7    VisuAlgo.net / /graphds Login Undirected/Unweighted U/W D/U D/W
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A graph is made up of vertices/nodes and edges/lines that connect those vertices.


A graph may be undirected (meaning that there is no distinction between the two vertices associated with each bidirectional edge) or a graph may be directed (meaning that its edges are directed from one vertex to another but not necessarily in the other direction).


A graph may be weighted (by assigning a weight to each edge, which represent numerical values associated with that connection) or a graph may be unweighted (either all edges have unit weight 1 or all edges have the same constant weight).


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Most graph problems that we discuss in VisuAlgo involves simple graphs.


In a simple graph, there is no (self-)loop edge (an edge that connects a vertex with itself) and no multiple/parallel edges (edges between the same pair of vertices). In another word: There can only be up to one edge between a pair of distinct vertices.


The number of edges E in a simple graph can only range from 0 to O(V2).


Graph algorithms on simple graphs are easier than on non-simple graphs.


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无向边e:(u,v)被认为是以其两个端点顶点入射:u和v。如果两个顶点与公共边缘一起入射,则它们被称为相邻(或邻居)。例如,边(0,2)入射到顶点0 + 2并且顶点0 + 2是相邻的。

如果两个边缘与公共顶点一起入射,则它们被称为相邻边缘。例如,边(0,2)和(2,4)是相邻的。

无向图中的顶点v的度数是与v一起入射的边的数量。度为0的顶点被称为孤立顶点。例如,顶点0/2/6分别具有2/3/1的度数。

图G的子图G'是包含G的顶点和边的子集的(较小的)图。例如,三角形{0,1,2}是当前显示的图的子图。

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A path (of length n) in an (undirected) graph G is a sequence of vertices {v0, v1, ..., vn-1, vn} such that there is an edge between vi and vi+1i ∈ [0..n-1] along the path.


If there is no repeated vertex along the path, we call such path as a simple path.


For example, {0, 1, 2, 4, 5} is one simple path in the currently displayed graph.

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An undirected graph G is called connected if there is a path between every pair of distinct vertices of G. For example, the currently displayed graph is not a connected graph.


An undirected graph C is called a connected component of the undirected graph G if 1). C is a subgraph of G; 2). C is connected; 3). no connected subgraph of G has C as a subgraph and contains vertices or edges that are not in C (i.e. C is the maximal subgraph that satisfies the other two criteria).


For example, the currently displayed graph have {0, 1, 2, 3, 4} and {5, 6} as its two connected components.

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在有向图中,前面提到的一些术语有小的调整。

如果我们有一个有向边e:(u→v),我们说v与u相邻但不一定在另一个方向上。例如,1在当前显示的有向图中与0相邻但0与1不相邻。

在有向图中,我们必须进一步将顶点v的度数区分为度数和度数。度数/出度是分别从v进入/离开的边数。例如,顶点1的入度/出度分别为2/1。

在有向图中,我们定义了强连通分量(SCC)的概念。在当前显示的有向图中,我们将{0},{1,2,3}和{4,5,6,7}作为其三个SCC。
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循环是以相同顶点开始和结束的路径。

非循环图是不包含循环的图。

在无向图中,每个无向边都会导致一个微不足道的循环,尽管我们通常不会将其归类为循环。

非循环的有向图具有特殊名称:有向无环图(DAG),如上所示。

我们可以在非循环图上执行有趣的算法,这些算法将在此可视化页面和VisuAlgo中的其他图形可视化页面中进行探索。
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A graph with specific properties involving its vertices and/or edges structure can be called with its specific name, like Tree (like the one currently shown), Complete Graph, Bipartite Graph, Directed Acyclic Graph (DAG), and also the less frequently used: Planar Graph, Line Graph, Star Graph, Wheel Graph, etc.


In this visualization, we will highlight the first four special graphs later.

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Graph appears very often in various form in real life. The most important part in solving graph problem is thus the graph modeling part, i.e. reducing the problem in hand into graph terminologies: vertices, edges, weights, etc.

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Social Network: Vertices can represent people, Edges represent connection between people (usually undirected and unweighted).


For example, see the undirected graph that is currently shown. This graph shows 7 vertices (people) and 8 edges (connection/relationship) between them. Perhaps we can ask questions like these:

  1. Who is/are the friend(s) of people no 0?
  2. Who has the most friend(s)?
  3. Is there any isolated people (those with no friend)?
  4. Is there a common friend between two strangers: People no 3 and people no 5?
  5. Etc...
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运输网络:顶点可以表示站点,边缘表示站点之间的连接(通常是加权的)。

例如,请参阅当前显示的有向加权图。该图显示了5个顶点(站点/位置)和6个边缘(站点之间的连接/道路,具有正的权重行进时间,如图所示)。假设我们正在开车。我们或许可以问一下从0号站到4号站的路径是什么,以便我们用最少的时间到达4号站?

讨论:想想其他一些可以建模为图形的现实生活场景。
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To toggle between the graph drawing modes, select the respective header. We have:

  1. U/U = Undirected/Unweighted,
  2. U/W = Undirected/Weighted,
  3. D/U = Directed/Unweighted, and
  4. D/W = Directed/Weighted.

We restrict the type of graphs that you can draw according to the selected mode.

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You can click any one of the example graphs and visualize the graph above.


You can also draw a graph directly in the visualization area:

  1. Click on empty space to add vertex,
  2. Click a vertex, hold, drag the drawn edge to another vertex, and drop it there to add an edge (PS: This action is not available for mobile users; you need a mouse),
  3. Select a vertex/edge and press 'Delete' key to delete that vertex/edge,
  4. Select an edge and press 'Enter' to change the weight of that edge [0..99],
  5. Press and hold 'Ctrl', then you can click and drag a vertex around.
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We limit the graphs discussed in VisuAlgo to be simple graphs. Refer to its discussion in this slide.


We also limit the number of vertices that you can draw on screen to be up to 10 vertices, ranging from vertex 0 to vertex 9. This, together with the simple graph constraint above, limit the number of undirected/directed edges to be 45/90, respectively.

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All example graphs can be found here.


For now, we provide at least two example graphs per category (U/U, U/W, D/U, D/W).


Note that after loading one of these example graphs, you can further modify the currently displayed graph to suit your needs.

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Tree, Complete, Bipartite, Directed Acyclic Graph (DAG) are properties of special graphs. As you modify the graph in the visualization/drawing area above, these properties are checked and updated instantly.


There are other less frequently used special graphs: Planar Graph, Line Graph, Star Graph, Wheel Graph, etc, but they are not currently auto-detected in this visualization when you draw them.

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Tree is a connected graph with V vertices and E = V-1 edges, acyclic, and has one unique path between any pair of vertices. Usually a Tree is defined on undirected graph.


An undirected Tree (see above) actually contains trivial cycles (caused by its bidirectional edges) but it does not contain non-trivial cycle. A directed Tree is clearly acyclic.


As a Tree only have V-1 edges, it is usually considered a sparse graph.


We currently show our U/U: Tree example. You can go to 'Exploration Mode' and draw your own trees.

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Not all Trees have the same graph drawing layout of having a special root vertex at the top and leaf vertices (vertices with degree 1) at the bottom. The (star) graph shown above is also a Tree as it satisfies the properties of a Tree.


Tree with one of its vertex designated as root vertex is called a rooted Tree.


We can always transform any Tree into a rooted Tree by designating a specific vertex (usually vertex 0) as the root, and run a DFS or BFS algorithm from the root.

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In a rooted tree, we have the concept of hierarchies (parent, children, ancestors, descendants), subtrees, levels, and height. We will illustrate these concepts via examples as their meanings are as with real-life counterparts:

  1. The parent of 0/1/7/2/4 are none/0/0/1/3, respectively,
  2. The children of 0/1/7 are {1,7}/{2,3,6}/{8,9}, respectively,
  3. The ancestors of 4/8 are {3,1,0}/{7,0}, respectively,
  4. The descendants of 1/7 are {2,3,4,5,6}/{8,9}, respectively,
  5. The subtree rooted at 1 includes 1, its descendants, and all associated edges,
  6. Level 0/1/2/3 members are {0}/{1,7}/{2,3,6,8,9}/{4,5}, respectively,
  7. The height of this rooted tree is its maximum level = 3.
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For rooted tree, we can also define additional properties:


A binary tree is a rooted tree in which a vertex has at most two children that are aptly named: left and right child. We will frequently see this form during discussion of Binary Search Tree and Binary Heap.


A full binary tree is a binary tree in which each non-leaf (also called the internal) vertex has exactly two children. The binary tree shown above fulfils this criteria.


A complete binary tree is a binary tree in which every level is completely filled, except possibly the last level may be filled as far left as possible. We will frequently see this form especially during discussion of Binary Heap.

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Complete graph is a graph with V vertices and E = V*(V-1)/2 edges (or E = O(V2)), i.e. there is an edge between any pair of vertices. Usually a Complete graph is denoted with KV.


Complete graph is the most dense simple graph.


We currently show our U/W: K5 example. You can go to 'Exploration Mode' and draw your own complete graphs (a bit tedious for larger V though).

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Bipartite graph is an undirected graph with V vertices that can be partitioned into two disjoint set of vertices of size m and n where V = m+n. There is no edge between members of the same set. Bipartite graph is also free from odd-length cycle.


We currently show our U/U: Bipartite example. You can go to 'Exploration Mode' and draw your own bipartite graphs.


A Bipartite Graph can also be complete, i.e. all m vertices from one disjoint set are connected to all n vertices from the other disjoint set. When m = n = V/2, such Complete Bipartite Graphs also have E = O(V2).

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Directed Acyclic Graph (DAG) is a directed graph that has no cycle, which is very relevant for Dynamic Programming (DP) techniques.


Each DAG has at least one Topological Sort/Order which can be found with a simple tweak to DFS/BFS Graph Traversal algorithm. DAG will be revisited again in DP technique for SSSP on DAG.


We currently show our D/W: Four 0→4 Paths example. You can go to 'Exploration Mode' and draw your own DAGs.

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There are many ways to store graph information into a graph data structure. In this visualization, we show three graph data structures: Adjacency Matrix, Adjacency List, and Edge List — each with its own strengths and weaknesses.

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Adjacency Matrix (AM) is a square matrix where the entry AM[i][j] shows the edge's weight from vertex i to vertex j. For unweighted graphs, we can set a unit weight = 1 for all edge weights. An 'x' means that that vertex does not exist (deleted).


We simply use a C++/Java native 2D array of size VxV to implement this data structure.

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Space Complexity Analysis: An AM unfortunately requires a big space complexity of O(V2), even when the graph is actually sparse (not many edges).


Discussion: Knowing the large space complexity of AM, when is it beneficial to use it? Or is AM always an inferior graph data structure and should not be used at all times?

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Adjacency List (AL) is an array of V lists, one for each vertex (usually in increasing vertex number) where for each vertex i, AL[i] stores the list of i's neighbors. For weighted graphs, we can store pairs of (neighbor vertex number, weight of this edge) instead.


We use a Vector of Vector pairs (for weighted graphs) to implement this data structure.
In C++: vector<vector<pair<int,int>>> AL;
In Java: Vector < Vector < IntegerPair > > AL;
// class IntegerPair in Java is like pair<int,int> in C++, next slide

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class IntegerPair implements Comparable<IntegerPair> {
Integer _f, _s;
public IntegerPair(Integer f, Integer s) { _f = f; _s = s; }
public int compareTo(IntegerPair o) {
if (!this.first().equals(o.first())) // this.first() != o.first()
return this.first() - o.first(); // is wrong as we want to
else // compare their values,
return this.second() - o.second(); // not their references
}
Integer first() { return _f; }
Integer second() { return _s; }
}
// IntegerTriple is similar to IntegerPair, just that it has 3 fields
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We use pairs as we need to store pairs of information for each edge: (neighbor vertex number, edge weight) where weight can be set to 0 or unused for unweighted graph.


We use Vector of Pairs due to Vector's auto-resize feature. If we have k neighbors of a vertex, we just add k times to an initially empty Vector of Pairs of this vertex (this Vector can be replaced with Linked List).


We use Vector of Vectors of Pairs for Vector's indexing feature, i.e. if we want to enumerate neighbors of vertex u, we use AL[u] (C++) or AL.get(u) (Java) to access the correct Vector of Pairs.

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Space Complexity Analysis: AL has space complexity of O(V+E), which is much more efficient than AM and usually the default graph DS inside most graph algorithms.


Discussion: AL is the most frequently used graph data structure, but discuss several scenarios when AL is actually not the best graph data structure?

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边缘列表(EL)是具有连接顶点及其权重的边的集合。通常,这些边缘通过增加重量来分类,例如, Kruskal最小生成树(MST)问题算法的一部分。但是,在此可视化中,我们通过增加第一个顶点数来对边进行排序,如果是连接,则通过增加第二个顶点数来对边进行排序请注意,无向/有向图中的双向边分别列出一次/两次。

我们使用三元组Vector来实现这种数据结构。在C ++中:vector <tuple <int,int,int >> EL;在Java中:Vector <integertriple> EL; // Java中的IntegerTriple类似于C ++中的tuple <int,int,int>
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Space Complexity Analysis: EL has space complexity of O(E), which is much more efficient than AM and as efficient as AL.


Discussion: Elaborate the potential usage of EL other than inside Kruskal's algorithm for Minimum Spanning Tree (MST) problem!

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After storing our graph information into a graph DS, we can answer a few simple queries.

  1. Counting V,
  2. Counting E,
  3. Enumerating neighbors of a vertex u,
  4. Checking the existence of edge (u, v), etc.
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In an AM and AL, V is just the number of rows in the data structure that can be obtained in O(V) or even in O(1) — depending on the actual implementation.


Discussion: How to count V if the graph is stored in an EL?


PS: Sometimes this number is stored/maintained in a separate variable so that we do not have to re-compute this every time — especially if the graph never/rarely changes after it is created, hence O(1) performance, e.g. we can store that there are 7 vertices (in our AM/AL/EL data structure) for the example graph shown above.

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In an EL, E is just the number of its rows that can be counted in O(E). Note that depending on the need, we may store a bidirectional edge just once in the EL but on other case, we store both directed edges inside the EL.


In an AL, E can be found by summing the length of all V lists and divide the final answer by 2 (for undirected graph). This requires O(V+E) computation time as each vertex and each edge is only processed once.


Discussion: How to count E if the graph is stored in an AM?


PS: Sometimes this number is stored/maintained in a separate variable for efficiency, e.g. we can store that there are 8 undirected edges (in our AM/AL/EL data structure) for the example graph shown above.

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In an AM, we need to loop through all columns of AM[u][j] ∀j ∈ [0..V-1] and report pair of (j, AM[u][j]) if AM[u][j] is not zero. This is O(V) — slow.


In an AL, we just need to scan AL[u]. If there are only k neighbors of vertex u, then we just need O(k) to enumerate them — this is called an output-sensitive time complexity and is already the best possible.


We usually list the neighbors in increasing vertex number. For example, neighbors of vertex 1 in the example graph above are {0, 2, 3}, in that increasing vertex number order.


Discussion: How to do this if the graph is stored in an EL?

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In an AM, we can simply check if AM[u][v] is non zero. This is O(1) — the fastest possible.


In an AL, we have to check whether AL[u] contains vertex v or not. This is O(k) — slower.


For example, edge (2, 4) exists in the example graph above but edge (2, 6) does not exist.


Note that if we have found edge (u, v), we can also access and/or update its weight.


Discussion: How to do this if the graph is stored in an EL?

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Quiz: So, what is the best graph data structure?

It Depends
Edge List
Adjacency Matrix
Adjacency List


Discussion: Why?

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You have reached the end of the basic stuffs of this relatively simple Graph Data Structures and we encourage you to explore further in the Exploration Mode by drawing your own graphs.


However, we still have a few more interesting Graph Data Structures challenges for you that are outlined in this section.


Note that graph data structures are usually just the necessary but not sufficient part to solve the harder graph problems like MST, SSSP, Max Flow, Matching, etc.

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For a few more interesting questions about this data structure, please practice on Graph Data Structures training module (no login is required).


However, for registered users, you should login and then go to the Main Training Page to officially clear this module and such achievement will be recorded in your user account.

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Try to solve two basic programming problems that somewhat requires the usage of graph data structure without any fancy graph algorithms:

  1. UVa 10895 - Matrix Transpose and,
  2. Kattis - flyingsafely.
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V=0, E=0 • Tree? No • Complete? No • Bipartite? No • DAG? No • 交叉? No
Adjacency Matrix
012
0010
1101
2010
Adjacency List
0: 1
1: 02
2: 1
Edge List
0: 01
1: 12

U/U: CP3 Fig 2.4

U/U: CP3 Fig 2.4, disjoint

U/U: Tree

U/U: Binary Tree

U/U: Bipartite

U/U: CP2.5C - Knight Jump

U/W: CP3 Fig 4.10

U/W: K5 (Complete)

U/W: Star

D/U: CP3 Fig 4.4

D/U: CP3 Fig 4.8

D/U: Cyclic

D/W: CP3 Fig 4.26B*

D/W: Four 0→4 Paths

>
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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。