图可以是无向的（意味着在与每个双向边相关联的两个顶点之间没有区别）或者可以指向图（意味着其边缘从一个顶点指向另一个顶点但不一定在另一个方向上）。
可以对图形进行加权（通过向每个边缘分配权重，其表示与该连接相关联的数值）或者图形可以是未加权的（所有边缘具有单位权重1或者所有边缘具有相同的恒定权重）。
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在一个简单图中，没有（自 - ）环边（连接顶点与自身的边），没有多边/平行边（同一对顶点之间的边）。换句话说：在一对不同的顶点之间最多只能有一条边。
简单图中的边E的数量范围仅为0到O（V2）。
简单图上的图算法比非简单图上的算法更容易。
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.
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。
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.
如果路径上没有重复的顶点，我们称这样的路径为简单路径。
例如，{0,1,2,4,5}是当前显示的图形中的一个简单路径。
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.
如果要把无向图C称为无向图G的连通分量，它必须符合以下的条件。 1）C是G的子图; 2） C是个连接图; 3）没有G的连通子图将C作为子图并且包含不在C中的顶点或边（即C是满足其他两个标准的最大子图）。
例如，当前显示的图形具有{0,1,2,3,4}和{5,6}作为其两个连接分量。
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。
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。
在此可视化中，我们将在稍后突出显示前四个特殊图表。
例如，请参阅当前显示的无向图。此图显示了它们之间的7个顶点（人）和8个边（连接/关系）。也许我们可以提出这样的问题：
- 谁是0号的朋友？
- 谁拥有最多的朋友？
- 有没有孤立的人（那些没有朋友的人）？
- 两个陌生人之间是否有共同的朋友：3号人和5号人？
- 等等...
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。
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.
要在图形绘制模式之间切换，请选择相应的标题。 我们有：
- U/U = 无向/不加权,
- U/W = 无向/加权,
- D/U = 有向/不加权, and
- D/W = 有向/加权.
我们根据所选模式限制您可以绘制的图形类型。
您可以单击任何一个示例图形并可视化上图。
您还可以直接在可视化区域中绘制图形：
- 点击空白区域添加顶点，
- 单击顶点，按住，将绘制的边缘拖动到另一个顶点，然后将其放在那里以添加边缘（PS：此操作不适用于移动用户;您需要鼠标），
- 选择顶点/边缘并按“删除”键删除该顶点/边，
- 选择边缘并按“Enter”以更改该边缘的权重[0..99]，
- 按住“Ctrl”，然后您可以单击并拖动顶点。
我们还将可以在屏幕上绘制的顶点数限制为最多10个顶点，范围从顶点0到顶点9.这与上面的简单图形约束一起，将无向/有向边的数量限制为45 /分别为90。
目前，我们为每个类别提供七个示例图表（U / U，U / W，D / U，D / W）。
请注意，在加载其中一个示例图表后，您可以进一步修改当前显示的图表以满足您的需要。
还有其他不太常用的特殊图形：平面图，线图，星图，轮图等，但在绘制它们时，它们当前在此可视化中不会被自动检测到。
无向树（见上文）实际上包含琐碎的循环（由其双向边缘引起），但它不包含非平凡循环。有针对性的树显然是非循环的。
由于树只有V-1边，因此它通常被认为是稀疏图。
我们目前显示我们的U / U：Tree示例。您可以进入“探索模式”并绘制自己的树木。
将其顶点之一指定为根顶点的树称为有根树。
我们总是可以通过将特定顶点（通常是顶点0）指定为根来将任何树转换为根树，并从根运行DFS或BFS算法。
- 0/1/7/9/4的父级分别为none / 0 / 1 / 8 / 3，
- 0/1/7的子女分别为{1,8} / {2,3,6,7} / none，
- 4/8的祖先分别为{3,1,0} / {0}，
- 1/8的后代分别是{2,3,4,5,6,7} / {9}，
- 以1为根的子树包括1，其后代和所有相关边，
- 0/1/2/3级成员分别为{0} / {1,8} / {2,3,6,7,9} / {4,5}，
- 这个有根树的高度是它的最高级别= 3。
二叉树是一个有根树，其中一个顶点最多有两个孩子，它们被恰当地命名为：left和right child（左子节点和右子节点）。在讨论二叉搜索树和二叉堆时，我们经常会看到这种形式。
满二叉树是一个其中每个非叶（也称为内部）顶点恰好有两个子节点的二叉树。上面显示的二叉树符合此标准。
一个完全二叉树的每个级别都被完全填充，除了最后一级可能尽可能地填充。我们经常会在讨论二叉堆时看到这种形式。
完整图是最密集的简单图。
我们目前展示了我们的U/W: K5示例。您可以进入“探索模式”并绘制自己的完整图形（虽然对于较大的V来说有点单调乏味）。
我们目前展示了我们的U / U：Bipartite示例。您可以进入“探索模式”并绘制自己的二分图。
二分图也可以是完整的，即来自一个不相交集的所有m个顶点都连接到来自另一个不相交集的所有n个顶点。当m = n = V / 2时，这样的完全二分图也具有E = O（V2）。
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.
我们简单地使用大小为VxV的C ++/Python/Java原生2D数组来实现这种数据结构。
讨论：了解AM的大空间复杂性，何时使用它是有益的？或者AM总是一个劣质的图形数据结构，永远不该使用？
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.
我们使用一个嵌套的Vector对（用于加权图）来实现此数据结构。在C++中：vector<vector<pair<int, int>>> AL; Python: AL = [[] for _ in range(N)] Java: Vector<Vector<IntegerPair>> AL; // Java 中的IntegerPair 类似于 C++中的pair<int, int>
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
由于Vector的自动调整大小功能，我们使用Vector来存储对。如果我们有一个顶点的k个邻居，我们只需将k个邻居添加到该顶点的最初为空的Vector对象（此Vector可以用Linked List替换）。
我们在外面再用一层Vector来储存Vector对，即如果我们想枚举顶点u的邻居，我们使用AL [u]（C ++/Python）或AL.get（u）（Java）来访问正确的Vector。
讨论：AL是最常用的图形数据结构，但讨论AL哪种情况实际上不是最佳的图形数据结构？
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.
边缘列表（EL）是具有连接顶点及其权重的边的集合。通常，这些边是按权重增加来排序的，例如， Kruskal's algorithm的一部分用于最小生成树（MST）的问题。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。C++：vector<tuple<int, int, int>> EL; Python: EL = [] Java: Vector<IntegerTriple> EL; // Java中的IntegerTriple类似于C++中的tuple<int, int, int>
讨论：除了Kruskal的最小生成树（MST）算法之外，详细说明EL的潜在用法！
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.
将图形信息存储到图形数据结构后，我们可以回答几个简单的问题。
- 计数 V,
- 计数 E,
- 枚举顶点u的邻居，
- 检查边缘（u，v）的存在等。
讨论：如果图存储在EL中，如何计算V？
PS：有时这个数字是存储/维护在一个单独的变量中，这样我们就不必每次都重新计算它 - 特别是如果图形在创建之后永远/很少改变，因此我们有O(1)性能，例如：对于上面显示的示例图，我们可以存储（在我们的AM / AL / EL数据结构中）有7个顶点。
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.
在邻接表（AL）中，可以通过将所有V列表的长度相加来找到 E，并将最终答案除以2（对于无向图）。这需要O(V+E) 计算时间，因为每个顶点和每个边缘仅被处理一次。
讨论：如果图表存储在邻接矩阵（AM）中，如何计算E？
PS：为了提高效率，有时将这个数字存储/维护在一个单独的变量中，例如：对于上面显示的示例图，我们可以记住（在我们的AM / AL / EL数据结构中）有8个无向边。
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.
在AL中，我们只需要遍历AL [u]。如果顶点u只有k个邻居，那么我们只需要O(k) 来枚举它们 - 这被称为输出敏感时间复杂度，已经是最好的了。
我们通常按顶点数量递增的方式列出邻居。例如，上面的示例图中的顶点1的邻居是{0,2,3}，按增加的顶点数顺序。
讨论：如果图形存储在EL中，如何执行此操作？
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.
在AL中，我们必须检查AL [u]是否包含顶点v。这是O（k） - 更慢。
例如，上面的示例图中存在边（2,4），但边不存在边（2,6）。
请注意，如果我们找到了edge（u，v），我们也可以访问和/或更新其权重。
讨论：如果图形存储在EL中，如何执行此操作？
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.
Quiz: So, what is the best graph data structure?
讨论: 为什么?
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.
然而，我们仍然有一些更有趣的图形数据结构的挑战，在本节中概述。
请注意，图形数据结构通常只是解决较难的图形问题（如MST, SSSP, MF, Matching, MVC, ST, or TSP）的必要部分，而不是充分部分。
尝试解决两个基本的编程问题，这些问题需要使用图形数据结构，而不需要任何花哨的图形算法：
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.
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.
| • V=0, E=0 | • 树? No | • 完成? No | • 二部? No | • DAG? No | • 交叉? No |
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邻接矩阵
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邻接列表
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边缘列表
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0.5x Scale
Edit Graph