add 940 solution

Hard/940.Distinct-Subsequences-II/description.md

@@ -0,0 +1,38 @@
+# 940. Distinct Subsequences II
+
+Given a string `s`, return *the number of **distinct non-empty subsequences** of*
+`s`. Since the answer may be very large, return it **modulo** `10^9 + 7`.
+
+A **subsequence** of a string is a new string that is formed from the original
+string by deleting some (can be none) of the characters without disturbing the
+relative positions of the remaining characters. (i.e., `"ace"` is a subsequence of
+`"abcde"` while `"aec"` is not.)
+
+## Example 1
+
+```text
+Input: s = "abc"
+Output: 7
+Explanation: The 7 distinct subsequences are "a", "b", "c", "ab", "ac", "bc", and "abc".
+```
+
+## Example 2
+
+```text
+Input: s = "aba"
+Output: 6
+Explanation: The 6 distinct subsequences are "a", "b", "ab", "aa", "ba", and "aba".
+```
+
+## Example 3
+
+```text
+Input: s = "aaa"
+Output: 3
+Explanation: The 3 distinct subsequences are "a", "aa" and "aaa".
+```
+
+## Constraints
+
+- `1 <= s.length <= 2000`
+- `s` consists of lowercase English letters.

Hard/940.Distinct-Subsequences-II/solution.md

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+# Intuition
+
+Sweep `s` left to right and carry one number: how many distinct subsequences the
+prefix has, counting the empty one. When the next character `c` arrives, every
+subsequence counted so far either ignores `c` or appends it, which suggests
+doubling the total. Appending really does produce `total` *distinct* strings —
+adding the same character to distinct strings cannot collide — so the only
+question is how many of those strings were already counted.
+
+The answer is exactly the strings that the **previous occurrence of `c`** created.
+Writing $$T_i$$ for the set of distinct subsequences of the first `i` characters
+and $$T_i \cdot c$$ for the set with `c` appended to each element:
+
+$$T_i = T_{i-1} \cup (T_{i-1} \cdot c), \qquad (T_{i-1} \cdot c) \cap T_{i-1} = T_{j-1} \cdot c$$
+
+where `j` is the previous index at which `c` occurred. A subsequence of the prefix
+that ends in `c` must have been formed at some earlier occurrence of `c`, and the
+most recent one already generated all of them. So
+
+$$|T_i| = 2\,|T_{i-1}| - |T_{j-1}|$$
+
+and remembering one number per letter — the total as it stood just before that
+letter was last consumed — is enough. Twenty-six counters replace the whole DP
+table.
+
+# Approach: Running Total With Per-Letter Memory
+
+1. Start with `total = 1` (the empty subsequence) and `last[26]` all zero, so a
+   letter never seen before subtracts nothing.
+2. For each character `c` of `s`:
+   - `new_total = 2 * total - last[c]`
+   - `last[c] = total` — the total *before* this step
+   - `total = new_total`
+3. Return `total - 1`, dropping the empty subsequence.
+
+## Why the temporary is not optional
+
+`last[c]` must record $$|T_{i-1}|$$, the total from *before* the update. Folding
+the two assignments together — updating `total` first and then storing it into
+`last[c]` — records $$|T_i|$$ instead and subtracts too much on the next
+occurrence of that letter. It fails on the smallest repeat:
+
+| `s`        | correct | with the assignments swapped |
+| ---------- | ------- | ---------------------------- |
+| `"aba"`    | `6`     | `5`                          |
+| `"aaa"`    | `3`     | `1`                          |
+| `"abab"`   | `11`    | `7`                          |
+
+That is what the `new_total` / `newTotal` variable is buying: the three lines are
+a simultaneous update, not a sequence of three independent ones.
+
+# Worked example: `s = "aba"` → `6`
+
+| step | `c` | `2 * total` | `last[c]` | new `total` | $$T_i$$                     |
+| ---- | --- | ----------- | --------- | ----------- | --------------------------- |
+| 0    | —   | —           | —         | `1`         | `{""}`                      |
+| 1    | `a` | `2`         | `0`       | `2`         | `{"", "a"}`                 |
+| 2    | `b` | `4`         | `0`       | `4`         | `{"", "a", "b", "ab"}`      |
+| 3    | `a` | `8`         | `1`       | `7`         | the four above, plus `"aa"`, `"ba"`, `"aba"` |
+
+Step 3 is the whole idea in miniature. Appending `a` to
+`{"", "a", "b", "ab"}` yields `{"a", "aa", "ba", "aba"}`, and `"a"` is already in
+the set — one duplicate, which is precisely `last[a] = 1`, the total back when the
+first `a` was consumed. The answer is `7 - 1 = 6`, matching Example 2.
+
+# Modular arithmetic: the `+ MOD` is load-bearing
+
+Two details make the difference between a correct submission and a subtly wrong
+one, and both are invisible on small inputs.
+
+**The running total can go negative in Go and Rust.** Those languages define `%`
+by truncation toward zero, so a negative dividend gives a negative remainder.
+Once `total` has wrapped past $$10^9 + 7$$ it can be small while `last[c]` — an
+older, unrelated residue — is large, making `2 * total - last[c]` negative and
+leaving `total` negative. This is not hypothetical: across the 10153-case corpus
+below it happens on 228 of them, with the running total dipping to
+`-999973602`. It cannot happen before the total first exceeds the modulus, which
+takes about 30 doublings, and a search over random 26-letter strings found it
+starting as early as the 31st character — far enough in that hand-checked
+examples never reach it. Python is immune: its `%` floors, so `total` stays in
+$$[0, 10^9 + 7)$$, and there the `+ MOD` is merely belt and braces.
+
+**One `+ MOD` is enough, but only just.** Because `%` bounds the magnitude,
+`total` always satisfies $$-(10^9 + 7) < total < 10^9 + 7$$, so `total - 1 + MOD`
+lands in $$[0, 2 \cdot (10^9 + 7))$$ and the final `% MOD` normalises it. Every
+operation on the way is an addition or subtraction, so each residue stays
+congruent to the true count and the normalised result is exact. Drop the `+ MOD`
+and a legitimate input whose count is a multiple of the modulus returns `-1`.
+
+**Width.** Since $$|2 \cdot total - last[c]| < 3 \cdot (10^9 + 7) \approx 3 \times 10^9$$,
+the arithmetic needs 64 bits. That is why the Rust version accumulates in `i64`
+and casts only at the end, where the normalised value is below $$2^{31}$$ and the
+cast is lossless; a bare `i32` accumulator would overflow. Go's `int` is 64-bit on
+the judge, so it is safe as written.
+
+# Complexity
+
+- Time complexity: $$O(n)$$, where `n` is the length of `s` — one pass, constant
+  work per character.
+- Space complexity: $$O(1)$$ — 26 counters, independent of `n`.
+
+# Code
+
+## Go
+
+```go
+const MOD = 1_000_000_000 + 7
+
+func distinctSubseqII(s string) int {
+    last := [26]int{}
+    total := 1
+    for _, ch := range s {
+        c := ch - 'a'
+        newTotal := (2 * total - last[c]) % MOD
+        last[c] = total
+        total = newTotal
+    }
+    return (total - 1 + MOD) % MOD
+}
+```
+
+`range` over a `string` yields runes, so `c` is a `rune` used directly as an array
+index — legal, and correct here because the constraints promise lowercase ASCII.
+
+## Rust
+
+```rust
+const MOD: i64 = 1_000_000_000 + 7;
+
+impl Solution {
+    pub fn distinct_subseq_ii(s: String) -> i32 {
+        let mut last = [0; 26];
+        let mut total = 1_i64;
+        for ch in s.as_bytes() {
+            let c = (ch - b'a') as usize;
+            let new_total = (2 * total - last[c]) % MOD;
+            last[c] = total;
+            total = new_total;
+        }
+        ((total - 1 + MOD) % MOD) as i32
+    }
+}
+```
+
+`last` is inferred as `[i64; 26]` from `last[c] = total`, which keeps the whole
+computation in 64 bits.
+
+## Python
+
+```python
+MOD = 1_000_000_000 + 7
+
+class Solution:
+    def distinctSubseqII(self, s: str) -> int:
+        total, last = 1, [0] * 26
+        for ch in s:
+            c = ord(ch) - ord('a')
+            new_total = (2 * total - last[c]) % MOD
+            last[c] = total
+            total = new_total
+        return (total - 1 + MOD) % MOD
+```
+
+# Test cases
+
+| `s`             | answer      | why                                          |
+| --------------- | ----------- | -------------------------------------------- |
+| `"abc"`         | `7`         | Example 1 — all distinct, $$2^3 - 1$$        |
+| `"aba"`         | `6`         | Example 2 — one duplicate removed            |
+| `"aaa"`         | `3`         | Example 3 — only `"a"`, `"aa"`, `"aaa"`      |
+| `"a"`           | `1`         | shortest possible input                      |
+| `"aa"`          | `2`         | first repeat, smallest subtraction           |
+| `"abab"`        | `11`        | catches swapped `last[c]` / `total` updates  |
+| `"a" * 2000`    | `2000`      | maximal input, minimal answer                |
+| `"ab" * 1000`   | `694708213` \* | maximal input where the total wraps repeatedly |
+
+\* the modular residue; the true count is astronomically larger.
+
+All three implementations were checked against an exact big-integer reference —
+the same recurrence with unbounded integers, reduced modulo $$10^9 + 7$$ only at
+the very end — on a shared corpus of 10153 cases: the three examples, every string
+over `{a,b,c}` of length up to 8, 300 random strings up to the maximum length 2000
+over alphabets of 1, 2, 3, 10 and 26 letters, and six structured maximal inputs.
+All agreed on every case. The reference was itself validated against a brute force
+that materialises the set of subsequences, on all 9851 cases short enough to
+enumerate. The Rust build had overflow checks enabled and never tripped them,
+confirming the 64-bit accumulator claim.

README.md

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 ## Solutions index
 
-Total: **213** problems with at least one solution file.
+Total: **214** problems with at least one solution file.
 
 Solution links use variant names when multiple approaches or languages exist (`main` = `solution.md`, others = `solution-<variant>.md`).
 
@@ -211,7 +211,7 @@ Solution links use variant names when multiple approaches or languages exist (`m
 | 3876. Construct Uniform Parity Array II                                            | [Link](https://leetcode.com/problems/construct-uniform-parity-array-ii/)                                          | [main](Medium/3876.Construct-Uniform-Parity-Array-II/solution.md)                                                                                                                                                                                                                                       |
 | 3904. Smallest Stable Index II                                                     | [Link](https://leetcode.com/problems/smallest-stable-index-ii/)                                                   | [main](Medium/3904.Smallest-Stable-Index-II/solution.md)                                                                                                                                                                                                                                                |
 
-### Hard (35)
+### Hard (36)
 
 | Problem                                                          | LeetCode                                                                                          | Solution                                                                                                                                                                                                                                                                             |
 | ---------------------------------------------------------------- | ------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |
@@ -231,6 +231,7 @@ Solution links use variant names when multiple approaches or languages exist (`m
 | 726. Number of Atoms                                             | [Link](https://leetcode.com/problems/number-of-atoms/)                                            | [go](Hard/726.Number-of-Atoms/solution-go.md) · [main](Hard/726.Number-of-Atoms/solution.md)                                                                                                                                                                                         |
 | 773. Sliding Puzzle                                              | [Link](https://leetcode.com/problems/sliding-puzzle/)                                             | [main](Hard/773.Sliding-Puzzle/solution.md)                                                                                                                                                                                                                                          |
 | 862. Shortest Subarray with Sum at Least K                       | [Link](https://leetcode.com/problems/shortest-subarray-with-sum-at-least-k/)                      | [main](Hard/862.Shortest-Subarray-with-Sum-at-Least-K/solution.md)                                                                                                                                                                                                                   |
+| 940. Distinct Subsequences II                                    | [Link](https://leetcode.com/problems/distinct-subsequences-ii/)                                   | [main](Hard/940.Distinct-Subsequences-II/solution.md)                                                                                                                                                                                                                                |
 | 995. Minimum Number of K Consecutive Bit Flips                   | [Link](https://leetcode.com/problems/minimum-number-of-k-consecutive-bit-flips/)                  | [main](Hard/995.Minimum-Number-of-K-Consecutive-Bit-Flips/solution.md)                                                                                                                                                                                                               |
 | 1106. Parsing A Boolean Expression                               | [Link](https://leetcode.com/problems/parsing-a-boolean-expression/)                               | [main](Hard/1106.Parsing-A-Boolean-Expression/solution.md)                                                                                                                                                                                                                           |
 | 1301. Number of Paths with Max Score                             | [Link](https://leetcode.com/problems/number-of-paths-with-max-score/)                             | [main](Hard/1301.Number-of-Paths-with-Max-Score/solution.md)                                                                                                                                                                                                                       |

SUMMARY.md

@@ -220,6 +220,7 @@
 * [726. Number of Atoms](Hard/726.Number-of-Atoms/solution.md)
 * [773. Sliding Puzzle](Hard/773.Sliding-Puzzle/solution.md)
 * [862. Shortest Subarray with Sum at Least K](Hard/862.Shortest-Subarray-with-Sum-at-Least-K/solution.md)
+* [940. Distinct Subsequences II](Hard/940.Distinct-Subsequences-II/solution.md)
 * [995. Minimum Number of K Consecutive Bit Flips](Hard/995.Minimum-Number-of-K-Consecutive-Bit-Flips/solution.md)
 * [1106. Parsing A Boolean Expression](Hard/1106.Parsing-A-Boolean-Expression/solution.md)
 * [1301. Number of Paths with Max Score](Hard/1301.Number-of-Paths-with-Max-Score/solution.md)

_sidebar.md

@@ -213,6 +213,7 @@
   - [726. Number of Atoms](Hard/726.Number-of-Atoms/solution.md)
   - [773. Sliding Puzzle](Hard/773.Sliding-Puzzle/solution.md)
   - [862. Shortest Subarray with Sum at Least K](Hard/862.Shortest-Subarray-with-Sum-at-Least-K/solution.md)
+  - [940. Distinct Subsequences II](Hard/940.Distinct-Subsequences-II/solution.md)
   - [995. Minimum Number of K Consecutive Bit Flips](Hard/995.Minimum-Number-of-K-Consecutive-Bit-Flips/solution.md)
   - [1106. Parsing A Boolean Expression](Hard/1106.Parsing-A-Boolean-Expression/solution.md)
   - [1301. Number of Paths with Max Score](Hard/1301.Number-of-Paths-with-Max-Score/solution.md)