Add subarray patterns reference guide (explanatory document)

org/questions/subarray-patterns.org

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+#+title: Subarray Sum — Pattern Recognition Guide
+#+AUTHOR: Noramyll
+
+* Core Trigger: Subarray + Sum
+
+When you see **"subarray"** and **"sum"** in the same problem description, **Prefix Sums** should almost always be your immediate first thought.
+
+The fundamental formula:
+
+  Sum(i..j) = PrefixSum[j] - PrefixSum[i-1]
+
+This algebraic rewrite turns a **range query** into a **difference between two points**. It completely eliminates the need to re-scan the array.
+
+The mental trigger is simple: subarray + sum → prefix sums. Always.
+
+**Variations of this pattern:**
+
+- **"Count of subarrays where sum equals K"**: Use the hash map approach, looking for `current_sum - K`.
+- **"Longest/Shortest subarray where sum equals K"**: Instead of storing the *frequency* of the prefix sum in your map, store its **first seen index** (`map[prefix_sum] = index`). This lets you track length via `current_index - map[current_sum - K]`.
+- **"Subarray sum divisible by K"**: Store `prefix_sum % K` in your hash map instead of the raw sum.
+
+* The Multiplicative Variant: Subarray + Product
+
+If the problem asks for a **subarray product** (e.g., "number of subarrays where product equals K"), the prefix sum pattern translates directly into a **Prefix Product**:
+
+  Product(i..j) = PrefixProduct[j] / PrefixProduct[i-1]
+
+Instead of subtracting, you divide.
+
+**Edge case:** Zeros reset the product to 0. Division by zero breaks the pattern. Handle by:
+1. Segmentation — split the array at zeros, solve each segment independently
+2. Track position of last seen zero to reset boundaries
+
+* The Counter-Pattern: When Is It *Not* Prefix Sums?
+
+While "subarray + sum" strongly hints at prefix sums, you need to look at the **constraints** and **types of numbers** to choose the absolute best tool.
+
+| If you see "Subarray + Sum" AND... | The Real Pattern Is... | Why? |
+|-------------------------------------|------------------------|------|
+| All numbers are strictly positive (>= 0) and you need to find a target sum or max length. | **Sliding Window (Two Pointers)** | Because the window sum is monotonic (expanding always increases the sum; shrinking always decreases it). Sliding window optimizes space to O(1) compared to O(n) for prefix sums. |
+| Numbers can be negative, or you need to find an exact target sum. | **Prefix Sum + Hash Map** | Monotonicity is broken. Adding an element could make the sum smaller, so sliding window fails. You *must* use a hash map to remember past states. |
+| The array is **constantly being updated** (dynamic updates) between sum queries. | **Segment Tree or Fenwick Tree (BIT)** | A standard prefix sum array takes O(n) to update if an element changes. Segment/Fenwick trees drop update and query times to O(log n). |
+
+* Generalizing the Concept: "Prefix State"
+
+Don't limit this trick just to addition. The core philosophy of a prefix sum is **accumulating history as you traverse linearly**. You can use a hash map to track the "prefix state" for things that don't look like math at first.
+
+| Problem | Mapping | Reduces To |
+|---------|---------|-----------|
+| Equal 0s and 1s | 0 → -1, 1 → +1 | Subarray sum = 0 |
+| Equal odd/even | even → +1, odd → -1 | Subarray sum = 0 |
+| Equal vowels/consonants | vowel → +1, consonant → -1 | Subarray sum = 0 |
+| Equal A/B/C counts | Track (c_A - c_B, c_B - c_C) | Prefix state tuple repeats |
+| Subarray product | Prefix product | Division (handle zeros) |
+| Subarray XOR | Prefix XOR | XOR is invertible |
+| Subarray sum | Prefix sum | Subtraction |
+
+The pattern:
+1. Define a state that accumulates as you traverse
+2. Find a way to "undo" or compare states (subtract, XOR, divide, compare tuples)
+3. Use a hash map to find when the same state (or target difference) appears
+
+* Summary Checklist
+
+1. Subarray + Sum + Negative Numbers → **Prefix Sum + Hash Map**
+2. Subarray + Sum + Only Positive Numbers → **Sliding Window**
+3. Subarray + Sum + Frequent Updates → **Segment/Fenwick Tree**