From 000c717980f1b53c285bccd8bac0fb17c95c2247 Mon Sep 17 00:00:00 2001 From: orzcyand1317 Date: Sun, 24 Mar 2019 00:21:25 +0800 Subject: [PATCH] add manhattan-dis.png --- docs/misc/distance.md | 19 +++++++++---------- docs/misc/images/manhattan-dis.png | Bin 0 -> 115067 bytes 2 files changed, 9 insertions(+), 10 deletions(-) create mode 100644 docs/misc/images/manhattan-dis.png diff --git a/docs/misc/distance.md b/docs/misc/distance.md index cb26bd3e..650413e4 100644 --- a/docs/misc/distance.md +++ b/docs/misc/distance.md @@ -40,7 +40,6 @@ $$ \end{gathered} $$ - [NOIP2017 提高组 奶酪](https://www.luogu.org/problemnew/show/P3958) 就运用了这一知识,可以作为欧氏距离的例题。 以此类推,我们就得到了 $n$ 维空间中欧氏距离的距离公式:对于 $\vec A(x_{11}, x_{12}, \cdots,x_{1n}) ,~ \vec B(x_{21}, x_{22}, \cdots,x_{2n})$ ,有 @@ -64,7 +63,7 @@ $$ 观察下图: -![https://cdn.luogu.org/upload/pic/47609.png](https://cdn.luogu.org/upload/pic/47609.png) +![](./images/manhattan-dis.png) 在 $A,B$ 间,黄线、橙线都表示曼哈顿距离,而红线、蓝线表示等价的曼哈顿距离,绿线表示欧氏距离。 @@ -93,25 +92,25 @@ $$ 曼哈顿距离是一个非负数。 - $d(i,j)\geq 0$ + $d(i,j)\geq 0$ - **统一性** 点到自身的曼哈顿距离为 $0$ 。 - $d(i,i) = 0​$ + $d(i,i) = 0​$ - **对称性** $A$ 到 $B$ 与 $B$ 到 $A$ 的曼哈顿距离相等,且是对称函数。 - $d(i,j) = d(j,i)$ + $d(i,j) = d(j,i)$ - **三角不等式** 从点 $i$ 到 $j$ 的直接距离不会大于途经的任何其它点 $k$ 的距离。 - $d(i,j)\leq d(i,k)+d(k,j)$ + $d(i,j)\leq d(i,k)+d(k,j)$ **例题:[洛谷 P5098](https://www.luogu.org/problemnew/show/P5098)** @@ -121,9 +120,9 @@ $$ 根据题意,对于式子 $|x_1-x_2|+|y_1-y_2|$ ,我们可以假设 $x_1 - x_2 \geq 0$ ,根据 $y_1 - y_2$ 的符号分成两种情况: -- $(y_1 - y_2 \geq 0)\rightarrow |x_1-x_2|+|y_1-y_2|=x_1 + y_1 - (x_2 + y_2)$ +- $(y_1 - y_2 \geq 0)\rightarrow |x_1-x_2|+|y_1-y_2|=x_1 + y_1 - (x_2 + y_2)$ -- $(y_1 - y_2 \lt 0)\rightarrow |x_1-x_2|+|y_1-y_2|=x_1 - y_1 - (x_2 - y_2)$ +- $(y_1 - y_2 \lt 0)\rightarrow |x_1-x_2|+|y_1-y_2|=x_1 - y_1 - (x_2 - y_2)$ 只要分别求出 $x+y, x-y$ 的最大值和最小值即能得出答案。 @@ -292,7 +291,7 @@ $$ 我们考虑将题目所求的 曼哈顿距离 转化为 切比雪夫距离,即把每个点的坐标 $(x,y)$ 变为 $(x + y, x - y)$ 。 -所求的答案就变为 $max_{i,j\in n}\begin{Bmatrix} max\begin{Bmatrix} |x_i - x_j|,|y_i - y_j|\end{Bmatrix}\end{Bmatrix}$ +所求的答案就变为 $max_{i,j\in n}\begin{Bmatrix} max\begin{Bmatrix} |x_i - x_j|,|y_i - y_j|\end{Bmatrix}\end{Bmatrix}$ 现要使得横坐标之差和纵坐标之差最大,只需要预处理出 $x,y$ 的最大值和最小值即可。 @@ -334,7 +333,7 @@ int main() { 一般地,我们定义平面上两点 $A(x_1, y_1)$ , $B(x_2, y_2)$ 之间的 $L_m$ 距离为 -$d(L_m) = (|x_1-x_2|^m+|y1-y2|^m)^{\frac{1}{m}}$ +$d(L_m) = (|x_1-x_2|^m+|y1-y2|^m)^{\frac{1}{m}}$ 特殊的, $L_2$ 距离就是欧几里得距离, $L_1$ 距离就是曼哈顿距离。 diff --git a/docs/misc/images/manhattan-dis.png b/docs/misc/images/manhattan-dis.png new file mode 100644 index 0000000000000000000000000000000000000000..500c37c6c997a95fb5f953ab09de11d86399c088 GIT binary patch literal 115067 zcmeFZc{r5++dtec32jnYTS>Me#F(j6+C=tU2w4V07|WP^Bt=MtBr{3&WwOlJjfg4R zkYzBKm~1n~J{YrjE}#3pzxVz9=z0El{`wuq?{*xvYk4o{`+c3S^L(AJYhD=K(-quv zY|oZ0TLf?2yk@v%%Z_#6o4tD%aAZVSL=X72)z?t>>XyQG@hRXRe(#$WzFW3vyxaKR zdJr@e0bF3=YHa@4T<@-?qZd@s{-KwHlOi1I4P3ot3ka?W{0Vh>Y%dLmdU*P3!nIHQ zeT63Q_r_tR6ViWQ^4MMbgt?x9w2qgLlk`PJRYm0!;62jP(jcFQ&YFhTZv5x%z<=5& 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