7    VisuAlgo.net / /ufds Login 联合 - 查找不相交集(UFDS)
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The Union-Find Disjoint Sets (UFDS) data structure is used to model a collection of disjoint sets, which is able to efficiently (i.e. in nearly constant time) determine which set an item belongs to, test if two items belong to the same set, and union two disjoint sets into one when needed. It can be used to find connected components in an undirected graph, and can hence be used as part of Kruskal's algorithm for the Minimum Spanning Tree (MST) problem.


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View the visualization of a sample Union-Find Disjoint Sets here!


Each tree represents a disjoint set (thus a collection of disjoint sets form a forest of trees) and the root of the tree is the representative item of this disjoint set.


Now stop and look at the currently visualized trees. How many items (N) are there overall? How many disjoint sets are there? What are the members of each disjoint set? What is the representative item of each disjoint set?


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As we fixed the default example for this e-Lecture, your answers should be: N=13 and there are 4 disjoint sets: {0,1,2,3,4,10}, {5,7,8,11}, {6,9}, {12} with the underlined members be the representative items (of their own disjoint set).


Another pro-tip: We designed this visualization and this e-Lecture mode to look good on 1366x768 resolution or larger (typical modern laptop resolution in 2017). We recommend using Google Chrome to access VisuAlgo. Go to full screen mode (F11) to enjoy this setup. However, you can use zoom-in (Ctrl +) or zoom-out (Ctrl -) to calibrate this.

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We can simply record this forest of trees with an array p of size N items where p[i] records the parent of item i and if p[i] = i, then i is the root of this tree and also the representative item of the set that contains item i.


Once again, look at the visualization above and determine the values inside this array p.

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On the same fixed example, your answers should be p = [1, 3, 3, 3, 3, 5, 6, 5, 5, 6, 4, 8,12] of size N = 13 ranging from p[0] to p[12].


You can check that p[3] = 3, p[5] = 5, p[6] = 6, and p[12] = 12, which are consistent with the fact that {3,5,6,12} are the representative items (of their own disjoint set).

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We also record one more information in array rank also of size N. The value of rank[i] is the upperbound of the height of subtree rooted at vertex i that will be used as guiding heuristic for UnionSet(i, j) operation. You will notice that after 'path-compression' heuristic (to be described later) compresses some path, the rank values no longer reflect the true height of that subtree.


As there are many items with rank 0, we set the visualization as follows to minimize clutter: Only when the rank of a vertex i is greater than 0, then VisuAlgo will show the value of rank[i] (abbreviated as a single character r) as a red text below vertex i.

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On the same fixed example, verify that {1,4,6,8} have rank 1 and {3,5} have rank 2, with the rest having rank 0 (not shown).


At this point of time, all rank values are correct, i.e. they really describe the height of the subtree rooted at that vertex. We will soon see that they will not be always correct in the next few slides.

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There are five available UFDS operations in this visualization page:
Examples, Initialize(N), FindSet(i), IsSameSet(i, j), and UnionSet(i, j).


The first operation (Examples) is trivial: List of example UFDS structures with various special characteristics for your starting point. This e-Lecture mode always use the 'Four disjoint sets' example as the starting point.


Also notice that none of the example contains a 'very tall' tree. You will soon understand the reason after we describe the two heuristics used.

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Initialize(N): Create N disjoint sets, all with p[i] = i and rank[i] = 0 (these rank values are initially not shown).


The time complexity of this operation is very clearly O(N).


Due to the limitation of screen size, we set 1 ≤ N ≤ 16.

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FindSet(i): From vertex i, recursively go up the tree. That is, from vertex i, we go to vertex p[i]) until we find the root of this tree, which is the representative item with p[i] = i of this disjoint set.


In this FindSet(i) operation, we employ path-compression heuristic after each call of FindSet(i) as now every single vertex along the path from vertex i to the root know that the root is their representative item and can point to it directly in O(1).

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If we execute FindSet(12), we will immediately get vertex 12. If we execute FindSet(9) we will get vertex 6 after 1 step and no other change.


Now try executing FindSet(0). If this is your first call on this default UFDS example, it will return vertex 3 after 2 steps and then modify the underlying UFDS structure due to path-compression in action (that is, vertex 0 points to vertex 3 directly). Notice that rank value of rank[1] = 1 is now wrong as vertex 1 becomes a new leaf. However, we will not bother to update its value.


Notice that the next time you execute FindSet(0) again, it will be much faster as the path has been compressed. For now, we assume that FindSet(i) runs in O(1).

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IsSameSet(i, j): Simply check if FindSet(i) == FindSet(j) or not. This function is used extensively in Kruskal's MST algorithm. As it only calls FindSet operation twice, we will assume it also runs in O(1).


Note that FindSet function is called inside IsSameSet function, so path-compression heuristic is also indirectly used.

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If we call IsSameSet(3, 5), we should get false as vertex 3 and vertex 5 are representative items of their respective disjoint sets and they are different.


Now try IsSameSet(0, 11) on the same default example to see indirect path-compression on vertex 0 and vertex 11. We should get false as the two representative items: vertex 3 and vertex 5, are different. Notice that the rank values at vertex {1, 5, 8} are now wrong. But we will not fix them.

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UnionSet(i, j): If item i and j come from two disjoint sets initially, we link the representative item of the shorter tree/disjoint set to the representative item of the taller tree/disjoint set (otherwise, we do nothing). This is also done in O(1).


This is union-by-rank heuristic in action and will cause the resulting tree to be relatively short. Only if the two trees are equally tall before union (by comparing their rank values heuristically — note that we are not comparing their actual heights), then the rank of the resulting tree will increase by one unit.

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Also note that FindSet function is called inside UnionSet function, so path-compression heuristic is also indirectly used. Each time path-compression heuristic compresses a path, at least one rank values will be incorrect. We do not bother fixing these rank values as they are only used as guiding heuristic for this UnionSet function.

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On the same default example, try UnionSet(9, 12). As the tree that represents disjoint set {6, 9} is currently taller (according to the value of rank[6] = 1), then the shorter tree that represents disjoint set {12} will be slotted under vertex 6, without increasing the height of the combined tree at all.


On the same default example, try UnionSet(0, 11). Notice that the ranks of vertex 3 and vertex 5 are the same rank[3] = rank[5] = 2. Therefore, we can either put vertex 3 under vertex 5 (our implementation) or vertex 5 under vertex 3 (both will increase the resulting height of the combined tree by 1). Notice the indirect path-compression heuristic in action.

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Quiz: Starting with N=8 disjoint sets, how tall (heuristically) can the resulting final tree if we call 7 UnionSet(i, j) operations strategically?

rank:4
rank:5
rank:1
rank:3
rank:2

Quiz: Starting with N=8 disjoint sets, how short (heuristically) can the resulting final tree if we call 7 UnionSet(i, j) operations strategically?

rank:2
rank:5
rank:4
rank:3
rank:1


Discussion: Why?

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e-Lecture: The content of this slide is hidden and only available for legitimate CS lecturer worldwide. Drop an email to visualgo.info at gmail dot com if you want to activate this CS lecturer-only feature and you are really a CS lecturer (show your University staff profile).

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So far, we say that FindSet(i), IsSameSet(i, j), and UnionSet(i, j) runs in O(1). Actually they run in O(α(N)) if the UFDS is implemented with both path-compression and union-by-rank heuristics.


This α(N) is called the inverse Ackermann function that grows extremely slowly. For practical usage of this UFDS data structure (assuming N ≤ 1M), we have α(1M) ≈ 1.

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You have reached the end of the basic stuffs of this UFDS data structure and we encourage you to go to Exploration Mode and explore this simple but interesting data structure using your own examples.


However, we still have a few more interesting UFDS challenges for you.

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You can download source code of our custom implementation of Union-Find Disjoint Sets data structure in Object-Oriented Programming (OOP) fashion here (please look for file ch2_unionfind_ds in cpp or java inside the zip file). You are free to customize this implementation to suit your needs as some harder problem requires customization of this basic implementation.


I do wish that one day C++ and/or Java will include this interesting data structure inside C++ STL and/or Java API.

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For a few more interesting questions about this data structure, please practice on UFDS training module (no login is required, but short and of medium difficulty setting only).


However, for registered users, you should login and then go to the Main Training Page to officially clear this module (after you have cleared the pre-requisite, which is Graph Data Structures, and such achievement will be recorded in your user account.

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Even after clearing the Online Quiz of this UFDS module, do you think that you have really mastered this data structure?


Let us challenge you by asking you to solve two programming problems that somewhat requires the usage of this Union-Find Disjoint Sets data structure: UVa 01329 - Corporative Network and Kattis - control.


Beware that both problems are actual ACM International Collegiate Programming Contest (ICPC) problems, i.e. they are "not trivial".

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当操作进行时,状态面板将会有每个步骤的描述。
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e-Lecture: The content of this slide is hidden and only available for legitimate CS lecturer worldwide. Drop an email to visualgo.info at gmail dot com if you want to activate this CS lecturer-only feature and you are really a CS lecturer (show your University staff profile).

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Control the animation with the player controls! Keyboard shortcuts are:

Spacebar: play/pause/replay
Left/right arrows: step backward/step forward
-/+: decrease/increase speed
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Return to 'Exploration Mode' to start exploring!


Note that if you notice any bug in this visualization or if you want to request for a new visualization feature, do not hesitate to drop an email to the project leader: Dr Steven Halim via his email address: stevenhalim at gmail dot com.

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Examples

Initialize(N)

FindSet(i)

IsSameSet(i, j)

UnionSet(i, j)

>

Three disjoint sets

Four disjoint sets

2 Trees of Rank 1

2 Trees of Rank 2

2 Trees of Rank 3

1 Tree of Rank 4

N =

执行

i =

执行

i = , j =

执行

i = , j =

执行

关于 团队 使用条款

关于

VisuAlgo在2011年由Steven Halim博士概念化,作为一个工具,帮助他的学生更好地理解数据结构和算法,让他们自己和自己的步伐学习基础。
VisuAlgo包含许多高级算法,这些算法在Steven Halim博士的书(“竞争规划”,与他的兄弟Felix Halim博士合作)和其他书中讨论。今天,一些高级算法的可视化/动画只能在VisuAlgo中找到。
虽然专门为新加坡国立大学(NUS)学生采取各种数据结构和算法类(例如CS1010,CS1020,CS2010,CS2020,CS3230和CS3230),作为在线学习的倡导者,我们希望世界各地的好奇心发现这些可视化也很有用。
VisuAlgo不是从一开始就设计为在小触摸屏(例如智能手机)上工作良好,因为需要满足许多复杂的算法可视化,需要大量的像素和点击并拖动手势进行交互。一个令人尊敬的用户体验的最低屏幕分辨率为1024x768,并且只有着陆页相对适合移动设备。
VisuAlgo是一个正在进行的项目,更复杂的可视化仍在开发中。
最令人兴奋的发展是自动问题生成器和验证器(在线测验系统),允许学生测试他们的基本数据结构和算法的知识。这些问题是通过一些规则随机生成的,学生的答案会在提交给我们的评分服务器后立即自动分级。这个在线测验系统,当它被更多的世界各地的CS教师采用,应该技术上消除许多大学的典型计算机科学考试手动基本数据结构和算法问题。通过在通过在线测验时设置小(但非零)的重量,CS教练可以(显着地)增加他/她的学生掌握这些基本问题,因为学生具有几乎无限数量的可以立即被验证的训练问题他们参加在线测验。培训模式目前包含12个可视化模块的问题。我们将很快添加剩余的8个可视化模块,以便VisuAlgo中的每个可视化模块都有在线测验组件。
另一个积极的发展分支是VisuAlgo的国际化子项目。我们要为VisuAlgo系统中出现的所有英语文本准备一个CS术语的数据库。这是一个很大的任务,需要众包。一旦系统准备就绪,我们将邀请VisuAlgo游客贡献,特别是如果你不是英语母语者。目前,我们还以各种语言写了有关VisuAlgo的公共注释:
zh, id, kr, vn, th.

团队

项目领导和顾问(2011年7月至今)
Dr Steven Halim, Senior Lecturer, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Software Engineer, Google (Mountain View)

本科生研究人员 1 (Jul 2011-Apr 2012)
Koh Zi Chun, Victor Loh Bo Huai

最后一年项目/ UROP学生 1 (Jul 2012-Dec 2013)
Phan Thi Quynh Trang, Peter Phandi, Albert Millardo Tjindradinata, Nguyen Hoang Duy

最后一年项目/ UROP学生 2 (Jun 2013-Apr 2014)
Rose Marie Tan Zhao Yun, Ivan Reinaldo

本科生研究人员 2 (May 2014-Jul 2014)
Jonathan Irvin Gunawan, Nathan Azaria, Ian Leow Tze Wei, Nguyen Viet Dung, Nguyen Khac Tung, Steven Kester Yuwono, Cao Shengze, Mohan Jishnu

最后一年项目/ UROP学生 3 (Jun 2014-Apr 2015)
Erin Teo Yi Ling, Wang Zi

最后一年项目/ UROP学生 4 (Jun 2016-Dec 2017)
Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir

List of translators who have contributed ≥100 translations can be found at statistics page.

致谢
这个项目是由来自NUS教学与学习发展中心(CDTL)的慷慨的教学增进赠款提供的。

使用条款

VisuAlgo是地球上的计算机科学社区免费。如果你喜欢VisuAlgo,我们对你的唯一的要求就是通过你知道的方式,比如:Facebook、Twitter、课程网页、博客评论、电子邮件等告诉其他计算机方面的学生/教师:VisuAlgo网站的神奇存在
如果您是数据结构和算法学生/教师,您可以直接将此网站用于您的课程。如果你从这个网站拍摄截图(视频),你可以使用屏幕截图(视频)在其他地方,只要你引用本网站的网址(http://visualgo.net)或出现在下面的出版物列表中作为参考。但是,您不能下载VisuAlgo(客户端)文件并将其托管在您自己的网站上,因为它是剽窃。到目前为止,我们不允许其他人分叉这个项目并创建VisuAlgo的变体。使用(客户端)的VisuAlgo的离线副本作为您的个人使用是很允许的。
请注意,VisuAlgo的在线测验组件本质上具有沉重的服务器端组件,并且没有简单的方法来在本地保存服务器端脚本和数据库。目前,公众只能使用“培训模式”来访问这些在线测验系统。目前,“测试模式”是一个更受控制的环境,用于使用这些随机生成的问题和自动验证在NUS的实际检查。其他感兴趣的CS教练应该联系史蒂文如果你想尝试这样的“测试模式”。
出版物名单
这项工作在2012年ACM ICPC世界总决赛(波兰,华沙)和IOI 2012年IOI大会(意大利Sirmione-Montichiari)的CLI讲习班上进行了简要介绍。您可以点击此链接阅读我们2012年关于这个系统的文章(它在2012年还没有被称为VisuAlgo)。
这项工作主要由我过去的学生完成。最近的最后报告是:Erin,Wang Zi,Rose,Ivan。
错误申报或请求添加新功能
VisuAlgo不是一个完成的项目。 Steven Halim博士仍在积极改进VisuAlgo。如果您在使用VisuAlgo并在我们的可视化页面/在线测验工具中发现错误,或者如果您想要求添加新功能,请联系Dr Steven Halim博士。他的联系邮箱是他的名字加谷歌邮箱后缀:StevenHalim@gmail.com。