From 005600b7fb520667e2782f3fbbc693be3298b0d8 Mon Sep 17 00:00:00 2001 From: Jikuhara Date: Tue, 3 Jun 2025 15:15:47 +0900 Subject: [PATCH] =?UTF-8?q?=E4=BD=9C=E6=A5=AD=E5=AE=8C=E4=BA=86=EF=BD=9E?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 53_maximum_subarray/memo.md | 204 ++++++++++++++++++ 53_maximum_subarray/step1.c | 41 ++++ 53_maximum_subarray/step2.c | 28 +++ 53_maximum_subarray/step3.c | 25 +++ ...{292A41D3-5729-4613-B35D-E42A39C15863}.png | Bin 0 -> 56895 bytes ...{36527AFF-BB15-4B4D-9ACB-B004040EBF56}.png | Bin 0 -> 54920 bytes ...{EA17A920-E9C3-44E7-9D6F-FE87CB3DE6DB}.png | Bin 0 -> 55255 bytes 7 files changed, 298 insertions(+) create mode 100644 53_maximum_subarray/memo.md create mode 100644 53_maximum_subarray/step1.c create mode 100644 53_maximum_subarray/step2.c create mode 100644 53_maximum_subarray/step3.c create mode 100644 53_maximum_subarray/{292A41D3-5729-4613-B35D-E42A39C15863}.png create mode 100644 53_maximum_subarray/{36527AFF-BB15-4B4D-9ACB-B004040EBF56}.png create mode 100644 53_maximum_subarray/{EA17A920-E9C3-44E7-9D6F-FE87CB3DE6DB}.png diff --git a/53_maximum_subarray/memo.md b/53_maximum_subarray/memo.md new file mode 100644 index 0000000..c5e6832 --- /dev/null +++ b/53_maximum_subarray/memo.md @@ -0,0 +1,204 @@ +# 53. maximum-subarray + +https://leetcode.com/problems/maximum-subarray/description/ + +## Comments +最大値を求める問題か。 +全ての部分配列を求めて、最大値を求めるのはO(n^2)かな。 +流石に無理そう。 + +Inputの数は、10^5までやし、 +O(n^2)ではなくて、O(n)で求める必要があるなぁ。 + +Maxを求める方法は? +まずは、1個1個のマスに対して、最大値を求める。 +最大値は、前のマスの最大値(自分自身を除く)OR 自分自身を含めた部分列の和、の2択かな。 + +もっと具体的に考えると、1個1個のマスに対して、最大値を求める方法は、 +1. そのマスを含む部分配列の最大値を求める。 +2. そのマスを含まない部分配列の最大値を求める。 +3. そのマスを含む部分配列の最大値と、含まない部分配列の最大値を比較して、最大値を求める。 + +直前の値を利用するのなら動的計画法かな。 +一旦まずは、dpにcopyした方が楽そう? +実装できそうやし手を動かしてみよう。 + + +Main関数で何個かテストしてみたところ、あってそうな感じする。提出してみよう。 + +![提出結果]({EA17A920-E9C3-44E7-9D6F-FE87CB3DE6DB}.png) + +OK! +けど計算時間もうちょい短く出来そう? +memcopyとdpと最大値で3回ループしてるからか。 + +1週に減らせへんかなぁ。 +とりあえず、memcopyは必要無さそうか。 +んで、下のwhile文は、一回目に統合しよう。 + +![二回目の提出結果]({36527AFF-BB15-4B4D-9ACB-B004040EBF56}.png) + +できた! +けど可読性は少し悪くなったかぁ。 +これなら前の実装の方が良いかも。 +てか、これmemcopyしてるのと同じか?? + +memory量もすこし減らせそう。dp配列2つしか使用してないし。 + +```c +int maxSubArray(int *nums, int numsSize) +{ + int *dp; + int i; + int max_sum; + + dp = malloc(sizeof(int) * 2); + if (!dp) + return (0); + dp[0] = nums[0]; + max_sum = dp[0]; + i = 1; + while (i < numsSize) + { + dp[1] = nums[i]; + if (dp[1] < dp[0] + nums[i]) + dp[1] = dp[0] + nums[i]; + if (max_sum < dp[1]) + max_sum = dp[1]; + dp[0] = dp[1]; + i++; + } + free(dp); + return (max_sum); +} +``` +![3回目の提出結果]({292A41D3-5729-4613-B35D-E42A39C15863}.png) + +うーん。 +改善できたけど、可読性は悪くなった。 +他の人の答案を見てみよう。 + +命名の仕方いい人見つけた。 +dpって命名良くないな。意味を込めたほうが良さそう。 +あと、直前の値(dp[0])もいらんのかなぁ。 + +```c +int maxSubArray(int* nums, int numsSize) { + int currentSum = nums[0]; + int maxSum = nums[0]; + for (int i = 1; i < numsSize; i++) { + if (currentSum < 0) + currentSum = nums[i]; + else + currentSum += nums[i]; + + if (currentSum > maxSum) + maxSum = currentSum; + } + return maxSum; +} +``` +参考にしたコード。 + +なんでこれでいけんのや?? +直前の値が負なら場合分け。 +うーん。いまいち分からん。 +ChatGPTに聞いてみるか。 + +```md +### 何をしているコードか ―― “最大部分配列問題” の高速解(Kadane 法) + +与えられた整数配列 `nums` の **連続した区間のうち、要素の合計が最大になるもの** を 1 回の走査で求めています。 +ポイントは + +| 変数 | 役割 | +| ------------ | ------------------------------- | +| `currentSum` | **いま見ている位置 i を終点とする区間**で取り得る最大和 | +| `maxSum` | これまでに見つかった **全区間のうちの最大和** | + +--- + +## 行ごとの意味 + +```c +int currentSum = nums[0]; // 最初の区間は要素0だけ +int maxSum = nums[0]; + +for (int i = 1; i < numsSize; i++) { + /* 直前までの区間が負なら、拡張すると必ず損をする */ + if (currentSum < 0) + currentSum = nums[i]; // 区間をリセット(ここから新しく始める) + else + currentSum += nums[i]; // 区間をそのまま右へ 1 要素伸ばす + + /* ここまでで得られた最大値を更新 */ + if (currentSum > maxSum) + maxSum = currentSum; +} +return maxSum; + + +--- + +## “なぜ負ならリセット?” を直感で + +* 連続区間の末尾に新しい要素 `nums[i]` を足すとき、 + もし **直前の合計 `currentSum` が負** なら + `currentSum + nums[i] < nums[i]` が必ず成り立つ ⟶ *必ず悪化* + → だったら “マイナスごと切り捨てて” `nums[i]` からやり直す方が得。 +* 正の合計を持っていれば、伸ばした方が大きくなる *可能性* があるので、そのまま足す。 + +--- + +## 具体例シミュレーション + +配列 `[-2, 1, -3, 4, -1, 2, 1, -5, 4]` を手で追うと: + +| i | nums\[i] | currentSum の計算 | currentSum | maxSum | +| - | -------- | --------------------- | ---------- | ------ | +| 0 | -2 | 初期値 | **-2** | **-2** | +| 1 | 1 | -2 < 0 → リセット → 1 | 1 | 1 | +| 2 | -3 | 1 ≥ 0 → 1 + (-3) = -2 | -2 | 1 | +| 3 | 4 | -2 < 0 → リセット → 4 | 4 | **4** | +| 4 | -1 | 4 ≥ 0 → 4 + (-1) = 3 | 3 | 4 | +| 5 | 2 | 3 ≥ 0 → 3 + 2 = 5 | 5 | **5** | +| 6 | 1 | 5 ≥ 0 → 5 + 1 = 6 | 6 | **6** | +| 7 | -5 | 6 ≥ 0 → 6 + (-5) = 1 | 1 | 6 | +| 8 | 4 | 1 ≥ 0 → 1 + 4 = 5 | 5 | 6 | + +最終結果 `maxSum = 6`(区間 `[4, -1, 2, 1]`)。 + +--- + +## 性能 + +* **時間計算量**:O(N) ― 配列を 1 回なめるだけ +* **追加メモリ**:O(1) ― 変数 2 つだけ + +--- + +## より読みやすく書き換える例 + +```c +int maxSubArray(int *a, int n) { + int best = a[0]; + int here = a[0]; + + for (int i = 1; i < n; ++i) { + here = (here < 0) ? a[i] : here + a[i]; + best = (best < here) ? here : best; + } + return best; +} + +同じ考え方で、全要素が負でも安全に動きます。 +理解の助けになれば幸いです! +``` + + +なるほど!! +めっちゃ単純な話で、直前が負なら、足した結果は必ず元の数字より小さくなるもんな。 + +Step3に再実装しといた。 +まあほぼfor文をwhile文に変えただけやけど。 + diff --git a/53_maximum_subarray/step1.c b/53_maximum_subarray/step1.c new file mode 100644 index 0000000..812fcc6 --- /dev/null +++ b/53_maximum_subarray/step1.c @@ -0,0 +1,41 @@ +#include +#include +#include + +int maxSubArray(int *nums, int numsSize) +{ + int *dp; + int i; + int max_sum; + + dp = malloc(sizeof(int) * numsSize); + if (!dp) + return (0); + memcpy(dp, nums, sizeof(int) * numsSize); + i = 1; + while (i < numsSize) + { + if (dp[i] < dp[i - 1] + nums[i]) + dp[i] = dp[i - 1] + nums[i]; + i++; + } + max_sum = dp[0]; + i = 0; + while (i < numsSize) + { + if (max_sum < dp[i]) + max_sum = dp[i]; + i++; + } + free(dp); + return (max_sum); +} + +int main(void) +{ + int nums[] = {1, 2, -3, 4}; + int numsSize = 4; + + printf("%i\n", maxSubArray(nums, numsSize)); + return (0); +} \ No newline at end of file diff --git a/53_maximum_subarray/step2.c b/53_maximum_subarray/step2.c new file mode 100644 index 0000000..afece4b --- /dev/null +++ b/53_maximum_subarray/step2.c @@ -0,0 +1,28 @@ +#include +#include +#include + +int maxSubArray(int *nums, int numsSize) +{ + int *dp; + int i; + int max_sum; + + dp = malloc(sizeof(int) * numsSize); + if (!dp) + return (0); + dp[0] = nums[0]; + max_sum = dp[0]; + i = 1; + while (i < numsSize) + { + dp[i] = nums[i]; + if (dp[i] < dp[i - 1] + nums[i]) + dp[i] = dp[i - 1] + nums[i]; + if (max_sum < dp[i]) + max_sum = dp[i]; + i++; + } + free(dp); + return (max_sum); +} \ No newline at end of file diff --git a/53_maximum_subarray/step3.c b/53_maximum_subarray/step3.c new file mode 100644 index 0000000..2c48272 --- /dev/null +++ b/53_maximum_subarray/step3.c @@ -0,0 +1,25 @@ +#include +#include +#include + +int maxSubArray(int *nums, int numsSize) +{ + int currentSum; + int i; + int maxSum; + + currentSum = nums[0]; + maxSum = nums[0]; + i = 1; + while 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