From b116b7203e788630ac045f560d5c54fbd361e00d Mon Sep 17 00:00:00 2001 From: CyC2018 <1029579233@qq.com> Date: Wed, 14 Mar 2018 22:44:13 +0800 Subject: [PATCH] auto commit --- notes/算法.md | 50 ++++++++++++------ pics/0e6cf8bd-b84e-4b3c-b79d-40d7dd54e120.png | Bin 0 -> 41984 bytes pics/0ed83061-9c1e-4df3-b15b-69aad5bfe9b8.png | Bin 0 -> 34816 bytes pics/1c237399-e322-4930-b5b4-a582b1ad8bda.png | Bin 0 -> 37888 bytes pics/1dc67ff6-d29b-4864-baac-fd6b23f9b2ac.png | Bin 0 -> 32768 bytes pics/521969c9-71f6-44a5-9c78-118530e5c135.png | Bin 0 -> 100352 bytes pics/83575315-20b5-44a6-bf58-94460a141ffa.jpg | Bin 0 -> 31744 bytes pics/8f1e2db5-a59b-4633-8b61-6b8b9505b8ea.png | Bin 0 -> 66560 bytes pics/9a30b932-f69f-40a1-9564-a1354ff8cf29.jpg | Bin 0 -> 33792 bytes pics/c11f5e3a-7a28-4db7-87b1-29ba1a2f2d72.jpg | Bin 0 -> 53248 bytes pics/c64f91e2-f5a8-436b-8663-b8f3fba3e098.png | Bin 0 -> 36864 bytes pics/cc3d855a-4281-445b-8f83-8f855458de19.png | Bin 0 -> 14336 bytes pics/e1cd89d1-8973-41d0-8ea9-940d94c314d9.jpg | Bin 0 -> 46080 bytes 13 files changed, 33 insertions(+), 17 deletions(-) create mode 100644 pics/0e6cf8bd-b84e-4b3c-b79d-40d7dd54e120.png create mode 100644 pics/0ed83061-9c1e-4df3-b15b-69aad5bfe9b8.png create mode 100644 pics/1c237399-e322-4930-b5b4-a582b1ad8bda.png create mode 100644 pics/1dc67ff6-d29b-4864-baac-fd6b23f9b2ac.png create mode 100644 pics/521969c9-71f6-44a5-9c78-118530e5c135.png create mode 100644 pics/83575315-20b5-44a6-bf58-94460a141ffa.jpg create mode 100644 pics/8f1e2db5-a59b-4633-8b61-6b8b9505b8ea.png create mode 100644 pics/9a30b932-f69f-40a1-9564-a1354ff8cf29.jpg create mode 100644 pics/c11f5e3a-7a28-4db7-87b1-29ba1a2f2d72.jpg create mode 100644 pics/c64f91e2-f5a8-436b-8663-b8f3fba3e098.png create mode 100644 pics/cc3d855a-4281-445b-8f83-8f855458de19.png create mode 100644 pics/e1cd89d1-8973-41d0-8ea9-940d94c314d9.jpg diff --git a/notes/算法.md b/notes/算法.md index 1f02fbc1..85d99464 100644 --- a/notes/算法.md +++ b/notes/算法.md @@ -91,11 +91,15 @@ ## 1. 函数转换 -指数函数可以转换为线性函数,从而在函数图像上显示的更直观。 +指数函数可以转换为线性函数,从而在函数图像上显示的更直观。例如 -
转换为

+


-

+可以在其两端同时取对数,得到: + +


+ +

## 2. 数学模型 @@ -103,13 +107,13 @@ 使用 \~f(N) 来表示所有随着 N 的增大除以 f(N) 的结果趋近于 1 的函数 , 例如 N3/6-N2/2+N/3 \~ N3/6。 -

+

**增长数量级** 增长数量级将算法与它的实现隔离开来,一个算法的增长数量级为 N3 与它是否用 Java 实现,是否运行于特定计算机上无关。 -

+

**内循环** @@ -121,7 +125,8 @@ ## 3. ThreeSum -ThreeSum 程序用于统计一个数组中三元组的和为 0 的数量。 +ThreeSum 用于统计一个数组中三元组的和为 0 的数量。 + ```java public class ThreeSum { @@ -142,7 +147,7 @@ public class ThreeSum { } ``` -该程序的内循环为 if (a[i] + a[j] + a[k] == 0) 语句,总共执行的次数为 N3/6 - N2/2 + N/3,因此它的近似执行次数为 \~N3/6,增长数量级为 N3。 +该算法的内循环为`if (a[i] + a[j] + a[k] == 0)`语句,总共执行的次数为 N(N-1)(N-2) = N3/6 - N2/2 + N/3,因此它的近似执行次数为 \~N3/6,增长数量级为 N3。 **改进** @@ -159,7 +164,7 @@ public class ThreeSumFast { for (int i = 0; i < N; i++) { for (int j = i + 1; j < N; j++) { // rank() 方法返回元素在数组中的下标,如果元素不存在,这里会返回 -1。 - // 应该注意这里的下标必须大于 j,这样就不会重复统计了。 + // 应该注意这里的下标必须大于 j,否则会重复统计。 if (BinarySearch.rank(-a[i] - a[j], a) > j) { cnt++; } @@ -172,11 +177,13 @@ public class ThreeSumFast { ## 4. 倍率实验 -如果 T(N) \~ aNblgN,那么 T(2N)/T(N) \~ 2b,例如对于暴力方法的 ThreeSum 算法,近似时间为 \~N3/6,对它进行倍率实验得到如下结果: +如果 T(N) \~ aNblgN,那么 T(2N)/T(N) \~ 2b。 -

+例如对于暴力方法的 ThreeSum 算法,近似时间为 \~N3/6。进行如下实验:多次运行该算法,每次取的 N 值为前一次的两倍,统计每次执行的时间,并统计本次运行时间与前一次运行时间的比值,得到如下结果: -可见 T(2N)/T(N)\~23,也就是 b 为 3。 +

+ +可以看到,T(2N)/T(N)\~23,因此可以确定 T(N) \~ aN2lgN。 ## 5. 注意事项 @@ -204,6 +211,10 @@ public class ThreeSumFast { ## 1. 栈 +first-in-last-out(FILO) + +

+ **数组实现** ```java @@ -312,6 +323,11 @@ public class Stack { ``` ## 2. 队列 +first-in-first-out(FIFO) + +

+ + 下面是队列的链表实现,需要维护 first 和 last 节点指针,分别指向队首和队尾。 这里需要考虑让哪个指针指针链表头部节点,哪个指针指向链表尾部节点。因为出队列操作需要让队首元素的下一个元素成为队首,就需要容易获取下一个元素,而链表的头部节点的 next 指针指向下一个元素,因此让队首指针 first 指针链表的开头。 @@ -365,11 +381,11 @@ public class Queue { 用于解决动态连通性问题,能动态连接两个点,并且判断两个点是否连接。 -

+

**API** -

+

**基本数据结构** @@ -416,7 +432,7 @@ public class UF { 在 union 时只将触点的 id 值指向另一个触点 id 值,不直接用 id 来存储所属的连通分量。这样就构成一个倒置的树形结构,根节点需要指向自己。在进行查找一个节点所属的连通分量时,要一直向上查找直到根节点,并使用根节点的 id 值作为本连通分量的 id 值。 -

+

```java public int find(int p) { @@ -434,7 +450,7 @@ public class UF { 这种方法可以快速进行 union 操作,但是 find 操作和树高成正比,最坏的情况下树的高度为触点的数目。 -

+

## 3. 加权 quick-union 算法 @@ -442,7 +458,7 @@ public class UF { 理论研究证明,加权 quick-union 算法构造的树深度最多不超过 lgN。 -

+

```java public class WeightedQuickUnionUF { @@ -489,7 +505,7 @@ public class WeightedQuickUnionUF { ## 5. 各种 union-find 算法的比较 -

+

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