Union-Find: add a new module in kernel library
This patch implements a union-find data structure in the kernel library, which includes operations for allocating nodes, freeing nodes, finding the root of a node, and merging two nodes. Signed-off-by: Xavier <xavier_qy@163.com> Signed-off-by: Tejun Heo <tj@kernel.org>
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Xavier
Tejun Heo
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102
Documentation/core-api/union_find.rst
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Documentation/core-api/union_find.rst
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.. SPDX-License-Identifier: GPL-2.0
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====================
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Union-Find in Linux
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====================
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:Date: June 21, 2024
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:Author: Xavier <xavier_qy@163.com>
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What is union-find, and what is it used for?
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------------------------------------------------
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Union-find is a data structure used to handle the merging and querying
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of disjoint sets. The primary operations supported by union-find are:
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Initialization: Resetting each element as an individual set, with
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each set's initial parent node pointing to itself.
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Find: Determine which set a particular element belongs to, usually by
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returning a “representative element” of that set. This operation
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is used to check if two elements are in the same set.
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Union: Merge two sets into one.
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As a data structure used to maintain sets (groups), union-find is commonly
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utilized to solve problems related to offline queries, dynamic connectivity,
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and graph theory. It is also a key component in Kruskal's algorithm for
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computing the minimum spanning tree, which is crucial in scenarios like
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network routing. Consequently, union-find is widely referenced. Additionally,
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union-find has applications in symbolic computation, register allocation,
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and more.
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Space Complexity: O(n), where n is the number of nodes.
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Time Complexity: Using path compression can reduce the time complexity of
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the find operation, and using union by rank can reduce the time complexity
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of the union operation. These optimizations reduce the average time
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complexity of each find and union operation to O(α(n)), where α(n) is the
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inverse Ackermann function. This can be roughly considered a constant time
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complexity for practical purposes.
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This document covers use of the Linux union-find implementation. For more
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information on the nature and implementation of union-find, see:
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Wikipedia entry on union-find
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https://en.wikipedia.org/wiki/Disjoint-set_data_structure
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Linux implementation of union-find
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-----------------------------------
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Linux's union-find implementation resides in the file "lib/union_find.c".
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To use it, "#include <linux/union_find.h>".
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The union-find data structure is defined as follows::
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struct uf_node {
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struct uf_node *parent;
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unsigned int rank;
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};
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In this structure, parent points to the parent node of the current node.
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The rank field represents the height of the current tree. During a union
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operation, the tree with the smaller rank is attached under the tree with the
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larger rank to maintain balance.
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Initializing union-find
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--------------------
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You can complete the initialization using either static or initialization
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interface. Initialize the parent pointer to point to itself and set the rank
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to 0.
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Example::
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struct uf_node my_node = UF_INIT_NODE(my_node);
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or
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uf_node_init(&my_node);
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Find the Root Node of union-find
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--------------------------------
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This operation is mainly used to determine whether two nodes belong to the same
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set in the union-find. If they have the same root, they are in the same set.
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During the find operation, path compression is performed to improve the
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efficiency of subsequent find operations.
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Example::
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int connected;
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struct uf_node *root1 = uf_find(&node_1);
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struct uf_node *root2 = uf_find(&node_2);
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if (root1 == root2)
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connected = 1;
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else
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connected = 0;
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Union Two Sets in union-find
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----------------------------
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To union two sets in the union-find, you first find their respective root nodes
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and then link the smaller node to the larger node based on the rank of the root
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nodes.
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Example::
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uf_union(&node_1, &node_2);
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87
Documentation/translations/zh_CN/core-api/union_find.rst
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87
Documentation/translations/zh_CN/core-api/union_find.rst
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.. SPDX-License-Identifier: GPL-2.0
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.. include:: ../disclaimer-zh_CN.rst
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:Original: Documentation/core-api/union_find.rst
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===========================
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Linux中的并查集（Union-Find）
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===========================
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:日期: 2024年6月21日
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:作者: Xavier <xavier_qy@163.com>
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何为并查集，它有什么用？
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---------------------
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并查集是一种数据结构，用于处理一些不交集的合并及查询问题。并查集支持的主要操作：
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初始化：将每个元素初始化为单独的集合，每个集合的初始父节点指向自身
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查询：查询某个元素属于哪个集合，通常是返回集合中的一个“代表元素”。这个操作是为
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了判断两个元素是否在同一个集合之中。
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合并：将两个集合合并为一个。
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并查集作为一种用于维护集合（组）的数据结构，它通常用于解决一些离线查询、动态连通性和
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图论等相关问题，同时也是用于计算最小生成树的克鲁斯克尔算法中的关键，由于最小生成树在
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网络路由等场景下十分重要，并查集也得到了广泛的引用。此外，并查集在符号计算，寄存器分
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配等方面也有应用。
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空间复杂度: O(n)，n为节点数。
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时间复杂度：使用路径压缩可以减少查找操作的时间复杂度，使用按秩合并可以减少合并操作的
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时间复杂度，使得并查集每个查询和合并操作的平均时间复杂度仅为O(α(n))，其中α(n)是反阿
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克曼函数，可以粗略地认为并查集的操作有常数的时间复杂度。
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本文档涵盖了对Linux并查集实现的使用方法。更多关于并查集的性质和实现的信息，参见：
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维基百科并查集词条
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https://en.wikipedia.org/wiki/Disjoint-set_data_structure
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并查集的Linux实现
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----------------
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Linux的并查集实现在文件“lib/union_find.c”中。要使用它，需要
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“#include <linux/union_find.h>”。
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并查集的数据结构定义如下::
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struct uf_node {
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struct uf_node *parent;
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unsigned int rank;
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};
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其中parent为当前节点的父节点，rank为当前树的高度，在合并时将rank小的节点接到rank大
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的节点下面以增加平衡性。
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初始化并查集
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---------
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可以采用静态或初始化接口完成初始化操作。初始化时，parent 指针指向自身，rank 设置
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为 0。
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示例::
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struct uf_node my_node = UF_INIT_NODE(my_node);
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或
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uf_node_init(&my_node);
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查找并查集的根节点
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----------------
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主要用于判断两个并查集是否属于一个集合，如果根相同，那么他们就是一个集合。在查找过程中
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会对路径进行压缩，提高后续查找效率。
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示例::
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int connected;
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struct uf_node *root1 = uf_find(&node_1);
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struct uf_node *root2 = uf_find(&node_2);
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if (root1 == root2)
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connected = 1;
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else
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connected = 0;
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合并两个并查集
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-------------
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对于两个相交的并查集进行合并，会首先查找它们各自的根节点，然后根据根节点秩大小，将小的
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节点连接到大的节点下面。
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示例::
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uf_union(&node_1, &node_2);
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@ -23458,6 +23458,15 @@ F: drivers/cdrom/cdrom.c
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F: include/linux/cdrom.h
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F: include/uapi/linux/cdrom.h
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UNION-FIND
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M: Xavier <xavier_qy@163.com>
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L: linux-kernel@vger.kernel.org
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S: Maintained
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F: Documentation/core-api/union_find.rst
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F: Documentation/translations/zh_CN/core-api/union_find.rst
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F: include/linux/union_find.h
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F: lib/union_find.c
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UNIVERSAL FLASH STORAGE HOST CONTROLLER DRIVER
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R: Alim Akhtar <alim.akhtar@samsung.com>
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R: Avri Altman <avri.altman@wdc.com>
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41
include/linux/union_find.h
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41
include/linux/union_find.h
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/* SPDX-License-Identifier: GPL-2.0 */
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#ifndef __LINUX_UNION_FIND_H
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#define __LINUX_UNION_FIND_H
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/**
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* union_find.h - union-find data structure implementation
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*
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* This header provides functions and structures to implement the union-find
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* data structure. The union-find data structure is used to manage disjoint
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* sets and supports efficient union and find operations.
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*
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* See Documentation/core-api/union_find.rst for documentation and samples.
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*/
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struct uf_node {
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struct uf_node *parent;
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unsigned int rank;
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};
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/* This macro is used for static initialization of a union-find node. */
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#define UF_INIT_NODE(node) {.parent = &node, .rank = 0}
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/**
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* uf_node_init - Initialize a union-find node
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* @node: pointer to the union-find node to be initialized
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*
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* This function sets the parent of the node to itself and
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* initializes its rank to 0.
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*/
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static inline void uf_node_init(struct uf_node *node)
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{
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node->parent = node;
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node->rank = 0;
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}
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/* find the root of a node */
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struct uf_node *uf_find(struct uf_node *node);
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/* Merge two intersecting nodes */
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void uf_union(struct uf_node *node1, struct uf_node *node2);
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#endif /* __LINUX_UNION_FIND_H */
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@ -34,7 +34,7 @@ lib-y := ctype.o string.o vsprintf.o cmdline.o \
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is_single_threaded.o plist.o decompress.o kobject_uevent.o \
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earlycpio.o seq_buf.o siphash.o dec_and_lock.o \
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nmi_backtrace.o win_minmax.o memcat_p.o \
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buildid.o objpool.o
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buildid.o objpool.o union_find.o
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lib-$(CONFIG_PRINTK) += dump_stack.o
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lib-$(CONFIG_SMP) += cpumask.o
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49
lib/union_find.c
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49
lib/union_find.c
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// SPDX-License-Identifier: GPL-2.0
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#include <linux/union_find.h>
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/**
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* uf_find - Find the root of a node and perform path compression
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* @node: the node to find the root of
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*
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* This function returns the root of the node by following the parent
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* pointers. It also performs path compression, making the tree shallower.
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*
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* Returns the root node of the set containing node.
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*/
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struct uf_node *uf_find(struct uf_node *node)
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{
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struct uf_node *parent;
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while (node->parent != node) {
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parent = node->parent;
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node->parent = parent->parent;
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node = parent;
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}
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return node;
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}
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/**
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* uf_union - Merge two sets, using union by rank
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* @node1: the first node
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* @node2: the second node
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*
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* This function merges the sets containing node1 and node2, by comparing
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* the ranks to keep the tree balanced.
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*/
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void uf_union(struct uf_node *node1, struct uf_node *node2)
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{
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struct uf_node *root1 = uf_find(node1);
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struct uf_node *root2 = uf_find(node2);
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if (root1 == root2)
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return;
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if (root1->rank < root2->rank) {
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root1->parent = root2;
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} else if (root1->rank > root2->rank) {
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root2->parent = root1;
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} else {
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root2->parent = root1;
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root1->rank++;
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}
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}
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