1. Introduction

图形由顶点/节点和连接这些顶点的边/线组成。
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)。

1-1. 简单图形

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.

1-2. 术语,第1部分

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。

1-3. 术语,第2部分

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.

1-4. 术语,第3部分

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.

1-5. 术语,第4部分

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。

1-6. 术语,第5部分

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。

1-7. 特殊图形

具有涉及其顶点和/或边结构的特定属性的图可以使用其特定名称来调用,如树(如当前所示),完整图,二分图,有向无环图(DAG),以及使用频率较低的图:平面图,线图,星图,轮图等

在此可视化中,我们将在稍后突出显示前四个特殊图表。

2. 图是普遍的

图表在现实生活中经常以各种形式出现。因此,解决图形问题的最重要部分是图形建模部分,即将手中的问题简化为图形术语:顶点,边,权重等。

2-1. 示例 - 更容易看到,1

社交网络:顶点可以代表人,边缘代表人与人之间的联系(通常是无向和未加权)。
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:
  1. 谁是0号的朋友?
  2. 谁拥有最多的朋友?
  3. 有没有孤立的人(那些没有朋友的人)?
  4. 两个陌生人之间是否有共同的朋友:3号人和5号人?
  5. 等等...

2-2. 示例 - 更容易看到,2

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。

2-3. 示例 - 更难看到

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3. 模式

要在图形绘制模式之间切换,请选择相应的标题。 我们有:

  1. U/U = 无向/不加权,
  2. U/W = 无向/加权,
  3. D/U = 有向/不加权, and
  4. 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。

4. 可视化

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:

  1. 点击空白区域添加顶点,
  2. 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),
  3. 选择顶点/边缘并按“删除”键删除该顶点/边,
  4. 选择边缘并按“Enter”以更改该边缘的权重[0..99],
  5. 按住“Ctrl”,然后您可以单击并拖动顶点。

4-1. 可视化约束

我们将VisuAlgo中讨论的图形限制为简单图形。请参阅本幻灯片中的讨论。

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。

5. 图形的示例

所有示例图都可以在这里找到。

目前,我们为每个类别提供至少两个示例图表(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。

6. 特殊图形

Tree,Complete,Bipartite,Directed Acyclic Graph(DAG)是特殊图的属性。在上面的可视化/绘图区域中修改图形时,会立即检查和更新这些属性。

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。

6-1. 特殊图形 - 树,第1部分

树是具有V顶点和E = V-1边的连通图,非循环,并且在任何顶点对之间具有一条唯一路径。通常,树在无向图上定义。

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。

6-2. 特殊图形 - 树,第2部分

并非所有树都具有相同的图形绘制布局,其顶部具有特殊根顶点,底部具有叶顶点(顶点为1)。上面显示的(星)图也是树,因为它满足树的属性。

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。

6-3. 特殊图形 - 树,第3部分

在根树中,我们有层次结构(父,子,祖先,后代),子树,级别和高度的概念。我们将通过示例说明这些概念,因为它们的含义与现实生活中的对应物一样:

0/1/7/2/4的父级分别为none / 0/0/1/3,
0/1/7的子女分别为{1,7} / {2,3,6} / {8,9},
4/8的祖先分别为{3,1,0} / {7,0},
1/7的后代分别是{2,3,4,5,6} / {8,9},
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} / {1,7} / {2,3,6,8,9} / {4,5},
这个有根树的高度是它的最高级别= 3。

6-4. 特殊图形 - 树,第4部分

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.

6-5. 特殊图表 - 完整

完整图是具有V个顶点和 E = V*(V-1)/2 个边(或E = O(V2)的图,即在任何顶点对之间存在边。通常,完整图表用KV表示。
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)。

6-6. 特殊图表 - 二分法

二分图是具有V个顶点的无向图,其可以被划分为大小为m和n的两个不相交的顶点集,其中V = m + n。同一组的成员之间没有边缘。二分图也没有奇数周期。

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)。

6-7. 特殊图表 - DAG

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.

7. 三个图形数据结构

有许多方法可以将图形信息存储到图形数据结构中。在此可视化中,我们显示了三个图形数据结构:邻接矩阵,邻接列表和边缘列表 - 每个都有自己的优点和缺点。

7-1. 邻接矩阵(AM)

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.

7-2. AM-继续

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?

7-3. 答案

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7-4. 邻接列表(AL)

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

7-5. Class IntegerPair (in Java)

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

7-6. Why Vector of Vector Pairs?

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.

7-7. AL-继续

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?

7-8. 答案

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7-9. 边缘列表(EL)

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>

7-10. EL-继续

空间复杂度分析:EL具有O(E)的空间复杂度,其比AM更有效并且与AL一样有效。

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!

7-11. 答案

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8. 简单应用

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.

8-1. 将V计数

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.

8-2. 答案

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8-3. 将 E 计数

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.

8-4. 答案

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8-5. 枚举顶点的邻居u

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?

8-6. 答案

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8-7. 检查边缘(u,v)的存在

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?

8-8. 答案

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8-9. 讨论

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

Adjacency List
It Depends
Edge List
Adjacency Matrix

讨论: 为什么?

8-10. 答案

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9. 附加功能

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.