From f0acd83923fd3857ca78b207d536ba39c551f9cd Mon Sep 17 00:00:00 2001 From: lifei <1058235935@qq.com> Date: Tue, 1 Dec 2020 23:37:44 +0800 Subject: [PATCH] =?UTF-8?q?=E5=AE=8C=E6=88=90=E6=A0=88=E5=92=8C=E9=98=9F?= =?UTF-8?q?=E5=88=97?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ch3/README.md | 43 +++++++++ ch3/matrix/README.md | 146 +++++++++++++++++++++++++++++++ ch3/matrix/cross-link.png | Bin 0 -> 606308 bytes ch3/stack-applications/README.md | 6 +- 4 files changed, 192 insertions(+), 3 deletions(-) create mode 100644 ch3/matrix/README.md create mode 100644 ch3/matrix/cross-link.png diff --git a/ch3/README.md b/ch3/README.md index 5002ec3..b7d1356 100644 --- a/ch3/README.md +++ b/ch3/README.md @@ -274,3 +274,46 @@ $$ - 太多层递归可能会导致栈溢出。 - 可能包含很多重复计算。 + +## 5. 队列的应用 + +- 树的层次遍历 +- 图的广度优先遍历 +- 多个进程争抢着使用优先的系统资源时,FCFS(First Come First Service,先来先服务)是一种常用策略。 + +## 6. [矩阵的压缩存储](matrix/README.md) + +数组的存储结构: + +- 一维数组 +- 二维数组 + +特殊矩阵: + +- 对称矩阵 + - 特点:对方阵中的任意一个元素,有 $a_{i,j}=a_{j,i}$。 + - 压缩:只存主对角线+下三角区(或主对角线+上三角区)。 +- 三角矩阵 + - 特点:上三角区全为常量(下三角矩阵);或下三角区全为常量(上三角矩阵)。 + - 压缩:按行优先/列优先规则依次存储非常量区域,并在最后一个位置存放常量 $c$。 +- 三对角矩阵 + - 特点:当 $|i-j|>1$ 时,$a_{i,j}=0$ ($1 \leq i, j \leq n$)。 + - 压缩:按行优先/列优先规则依次存储带状区域。 +- 稀疏矩阵 + - 特点:非零元素个数远少于零元素个数。 + - 压缩:只存储非零元素 + - 顺序存储:顺序存储三元组 `<行,列,值>`。 + - 链式存储:十字链表法。 + +常见考题: + +- 矩阵的压缩存储需要多长的数组。 +- 由矩阵行列号 $$ 推出对应的数组下标号 $k$。 +- 由 $k$ 推出 $$ + - 如何处理不等式中的“刚好大于等于/小于等于”。 + - “向上取整/向下取整”。 +- 易错点: + - 存储上三角?下三角? + - 行优先?列优先? + - 矩阵元素的下标从 0?1?开始 + - 数组下标从 0?1?开始 diff --git a/ch3/matrix/README.md b/ch3/matrix/README.md new file mode 100644 index 0000000..7ba1686 --- /dev/null +++ b/ch3/matrix/README.md @@ -0,0 +1,146 @@ +# 矩阵的压缩存储 + +## 1. 一维数组 + +个数组元素大小相同,且物理上连续存放。 + +数据元素 $s[i]$ 存放地址为 $LOC+i*sizeof(ElemType)$。 + +> 除非题目特别说明,否则数组下标默认从 0 开始。 + +## 2. 二维数组 + +$b[M][N]$ + +- 行优先存储。$b[i][j]$ 的存储地址为 $LOC+(i*N+j)*sizeof(ElemType)$。 +- 列优先存储。$b[i][j]$ 的存储地址为 $LOC+(j*M+i)*sizeof(ElemType)$。 + +## 3. 对称矩阵 + +若 $n$ 阶**方阵**中任意一个元素 $a_{i,j}$ 都有 $a_{i,j}=a_{j,i}$,则该矩阵为对称矩阵。 + +- 主对角线 +- 上三角区 +- 下三角区 + +存储: + +- 普通存储:$n \times n$ 二维数组 +- 压缩存储策略: + - 只存储主对角线+下三角区 + - 只存储主对角线+上三角区 + +按照**行优先**原则将各个元素存入一维数组中,该一维数组长度为 $\frac{n \times (n+1)}{2}$。 + +$$ +B[0],B[1],B[2],B[3],...,B[\frac{n \times (n+1)}{2}-1] +$$ + +映射函数: + +- $i \geq j$ 情况,$a_{i,j}$ 对应 $B[\frac{i \times (i-1)}{2}+j-1]$ +- $i \lt j$ 情况,$a_{i,j}$ 对应 $B[\frac{j \times (j-1)}{2}+i-1]$ + +出题方法: + +- 存储上三角?下三角? +- 行优先?列优先? +- 矩阵元素的下标从 0?1?开始 +- 数组下标从 0?1?开始 + +## 4. 三角矩阵 + +- 下三角矩阵:除了主对角线和下三角区,其余的元素都相同。 +- 上三角矩阵:除了主对角线和上三角区,其余的元素都相同。 + +### 4.1. 下三角矩阵的映射函数 + +$$ +B[0],B[1],B[2],B[3],...,B[\frac{n \times (n+1)}{2}-1],B[\frac{n \times (n+1)}{2}] +$$ + +- $i \geq j$ 情况,$a_{i,j}$ 对应 $B[\frac{i \times (i-1)}{2}+j-1]$ +- $i \lt j$ 情况,$a_{i,j}$ 对应 $B[\frac{n \times (n+1)}{2}]$ + +### 4.2. 上三角矩阵的映射函数 + +- $i \leq j$ 情况,$a_{i,j}$ 对应 $B[\frac{(i-1) \times (2n-i+2)}{2}+(j-i)]$ +- $i \gt j$ 情况,$a_{i,j}$ 对应 $B[\frac{n \times (n+1)}{2}]$ + +## 5. 三对角矩阵 + +三对角矩阵,又称带状矩阵。 + +当 $|i-j|>1$ 时,$a_{i,j}=0$ ($1 \leq i, j \leq n$)。 + +### 5.1. 从矩阵到数组 + +行优先原则,只存储带状部分。数组大小为 $3n-2$。 + +$$ +a_{1,1},a_{1,2}, +a_{2,1},a_{2,2},a_{2,3}, +a_{3,2},a_{3,3},a_{3,4}, +..., +a_{n,n-1},a_{n,n}, +$$ + +$$ +B[0],B[1],B[2],B[3],...,B[3n-3] +$$ + +- 前 $i-1$ 行共 $3(i-1)-1$ 个元素。 +- $a_{i,j}$ 是第 $i$ 行第 $j-i+2$ 个元素。 +- $a_{i,j}$ 是第 $2i+j-2$ 个元素。 +- 数组下标从 $0$ 开始,所以 $a_{i,j}$ 对应 $B[2i+j-3]$ + +### 5.2. 从数组到矩阵 + +若已知数组下标 $k$,如何得到 $i,j$? + +数组下标从 $0$ 开始,所以该问题是问第 $k+1$ 个元素,在第几行?第几列? + +- 前 $i-1$ 行共 $3(i-1)-1$ 个元素。 +- 前 $i$ 行共 $3i-1$ 个元素。 +- 显然:$3(i-1)-1 \lt k+1 \leq 3i-1$ + +$$ +i \geq (k+2)/3 +$$ + +向上取整得到 $i$。 + +> 或 $3(i-1)-1 \leq k \lt 3i-1$,结果为 $i \leq (k+1)/3+1$,向下取整得到 $i$。 + +## 6. 稀疏矩阵 + +稀疏矩阵:非零元素远远少于矩阵元素的个数。 + +### 6.1. 顺序存储 + +三元组 `<行,列,值>`。 + +$$ +\left( +\begin{matrix} +0 & 0 & 4 & 0 & 0 & 0 & 5 \\ +0 & 3 & 0 & 0 & 9 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 & 7 & 0 \\ +0 & 2 & 0 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 +\end{matrix} +\right) +$$ + +| i(行) | j(列) | v(值) | +| ------- | ------- | ------- | +| 1 | 3 | 4 | +| 1 | 6 | 5 | +| 2 | 2 | 3 | +| 2 | 4 | 9 | +| 3 | 5 | 7 | +| 4 | 2 | 2 | + +### 6.2. 十字链表法 + +![十字链表法](cross-link.png) diff --git a/ch3/matrix/cross-link.png b/ch3/matrix/cross-link.png new file mode 100644 index 0000000000000000000000000000000000000000..d4fa541660922f97e16158ae4fd3458915a2383a GIT binary patch literal 606308 zcmeGD1#_dzvNZ^s8QRRu%*<@JnVGT8%*>2!YO~#DW@ct;Gqc@hmW<&Qc{pYgu{gc0Rcgjkrr100f7z#0fFF#0sH#IvR(uI>jl(B zMM@N;W(NQ4>jJ`3L|z00q&@-u%^33Q8rD%-+XVy!Y4D#X=%_=P83@Q{v5dHgny10V zJG2kpXv1^NHj~>d&(5LX_5D@jlg^~ZZ!ijF9_?fbXl*1oIq4}AC5stYa%bWUHtZyq zc)!XHP?+D;+AS0t*9MO+Ufic=XEjLJ2p?8UZvh{*pVyyub2wVz_4H&g!NmXlh!l#S zH~OC@5fE1Y{t9LvO$>}(&L4W6{`Wv%?*-w> zh0*@io5;63NYs4Te~%kE3kEy)|JQ8DGU)$4y++~xZ+ic)S-pJ+s3~FNPgNB8A}-v2 zA&v+Hc5Y}Vihk0f!RQkOjJ|Ax13d~$qz;KA6}geIB7DL4QHQ84CMda 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