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---
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## Template — prefix + hashmap (LC 560)
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```js
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function subarraySum(nums, k) {
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const count = new Map([[0, 1]]); // empty prefix
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let sum = 0;
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let total = 0;
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for (const x of nums) {
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sum += x;
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total += count.get(sum - k) ?? 0; // earlier prefix to cut off
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count.set(sum, (count.get(sum) ?? 0) + 1);
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}
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return total;
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}
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```
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sum(l..r) = prefix[r] − prefix[l−1] — so look up `sum - k`.
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---
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## Pitfalls
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- Forgetting the seed `(0, 1)` misses subarrays starting at index 0
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- Count **before** inserting the current prefix (a subarray has length ≥ 1)
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- Sliding window fails here when values can be negative
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---
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## Recall
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- Why does LC 560 need a hashmap instead of a window?
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- LC 238 Product Except Self: how do prefix and suffix passes combine?
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- LC 525 Contiguous Array: what do you turn 0s into, and why?
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---
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## Drill
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- 303 Range Sum Query · 1480 Running Sum
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- 238 Product of Array Except Self
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- 560 Subarray Sum Equals K
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- 525 Contiguous Array · 974 Subarray Sums Divisible by K
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