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Maximum (Max) Flow is one of the problems in the family of problems involving flow in networks.


In Max Flow problem, we aim to find the maximum flow from a particular source vertex s to a particular sink vertex t in a directed weighted graph G.


There are several algorithms for finding the maximum flow including Ford-Fulkerson method, Edmonds-Karp algorithm, and Dinic's algorithm (there are a few others, but they are not included in this visualization yet).


The dual problem of Max Flow is Min Cut, i.e., by finding the max s-t flow of G, we also simultaneously find the min s-t cut of G, i.e., the set of edges with minimum weight that have to be removed from G so that there is no path from s to t in G.


Remarks: By default, we show e-Lecture Mode for first time (or non logged-in) visitor.
If you are an NUS student and a repeat visitor, please login.

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Max-Flow (or Min-Cut) problems arise in various applications, e.g.,

  1. Transportation-related problems (what is the best way to send goods/material from s (perhaps a factory) to t (perhaps a super-sink of all end-users)
  2. Network attacks problems (sabotage/destroy some edges to disconnect two important points s and t)
  3. (Bipartite) Matching and Assignment problems (that also has specialized algorithms, see Graph Matching visualization
  4. Sport teams prospects
  5. Image segmentation, etc...

Pro-tip 1: Since you are not logged-in, you may be a first time visitor (or not an NUS student) who are not aware of the following keyboard shortcuts to navigate this e-Lecture mode: [PageDown]/[PageUp] to go to the next/previous slide, respectively, (and if the drop-down box is highlighted, you can also use [→ or ↓/← or ↑] to do the same),and [Esc] to toggle between this e-Lecture mode and exploration mode.

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This visualization page will show the execution of a chosen Max Flow algorithm running on a flow (residual) graph.


To make the visualization of these flow graphs consistent, we enforce a graph drawing rule for this page whereby the source vertex s/sink vertex t is always vertex 0/V-1 and is always drawn on the leftmost/rightmost side of the visualization, respectively. Another visualization-specific constraint is that the edge capacities are integers between [1..99].


These visualization-specific constraints do not exist in the standard max flow problems.


Pro-tip 2: We designed this visualization and this e-Lecture mode to look good on 1366x768 resolution or larger (typical modern laptop resolution in 2021). 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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The input for a Max Flow algorithm is a flow graph (a directed weighted graph G = (V, E) where edge weight of edge e represent the capacity c(e) (the unit is problem-dependent, e.g., liters/second, person/hour, etc) of flow that can go through that edge) with two distinguished vertices: The source vertex s (with in-degree 0) and the sink/target/destination vertex t (with out-degree 0). The flow graph is usually s-t connected, i.e., there is at least one path from s to t (otherwise the max flow is trivially 0).


In this visualization, these two additional inputs of s (usually vertex 0) and t (usually vertex V-1) are asked before the execution of the chosen Max Flow algorithm and can be customized by the user.


Pro-tip 3: Other than using the typical media UI at the bottom of the page, you can also control the animation playback using keyboard shortcuts (in Exploration Mode): Spacebar to play/pause/replay the animation, / to step the animation backwards/forwards, respectively, and -/+ to decrease/increase the animation speed, respectively.

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The output for a Max Flow algorithm is the max flow value and an assignment of flow f to each edge that satisfies two important constraints:

  1. Capacity constraints (flow on each edge (f(e)) is between 0 and its (unit) capacity (c(e)), i.e., 0 ≤ f(e)c(e) — not negative and not more than the capacity), and
  2. Equilibrium constraints (for every vertex except s and t, flow-in = flow-out)
so that the value of the flow (value(f) = ∑v: (s, v) ∈ E f(s,v)) is maximum.


In this visualization, we focus on showing the final max flow value and the final ST-min cut components at the end of each max flow algorithm execution, instead of the precise assignment of flow f to each edge, i.e., f(e) must be computed manually from the initial capacity c(e) (first frame of the animation) minus the final residual capacity of that edge e (last frame of the animation). This missing feature will likely be added in the next iteration of this visualization page.


Discussion: Is there other ways to compute the value of the flow value(f)?

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The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

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At the start of the three Max Flow algorithms discussed in this visualization (Ford-Fulkerson method, Edmonds-Karp algorithm, and Dinic's algorithm), the initial flow graph is converted into residual graph (with potential addition of back flow edges with initial capacity of zeroes).


The edges in the residual graph store the remaining capacities of those edges that can be used by future flow(s). At the beginning, these remaining capacities equal to the original capacities as specified in the input flow graph.


A Max Flow algorithm will send flows to use some (or all) of these available capacities, iteratively.


Once the remaining capacity of an edge reaches 0, that edge can no longer admit any more flow. In the near future, we will update this visualization so that any edge in the residual graph that has capacity 0 (including the initial zeroes of the back flow edges) is not shown in the visualization.

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There are three different sources for specifying an input flow graph:

  1. Draw Graph: You can draw any directed weighted (weight ∈ [1..99]) graph as the input flow graph with vertex 0 as the default source vertex (the left side of the screen) and vertex V-1 as the default sink vertex (the right side of the screen),
  2. Modeling: Several graph problems can be reduced into a Max Flow problem. In this visualization, we have the modeling examples for the famous Maximum Cardinality Bipartite Matching (MCBM) problem, Rook Attack problem (currently disabled), and Baseball Elimination problem (currently disabled),
  3. Example Graphs: You can select from the list of our selected example flow graphs to get you started.
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There are three different max flow algorithms in this visualization:

  1. The slow O(mf × E) Ford-Fulkerson method,
  2. The O(V × E^2) Edmonds-Karp algorithm, or
  3. The O(V^2 × E) Dinic's algorithm.

There are a few other max flow algorithms out there, but they are not available in this visualization yet.

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For the three Max Flow algorithms discussed in this visualization, successive flows are sent from the source vertex s to the sink vertex t via available augmenting paths (augmenting path is a path from s to t that goes through edges with positive weight residual capacity (c(e)-f(e)) left).


The three Max Flow algorithms in this visualization have different behavior on how they find augmenting paths.


However, all three Max Flow algorithms in this visualization stop when there is no more augmenting path possible and report the max flow value (and the assignment of flow on each edge in the flow graph).


Later we will discuss that this max flow value is also the min cut value of the flow graph (that famous Max-Flow/Min-Cut Theorem).

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start with 0 flow
while there exists an augmenting path: // iterative algorithm
  find an augmenting path (for now, 'any' graph traversal will do)
  compute bottleneck capacity
  increase flow on the path by the bottleneck capacity
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This famous theorem states that in a flow network, the maximum flow from s to t is equal to the total weight of the edges in a minimum cut, i.e., the smallest total weight of the edges that have to be removed to disconnect s from t.


In a typical Computer Science classes, the lecturer will usually spend some time to properly explain this theorem (explaining what is an st-cut, capacity of an st-cut, net flow across an st-cut equals to current flow f assignment that will never exceed the capacity of the cut, and finally that Max-Flow/Min-Cut Theorem). For this visualization, we just take this statement as it is.


Discussion: For live class in NUS, we will actually discuss these theorem.

🕑

The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

Using the Max-Flow/Min-Cut Theorem, we can then prove that flow f is a maximum flow if and only if there is no (more) augmenting path remaining in the residual graph.


As this is what Ford-Fulkerson Method is doing, we can conclude the correctness of this Ford-Fulkerson Method, i.e., if Ford-Fulkerson Method terminates, then there is no augmenting path left and thus the resulting flow is maximum (and we can also construct the equivalent Min-Cut, next slide).

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We can constructively identify the edges in the Min-Cut as follows:

  1. Run Ford-Fulkerson (or any other Max Flow) algorithm until it terminates.
  2. Let S be the set of vertices that are still reachable from the source s.
    We can run DFS (or BFS) in the residual graph from the source vertex s.
    All the vertices that are still reachable are in S.
    Let T be the remaining vertices, i.e., T = V \ S.
  3. For every edge in S, enumerate outgoing edges:
    If edge exits S (and into T), add to min-cut.
    If both ends of edge are in S, then continue.

That's it, (S,T) is an st-cut, edges from (S → T) are the minimum cut, and the flow that goes through this minimum cut (S,T) is the maximum possible.

🕑

The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

Ford-Fulkerson method always terminates if the capacities are integers.


This is because every iteration of Ford-Fulkerson method always finds a new augmenting path and each augmenting path must has bottleneck capacity at least 1 (due to that integer constraint). Therefore, each iteration increases the flow of at least one edge by at least 1, edging the Ford-Fulkerson closer to termination.


As the number of edges is finite (as well as the finite max capacity per edge), this guarantees the eventual termination of Ford-Fulkerson method when the max flow mf is reached and there is no more augmenting path left.


In the worst case, Ford-Fulkerson method runs for mf iterations, and each time it uses O(E) DFS. The rough overall runtime is thus O(mf × E) — this is actually not desirable especially if the value of mf is a huge number.


Discussion: What if the capacities are rational numbers? What if the capacities are floating-point numbers?

🕑

The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

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Idea: What if we don't consider any augmenting paths but consider augmenting paths with the smallest number of edges involved first (so we don't put flow on more edges than necessary).

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Implementation: We first ignore capacity of the edges first (assume all edges in the residual graph have weight 1), and we run O(E) BFS to find the shortest (in terms of # of edges used) augmenting path. Everything else is the same as the basic Ford-Fulkerson Method outlined earlier.


It can be proven that Edmonds-Karp will use at most O(VE) iterations thus it runs in at most in O(VE * E) = O(VE^2) time.

🕑

The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.


If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.


FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

Dinic's algorithm also uses similar strategy of finding shortest augmenting paths first.


But Dinic's algorithm runs in a faster time of O(V^2 × E) due to the more efficient usage of BFS shortest path information.


This slide will be expanded.

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When you are presented with a Max Flow (or a Min Cut)-related problem, we do not have to reinvent the wheel every time.


You are allowed to use/modify/adapt/enhance our implementation code for Max Flow Algorithms (Edmonds-Karp/Dinic's): maxflow.cpp | py | java | ml.


You have reached the last slide. 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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Visualisation Scale

Toggle V. Number for 0.5x

编辑图表

Modeling

图示

Ford-Fulkerson

Edmonds-Karp

Dinic

Min-Cost-Max-Flow

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1.0x (Default)

0.5x (Minimal Details)

二分图匹配

Corner Case

Special Case

CS4234 MF Demo

CP4 8.15* (Dinic Showcase)

Matching with Capacity

waif (AC)

Reduction

MCMF

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t =

前进

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前进

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前进

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前进

关于 团队 使用条款
隐私政策

关于

VisuAlgo最初由副教授Steven Halim于2011年构思,旨在通过提供自学、互动式学习平台,帮助学生更深入地理解数据结构和算法。

VisuAlgo涵盖了Steven Halim博士与Felix Halim博士、Suhendry Effendy博士合著的书《竞技编程》中讨论的许多高级算法。即使过去十年,VisuAlgo仍然是可视化和动画化这些复杂算法的独家平台。

虽然VisuAlgo主要面向新加坡国立大学(NUS)的学生,包括各种数据结构和算法课程(例如CS1010/等价课程,CS2040/等价课程(包括IT5003),CS3230,CS3233和CS4234),但它也是全球好奇心的宝贵资源,促进在线学习。

最初,VisuAlgo并不适用于智能手机等小触摸屏,因为复杂的算法可视化需要大量的像素空间和点击拖动交互。为了获得最佳用户体验,建议使用最低分辨率为1366x768的屏幕。然而,自2022年4月以来,VisuAlgo的移动(精简)版本已经推出,使得在智能手机屏幕上使用VisuAlgo的部分功能成为可能。

VisuAlgo仍然在不断发展中,正在开发更复杂的可视化。目前,该平台拥有24个可视化模块。

VisuAlgo配备了内置的问题生成器和答案验证器,其“在线测验系统”使学生能够测试他们对基本数据结构和算法的理解。问题根据特定规则随机生成,并且学生提交答案后会自动得到评分。随着越来越多的计算机科学教师在全球范围内采用这种在线测验系统,它可以有效地消除许多大学标准计算机科学考试中手工基本数据结构和算法问题。通过给通过在线测验的学生分配一个小但非零的权重,计算机科学教师可以显著提高学生对这些基本概念的掌握程度,因为他们可以在参加在线测验之前立即验证几乎无限数量的练习题。每个VisuAlgo可视化模块现在都包含自己的在线测验组件。

VisuAlgo已经被翻译成三种主要语言:英语、中文和印尼语。此外,我们还用各种语言撰写了关于VisuAlgo的公开笔记,包括印尼语、韩语、越南语和泰语:

id, kr, vn, th.

团队

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

本科生研究人员 1
CDTL TEG 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
Jun 2013-Apr 2014 Rose Marie Tan Zhao Yun, Ivan Reinaldo

本科生研究人员 2
CDTL TEG 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 2
Jun 2014-Apr 2015: Erin Teo Yi Ling, Wang Zi
Jun 2016-Dec 2017: Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir
Aug 2021-Apr 2023: Liu Guangyuan, Manas Vegi, Sha Long, Vuong Hoang Long, Ting Xiao, Lim Dewen Aloysius

本科生研究人员 3
Optiver: Aug 2023-Oct 2023: Bui Hong Duc, Oleh Naver, Tay Ngan Lin

最后一年项目/ UROP学生 3
Aug 2023-Apr 2024: Xiong Jingya, Radian Krisno, Ng Wee Han

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

致谢
NUS教学与学习发展中心(CDTL)授予拨款以启动这个项目。在2023/24学年,Optiver的慷慨捐赠将被用来进一步开发 VisuAlgo。

使用条款

VisuAlgo慷慨地向全球计算机科学界提供免费服务。如果您喜欢VisuAlgo,我们恳请您向其他计算机科学学生和教师宣传它的存在。您可以通过社交媒体平台(如Facebook、YouTube、Instagram、TikTok、Twitter等)、课程网页、博客评论、电子邮件等方式分享VisuAlgo。

数据结构与算法(DSA)的学生和教师可以直接在课堂上使用本网站。如果您从本网站截取屏幕截图或视频,可以在其他地方使用,但请引用本网站的URL(https://visualgo.net)和/或下面的出版物列表作为参考。但请不要下载VisuAlgo的客户端文件并将其托管在您的网站上,因为这构成了抄袭行为。目前,我们不允许他人分叉此项目或创建VisuAlgo的变体。个人使用离线副本的客户端VisuAlgo是可以接受的。

请注意,VisuAlgo的在线测验组件具有重要的服务器端元素,保存服务器端脚本和数据库并不容易。目前,普通公众只能通过“培训模式”访问在线测验系统。“测试模式”提供了一个更受控制的环境,用于在新加坡国立大学的真实考试中使用随机生成的问题和自动验证。


出版物列表

这项工作曾在2012年国际大学生程序设计竞赛(波兰,华沙)的CLI研讨会上和2012年国际信息学奥林匹克竞赛(意大利,锡尔米奥内-蒙蒂基亚里)的IOI会议上展示过。您可以点击此链接阅读我们2012年关于该系统的论文(当时还没有称为VisuAlgo),以及此链接阅读2015年的简短更新(将VisuAlgo与之前的项目关联起来)。


错误报告或新功能请求

VisuAlgo并不是一个完成的项目。Steven Halim副教授仍在积极改进VisuAlgo。如果您在使用VisuAlgo时发现任何可视化页面/在线测验工具中的错误,或者您想要请求新功能,请联系Steven Halim副教授。他的联系方式是将他的名字连接起来,然后加上gmail dot com。

隐私政策

版本 1.2 (更新于2023年8月18日星期五)。

自2023年8月18日(星期五)起,我们不再使用 Google Analytics。因此,我们现在使用的所有 cookies 仅用于此网站的运营。即使是首次访问的用户,烦人的 cookie 同意弹窗现在也已关闭。

自2023年6月7日(星期五)起,由于 Optiver 的慷慨捐赠,全世界的任何人都可以自行创建一个 VisuAlgo 账户,以存储一些自定义设置(例如,布局模式,默认语言,播放速度等)。

此外,对于 NUS 学生,通过使用 VisuAlgo 账户(一个 NUS 官方电子邮件地址,课堂名册中的学生姓名,以及在服务器端加密的密码 - 不存储其他个人数据),您同意您的课程讲师跟踪您的电子讲义阅读和在线测验培训进度,这是顺利进行课程所必需的。您的 VisuAlgo 账户也将用于参加 NUS 官方的 VisuAlgo 在线测验,因此,将您的账户凭据传递给他人代您进行在线测验构成学术违规。课程结束后,您的用户账户将被清除,除非您选择保留您的账户(OPT-IN)。访问完整的 VisuAlgo 数据库(包含加密密码)的权限仅限于 Halim 教授本人。

对于全球其他已经给 Steven 写过信的 CS 讲师,需要一个 VisuAlgo 账户(您的(非 NUS)电子邮件地址,您可以使用任何显示名称,以及加密密码)来区分您的在线凭据与世界其他地方。您的账户将具有 CS 讲师特定的功能,即能够查看隐藏的幻灯片,这些幻灯片包含了在隐藏幻灯片之前的幻灯片中提出的问题的(有趣的)答案。您还可以访问 VisuAlgo 在线测验的 Hard 设置。您可以自由地使用这些材料来增强您的数据结构和算法课程。请注意,未来可能会有其他 CS 讲师特定的功能。

对于任何拥有 VisuAlgo 账户的人,如果您希望不再与 VisuAlgo 工具有关联,您可以自行删除您的账户。