From 7776b90acfef6e406f79725e7282dcc88927c0ba Mon Sep 17 00:00:00 2001 From: kazukiii Date: Mon, 30 Sep 2024 00:47:40 -0700 Subject: [PATCH] =?UTF-8?q?step1,=20step2,=20step3=E3=82=92=E8=BF=BD?= =?UTF-8?q?=E5=8A=A0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- arai60/paint-fence/README.md | 117 ++++++++++++++++++++++++++++ arai60/paint-fence/images/image.png | Bin 0 -> 185885 bytes 2 files changed, 117 insertions(+) create mode 100644 arai60/paint-fence/README.md create mode 100644 arai60/paint-fence/images/image.png diff --git a/arai60/paint-fence/README.md b/arai60/paint-fence/README.md new file mode 100644 index 0000000..5dc3ce4 --- /dev/null +++ b/arai60/paint-fence/README.md @@ -0,0 +1,117 @@ +## Step1 +- 以前はファイルを分けていたが、今回から全ての思考過程を一つのmdファイルにまとめてみる + +- 考えたこと + - 場合分けして、漸化式を立式 + - あとはテーブルを作って bottom-up DP + - いつも書く書き方だと2行のテーブルを作るが、可読性を考えてそれぞれ名前を付けて工夫してみた + - 変数名は迷ったが、`num_ways_same` と `num_ways_diff` としてみた。`same` と `diff` だけでもいいかもしれない +```python +class Solution: + def numWays(self, n: int, k: int) -> int: + # when the (i-1)-th and i-th posts are same colors + num_ways_same = [0] * (n + 1) + # when the (i-1)-th and i-th posts are different colors + num_ways_diff = [0] * (n + 1) + + # recurrence relation + # num_ways_same[i + 1] = num_ways_diff[i], num_ways_diff[i + 1] = (num_ways_same[i] + num_ways_diff[i]) * (k - 1) + # initial conditions + # num_ways_diff[1] = k, num_ways_same[1] = 0 + + num_ways_same[1] = 0 + num_ways_diff[1] = k + for i in range(1, n): + num_ways_same[i + 1] = num_ways_diff[i] + num_ways_diff[i + 1] = (num_ways_same[i] + num_ways_diff[i]) * (k - 1) + return num_ways_same[-1] + num_ways_diff[-1] +``` + +- 考えたこと2 + - 今回のarai60のDP演習では top-down DP も簡単に書けるように練習したい + - 以前、Pramp で模擬面接をしたときに bottom-up DP を書いたことがあるが説明に苦労した + - 可読性の観点では、top-down DP(メモ化再帰)の方に分があると思っている + +```python +class Solution: + def numWays(self, n: int, k: int) -> int: + @cache + def num_ways_helper(n, k) -> tuple[int]: + if n == 1: + return 0, k + same, diff = num_ways_helper(n - 1, k) + new_same = diff + new_diff = (same + diff) * (k - 1) + return new_same, new_diff + + return sum(num_ways_helper(n, k)) +``` + +- 考えたこと3 + - ここで今回の場合だと、n -> n + 1 の遷移しかないため、配列の使いまわしが出来ることに気づいた + - メモリ最適化までは面接で求められている気がするので、初手の段階でちゃんと書けるようにしておきたい + - 変数名を `same` と `diff` にしてシンプルにしてみた + +```python +class Solution: + def numWays(self, n: int, k: int) -> int: + same = 0 + diff = k + for _ in range(n - 1): + next_same = diff + next_diff = (same + diff) * (k - 1) + same = next_same + diff = next_diff + return same + diff +``` + + +## Step2 +- 過去ログを参照 + - https://github.com/TORUS0818/leetcode/pull/32 + - https://github.com/Yoshiki-Iwasa/Arai60/pull/44 + >あー、あと、この問題、行列の n 乗なので、そういう方法でも計算できるはずです。 + - たしかに、行列のn乗として表現出来る + - 行列累乗は繰り返し二乗法で O(log n) で計算出来るので本質的な計算量削減につながりそう + - https://github.com/goto-untrapped/Arai60/pull/44 + >前と違う色を塗り分ける時は、1個前までの組み合わせ数×(k-1)通りで、前と同じ色を塗る時は2個前までの組み合わせ数×(k-1)×1通り、の和 + - 実は、fibonacci numbers pattern に落とし込める + - いきなり思いつくことは難しそうな印象を受けた + + >フィボナッチ数列に関連して、 時間計算量 O(log n) の解法は思いつきますか? + - こちらも行列累乗の話かな + - 行列演算でもやってみる + +```python +import numpy as np + +class Solution: + def numWays(self, n: int, k: int) -> int: + A = np.array([[0, 1], [k - 1, k - 1]]) + v = np.array([0, k]) + return np.sum(np.linalg.matrix_power(A, n - 1) @ v) +``` + +![](images/image.png) + +- 行列累乗はnumpyで + - matrix_power + - https://numpy.org/doc/2.0/reference/generated/numpy.linalg.matrix_power.html +- 計算量的には O(n) -> O(log n) と落ちるはずだがleetcode上ではむしろ遅くなった + - n <= 50 と小さいからかな + +## Step3 +- メモリ最適化した bottom-up DP で +```python +class Solution: + def numWays(self, n: int, k: int) -> int: + same = 0 + diff = k + for _ in range(n - 1): + next_same = diff + next_diff = (k - 1) * (same + diff) + same = next_same + diff = next_diff + return same + diff +``` + diff --git a/arai60/paint-fence/images/image.png b/arai60/paint-fence/images/image.png new file mode 100644 index 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