n 为主串长度，m 为要匹配的子串长度。
KMP算法是由D.E. Knuth、J.H. Morris和V.R. Pratt(其中Knuth和Pratt共同研究, Mor-ris独立研究)发表一个模式匹配算法，KMP算法的最大特点使得它在处理大量文本匹配的问题时，比暴力枚举算法有更好的性能。
关于字符串匹配，是字符串很重要的知识点，也是面试笔试的高频考点。Leetcode的第28题就是考查字符串匹配算法。另外本文是查看了《大话数据结构》这本书做的总结，同时next数组部分也参考了这篇博客KMP算法中next数组的计算。
KMP是由基础的字符串匹配BF算法改进而来的
KMP模式匹配算法原理
目标串（主串） S = "heloohello", 模式串（子串） T = "hello"。我们要从主串 S 中找到子串 T 的位置。如果使用BF算法，步骤如下图所示。
在 ① 中当 S 和 T 不匹配时，BF算法的操作是主串 S 回退到 i-j+1（从本次匹配的初始位置后移一位，图中步骤②的位置），匹配串 T 回退到 j = 0 (初始位置）, 然后执行 ②③④⑤⑥ 步依次进行匹配，但所有这些步骤一定都是必需的吗？
在步骤 ① 中，单看模式串 T，前三个字符均不相等（'h' != 'e' != 'l'），同时S 和 T 串前三个字符又相匹配，那么模式串 T 的首字符'h' 自然不可能和主串 S 的第二、三位字符相等。所以步骤②③都是多余的。这是KMP算法的关键所在，如果我们知道 T 串中哪些字符相等（也是关键点，后续会讲），那么有些步骤就可以省略。
因此只需要保留①④⑤⑥的步骤即可。从下图可以看出，指针 i 是不是一直没有回溯？这就是KMP的妙处所在。在KMP中，指针 i 永远不会回溯，只有指向模式串的指针 j 会发生回溯。在本例中 j 每次都回退到首元素，我们再举一个例子，看看 j 会怎么变化。
第二个例子：
目标串（主串） S = "abcababca", 模式串（子串） T = "abcabx"。BF算法执行过程如下图所示，在步骤 ① 中前5个字符完全相等，根据上一个例子的经验，已知模式串T中第一位字符与第二位、第三位不等（后续会根据next[]数组计算得出），步骤 ②③ 都是多余的，直接省略。
与上个例子不同的是，这里的模式串 T 的首位字符 'a' 与 T 第四位的 'a' 相等，第二位的 'b' 与第五位的 'b' 相等，而在 ① 中,第四位的 'a' 第五位的 'b' 已经与主串 S 中的相应位置比较过了，是相等的。因此可以断定,T 的首字符 'a'、第二位的字符 'b' 与 S 的第四位字符和第五位字符肯定也是相等的，所以 ④⑤ 这两个比较得出字符相等的步骤也可以省略。
总结上面两个例子，在KMP算法中，主串的 i 值是不需要回溯的。所以我们只需要考虑变化的 j 值，模式串 j 值的变化通过观察可以发现，当主串和模式串不匹配时，下一步 j 该指向哪个元素只与模式串 T 本身有关系。当发现有相同的字符，j 的变化也就会不同。
在KMP中，当主串和模式串不匹配时, 下一步 j 值的多少取决于当前字符之前的串的前后缀的相似度。
next数组
基础知识
在我们计算next数组之前，我们先讲解一些基础知识。
- 前缀：字符串的开头，例如字符串abcd的前缀为a, ab, abc, abcd。在KMP算法中使用的前缀为真前缀，既不包括原字符串abcd的前缀。（真前缀：a, ab, abc）
- 后缀：字符串的结尾，在KMP算法中同样使用的是真后缀(bcd,cd,d)。
- 最长公共前后缀：最长的相等的前缀与后缀，例如字符串ABCxyzABC的最长公共前后缀为ABC
- ABCXYABC的真前缀：A，AB, ABC, ABCx, ABCxy, ABCxyz, ABCxyzA, ABCxyzAB
- ABCXYABC的真后缀：BCxyzABC, CxyzABC, xyzABC, yzABC, zABC, ABC, BC, C
- 前缀表：存储每一个前缀的最长公共前后缀的长度。
举例：若模式串 T="abaaabaaca"。
- next数组：把 T 串各个位置的 j 的变化定义为数组 next，next 的长度就是 T 串的长度。主串和模式串不匹配时，下一步 j 的值由 next[j] 决定。例如目标串 S="abcaba", T="aba", 根据前缀表求出 next=[-1,0 0], 当 j=2 时发生不匹配, next[2]=0, 下一步 j 将等于 0 进行字符匹配。
前缀表和next数组的关系
前缀表存储每一个前缀的最长公共前后缀的长度，next数组存储的是模式串向右移动到next值的位置，这个值与前缀的最长公共前后缀的长度有关，所以next数组是可以由前缀表生成的。
用前缀表生成一个next数组很容易，将前缀表每一位都向后移动1位（最后一位舍去）并在第一位补一个-1就得到了next数组。
如果有同学不理解这个关系还可以看一下手动推理过程：
T="abaaabaaca"
- 位置0上的元素a前面没有子串，令next[0]=-1
- 位置1上的元素b前面的字符串为"a"，字符串"a"没有最长公共前后缀，next[1]=0
- 位置2上的元素a前面的字符串为"ab","ab"没有最长公共前后缀，next[2]=0
- 位置3上的元素a前面的字符串为"aba"，最长公共前后缀为"a"，next[3]=1
- 位置4上的元素a前面的字符串为"abaa"，最长公共前后缀为"a"，next[4]=1
- 位置5上的元素b前面的字符串为"abaaa"，最长公共前后缀为"a"，next[5]=1
- 位置6上的元素a前面的字符串为"abaaab"，最长公共前后缀为"ab"，next[6]=2
- 位置7上的元素a前面的字符串为"abaaaba"，最长公共前后缀为"aba"，next[7]=3
- 位置8上的元素a前面的字符串为"abaaabaa"，最长公共前后缀为"abaa"，next8]=4
- 位置9上的元素a前面的字符串为"abaaabac"，没有最长公共前后缀，next[9]=0
同时在KMP算法中next数组的计算这篇博客中提到了一个地方：为什么有些next数组是0,1开头，而有些next数组是-1,0开头？
-1,0开头与0, 1开头的next数组本质是一样的。实际上，以0, 1开头的next数组就是以-1,0开头的next数组每一项加1得到的。出现这种情况的原因在于模式串起始的索引值：在程序中，一个数组的索引的起始值为0；然而在考试和书中给的模式串起始值是多从1开始。所以在考试中遇到的next数组通常是以0, 1开头；而一些程序或教程中的next数组是以-1, 0开头。
注：在考试中通常会给模式串的索引，或者会给next值的前两项，在答题时要按照题目中的要求写next数组。
代码实现
next数组的代码实现, 可以计算出当前匹配串 T 的 next 数组
void get_next(string T, int *next) {next[0] = -1;int i = 0;int j = -1;while(i < T.size() - 1) {//T[i]表示后缀的单个字符//T[j]表示前缀的单个字符if(j == -1 || T[i] == T[j]){//++i;++j;next[i] = j;} else {//如果字符不相同，则j值回溯j = next[j];}}
}
KMP代码实现
int KMP(string S, string T) {int ans = -1;// i用于遍历主串Sint i = 0;// j用于遍历匹配串Tint j = 0;int next[255]; // 这里初始长度为255,需自行调整// 对T做分析，得到next数组get_next(T, next);while (i < S.size()) {// 匹配成功则继续向下一个字符进行匹配if (j == -1 || S[i] == T[j]) {++i;++j;}// 匹配失败进行回溯else {// j回溯到合适的位置j = next[j];}if (j == T.size()) {ans = i - T.size();break;}}return ans;
}
时间复杂度
令 n 为主串长度，m 为要匹配的子串长度。
对于Get_next函数来说，时间复杂度为O(m)，因为i值不回溯，所以使得KMP算法效率得到提高，在KMP函数中while循环的时间复杂度为O(n)，因此整个算法的时间复杂度为O(n + m)。
KMP算法仅当模式与主串之间存在许多“部分匹配”时，才会体现出它的优势，否则两者差异不明显。
替代方案：使用 CephFS（更适合 Docker 多容器共享场景）
若目标是多个容器同时读写同一份数据，RBD 的块设备语义受限，CephFS 是更自然的选择：
部署 CephFS（启用 MDS 服务），创建 filesystem 和 auth 密钥
在宿主机 mount CephFS 到本地路径（如 /mnt/cephfs），再用 docker run -v /mnt/cephfs:/data 挂载
或使用 docker-volume-cephfs 插件，支持动态创建子目录作为独立卷
CephFS 提供 POSIX 语义，天然支持多读多写，延迟略高于 RBD，但运维更简单
如何在docker内挂载ceph块存储作为容器数据卷存储后端
在 Docker 中使用 Ceph 块设备（RBD）作为容器数据卷后端，核心是通过 rbd 插件或 ceph-csi 驱动实现持久化存储挂载。Docker 原生不直接支持 RBD 卷类型，需借助第三方插件或转向更主流的 CSI 方案。以下提供两种切实可行、生产可用的方式：
方式一：使用 docker-volume-rbd 插件（轻量级，适合小规模部署）
该插件将 RBD 映射为本地块设备，并创建 ext4/xfs 文件系统供容器挂载。
确保宿主机已安装 ceph-common 和 rbd 工具，并能访问 Ceph 集群（/etc/ceph/ceph.conf 与 /etc/ceph/ceph.client.admin.keyring 可用）
安装插件：docker plugin install --grant-all-permissions ceph/rbd（官方镜像已归档，推荐使用社区维护版如 linode/docker-volume-rbd 或自建构建）
创建 RBD 镜像（如 rbd create rbd/myvol --size 10G --image-feature layering），并确保启用 layering 特性以支持克隆
启动容器时指定卷：docker run -v myvol:/data --volume-driver rbd -it alpine sh，插件会自动映射 RBD 镜像、格式化（首次）、挂载到容器内
方式二：通过 ceph-csi + Docker swarm 或 Kubernetes（推荐用于生产环境）
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