From 4a4ae1242fbc2b5fe86fe61aa5677523a683682a Mon Sep 17 00:00:00 2001 From: Siger Young Date: Mon, 8 Oct 2018 00:53:00 +0800 Subject: [PATCH] feat: Update the content of DP on DAG --- docs/dp/dag.md | 115 +++++++++++++++++++++++++++++++++++++++++ docs/dp/images/dag-babylon.png | Bin 0 -> 73868 bytes 2 files changed, 115 insertions(+) create mode 100644 docs/dp/images/dag-babylon.png diff --git a/docs/dp/dag.md b/docs/dp/dag.md index 8b137891..843ee97a 100644 --- a/docs/dp/dag.md +++ b/docs/dp/dag.md @@ -1 +1,116 @@ +DAG 即[有向无环图](/graph/dag),一些实际问题中的二元关系都可使用 DAG 来建模。 +## 例子 +以这道题为例子,来分析一下 DAG 建模的过程。 + +??? note "例题 [UVa 437 巴比伦塔 The Tower of Babylon](https://cn.vjudge.net/problem/UVA-437)" + 有 $n (n\leqslant 30)$ 种砖块,已知三条边长,每种都有无穷多个。要求选一些立方体摞成一根尽量高的柱子(每个砖块可以自行选择一条边作为高),使得每个砖块的底面长宽分别严格小于它下方砖块的底面长宽,求塔的最大高度。 + +### 建立 DAG + +由于每个砖块的底面长宽分别严格小于它下方砖块的底面长宽,因此不难将这样一种关系作为建图的依据,而本题也就转化为最长路问题。 + +也就是说如果砖块 $j$ 能放在砖块 $i$ 上,那么 $i$ 和 $j$ 之间存在一条边 $(i, j)$,且边权就是砖块 $j$ 所选取的高。 + +本题的另一个问题在于每个砖块的高有三种选法,怎样建图更合适呢? + +不妨将每个砖块拆解为三种堆叠方式,即将一个砖块分解为三个砖块,每一个拆解得到的砖块都选取不同的高。 + +初始的起点是大地,大地的底面是无穷大的,则大地可达任意砖块,当然我们写程序时不必特意写上无穷大。 + +假设有两个砖块,三条边分别为 $31, 41, 59$ 和 $33, 83, 27$ ,那么整张 DAG 应该如下图所示。 +![](./images/dag-babylon.png) + +图中蓝实框所表示的是一个砖块拆解得到的一组砖块,之所以用 $\{\}$ 表示底面边长,是因为砖块一旦选取了高,底面边长就是无序的。 + +图中黄虚框表示的是重复计算部分,为下文做铺垫。 + +### 转移 + +题目要求的是塔的最大高度,已经转化为最长路问题,其起点上文已指出是大地,那么终点呢? + +显然终点已经自然确定,那就是某砖块上不能再搭别的砖块的时候。 + +之前在图上标记的黄虚框表明有重复计算,下面我们开始考虑转移方程。 + +显然,砖块一旦选取了高,那么这块砖块上最大能放的高度是确定的。 + +某个砖块 $i$ 有三种堆叠方式分别记为 $0, 1, 2$,那么对于砖块 $i$ 和其堆叠方式 $r$ 来说则有如下转移方程 + +$d(i, r) = \max\left\{d(j, r') + h'\right\}$ + +其中 $j$ 是所有那些在砖块 $i$ 以 $r$ 方式堆叠时可放上的砖块,$r'$ 对应 $j$ 此时的摆放方式,也就确定了此时唯一的高度 $h'$。 + +在实际编写时,将所有 $d(i, r)$ 都初始化为 $-1$,表示未计算过。 + +在试图计算前,如果发现已经计算过,直接返回保存的值;否则就按步计算,并保存。 + +最终答案是所有 $d(i, r)$ 的最大值。 + +### 题解 +```c++ +#include +#include +#define MAXN (30+5) +#define MAXV (500+5) +#define MAX(a,b) (((a)>(b))?(a):(b)) +int d[MAXN][3]; +int x[MAXN], y[MAXN], z[MAXN]; +int babylon_sub(int c, int rot, int n) +{ + if(d[c][rot] != -1) + { + return d[c][rot]; + } + d[c][rot] = 0; + int base1, base2; + if(rot == 0) { base1 = x[c]; base2 = y[c]; } + if(rot == 1) { base1 = y[c]; base2 = z[c]; } + if(rot == 2) { base1 = x[c]; base2 = z[c]; } + for(int i = 0; i < n; i++) + { + if((x[i] < base1 && y[i] < base2) || (y[i] < base1 && x[i] < base2)) + d[c][rot] = MAX(d[c][rot], babylon_sub(i, 0, n) + z[i]); + if((y[i] < base1 && z[i] < base2) || (z[i] < base1 && y[i] < base2)) + d[c][rot] = MAX(d[c][rot], babylon_sub(i, 1, n) + x[i]); + if((x[i] < base1 && z[i] < base2) || (z[i] < base1 && x[i] < base2)) + d[c][rot] = MAX(d[c][rot], babylon_sub(i, 2, n) + y[i]); + } + return d[c][rot]; +} +int babylon(int n) +{ + for(int i = 0; i < n; i++) + { + d[i][0] = -1; + d[i][1] = -1; + d[i][2] = -1; + } + int r = 0; + for(int i = 0; i < n; i++) + { + r = MAX(r, babylon_sub(i, 0, n) + z[i]); + r = MAX(r, babylon_sub(i, 1, n) + x[i]); + r = MAX(r, babylon_sub(i, 2, n) + y[i]); + } + return r; +} +int main() +{ + int t = 0; + while(true) + { + int n; + std::cin >> n; + if(n == 0) break; + t++; + for(int i = 0; i < n; i++) + { + std::cin >> x[i] >> y[i] >> z[i]; + } + std::cout << "Case " << t << ":" << " maximum height = " << babylon(n); + std::cout << std::endl; + } + return 0; +} +``` \ No newline at end of file diff --git a/docs/dp/images/dag-babylon.png b/docs/dp/images/dag-babylon.png new file mode 100644 index 0000000000000000000000000000000000000000..e8c4ad4865aaea59f551e7bcb4043381f25f5c16 GIT binary patch literal 73868 zcmbTd1yGw^^#2Kk0>z4ZaEe252(H1cIFwS{H4wCv7MDQKqQxo2T>`YYyA%r!Mbjb$ zT5QAn|LyMV?(FQ$W+uZxhUdA@J?9?zo=;+QwLt{9G`MJJXas7iO8RJMPu`&3#W+t; zf2oTePeT1b_t6I_pjA)M9iV=}bdcAQM}3rtcl-7!>i1`!s-`|@Xe8jrH~N}AgFhPD zLy4M_yn(;v@ttWBi*eR^%-3S0MB!v7R~1pg*W`4jTnl9y>)vss@mF%-g@)%{%!&?aq4qFg2(sYtNLyWytDQAa;7ZX z-f=W0vht_?p%6r z?y#7_VZzxnCy)&F=WlqBXPT3Fa{Y7YufF!T5D@9*O}hy%!(r%tm&PhT2- zNpl_tma$O&Sd5+ZitpbysS?X&(gu}8PYb|m(!Rz+djk3Rs!vJEwyEM~^}-6l!AU%G 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