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邻接表通常是首选的，因为它们能有效地表示'''<font color="#ff8000">稀疏图 Sparse Graph</font>'''。如果图是'''<font color="#ff8000">稠密图 Dense Graph</font>'''的，那么邻接矩阵是首选的，即边的数目 |<big>E</big>| 接近于顶点的平方数，|<big>V</big>|² ，或者说如果有一条边连接两个顶点，那么所选取的数据结构必须能满足快速查找到数据才行。

邻接表通常是首选的，因为它们能有效地表示'''<font color="#ff8000">稀疏图 Sparse Graph</font>'''。如果图是'''<font color="#ff8000">稠密图 Dense Graph</font>'''的，那么邻接矩阵是首选的，即边的数目 |<big>E</big>| 接近于顶点的平方数，|<big>V</big>|² ，或者说如果有一条边连接两个顶点，那么所选取的数据结构必须能满足快速查找到数据才行。

== Parallel Graph Representations ==

== Parallel Graph Representations 图的并行化表示==

图的并行化表示<br>

<br>

The parallelization of graph problems faces significant challenges: Data-driven computations, unstructured problems, poor locality and high data access to computation ratio.<ref name=":1">{{Cite book|last=Bader|first=David|url=http://www.ams.org/conm/588/|title=Graph Partitioning and Graph Clustering|last2=Meyerhenke|first2=Henning|last3=Sanders|first3=Peter|last4=Wagner|first4=Dorothea|date=January 2013|publisher=American Mathematical Society|isbn=978-0-8218-9038-7|series=Contemporary Mathematics|volume=588|language=en|doi=10.1090/conm/588/11709}}</ref><ref>{{Cite journal|last=LUMSDAINE|first=ANDREW|last2=GREGOR|first2=DOUGLAS|last3=HENDRICKSON|first3=BRUCE|last4=BERRY|first4=JONATHAN|date=March 2007|title=CHALLENGES IN PARALLEL GRAPH PROCESSING|url=http://dx.doi.org/10.1142/s0129626407002843|journal=Parallel Processing Letters|volume=17|issue=01|pages=5–20|doi=10.1142/s0129626407002843|issn=0129-6264}}</ref> The graph representation used for parallel architectures plays a significant role in facing those challenges. Poorly chosen representations may unnecessarily drive up the communication cost of the algorithm, which will decrease its [[scalability]]. In the following, shared and distributed memory architectures are considered.

The parallelization of graph problems faces significant challenges: Data-driven computations, unstructured problems, poor locality and high data access to computation ratio.<ref name=":1">{{Cite book|last=Bader|first=David|url=http://www.ams.org/conm/588/|title=Graph Partitioning and Graph Clustering|last2=Meyerhenke|first2=Henning|last3=Sanders|first3=Peter|last4=Wagner|first4=Dorothea|date=January 2013|publisher=American Mathematical Society|isbn=978-0-8218-9038-7|series=Contemporary Mathematics|volume=588|language=en|doi=10.1090/conm/588/11709}}</ref><ref>{{Cite journal|last=LUMSDAINE|first=ANDREW|last2=GREGOR|first2=DOUGLAS|last3=HENDRICKSON|first3=BRUCE|last4=BERRY|first4=JONATHAN|date=March 2007|title=CHALLENGES IN PARALLEL GRAPH PROCESSING|url=http://dx.doi.org/10.1142/s0129626407002843|journal=Parallel Processing Letters|volume=17|issue=01|pages=5–20|doi=10.1142/s0129626407002843|issn=0129-6264}}</ref> The graph representation used for parallel architectures plays a significant role in facing those challenges. Poorly chosen representations may unnecessarily drive up the communication cost of the algorithm, which will decrease its [[scalability]]. In the following, shared and distributed memory architectures are considered.

The parallelization of graph problems faces significant challenges: Data-driven computations, unstructured problems, poor locality and high data access to computation ratio. The graph representation used for parallel architectures plays a significant role in facing those challenges. Poorly chosen representations may unnecessarily drive up the communication cost of the algorithm, which will decrease its scalability. In the following, shared and distributed memory architectures are considered.

The parallelization of graph problems faces significant challenges: Data-driven computations, unstructured problems, poor locality and high data access to computation ratio. The graph representation used for parallel architectures plays a significant role in facing those challenges. Poorly chosen representations may unnecessarily drive up the communication cost of the algorithm, which will decrease its scalability. In the following, shared and distributed memory architectures are considered.

图问题的并行化面临着重大的挑战: 数据驱动的计算、非结构化问题、局部性差和计算数据访问率高。用于并行架构的图表示在面对这些挑战时扮演着重要的角色。选择不当的表示可能会不必要地增加算法的通信代价，从而降低算法的可扩展性。在下面，我们将考虑共享和分布式的内存架构。

图问题的并行化面临着重大的挑战: 数据驱动的计算、非结构化问题、局部性差和计算数据访问率高。用于并行架构的图表示在面对这些挑战时扮演着重要的角色。选择的表示方式不当可能会增加不必要的算法通信代价，从而降低算法的可扩展性。在下面，我们将考虑共享和分布式的内存架构。

=== Shared memory ===

=== Shared memory 共享内存===

共享内存<br>

<br>

In the case of a [[shared memory]] model, the graph representations used for parallel processing are the same as in the sequential case,<ref name=":0">{{Cite book|last=Sanders|first=Peter|url=https://www.springer.com/gp/book/9783030252083|title=Sequential and Parallel Algorithms and Data Structures: The Basic Toolbox|last2=Mehlhorn|first2=Kurt|last3=Dietzfelbinger|first3=Martin|last4=Dementiev|first4=Roman|date=2019|publisher=Springer International Publishing|isbn=978-3-030-25208-3|language=en}}</ref> since parallel read-only access to the graph representation (e.g. an [[adjacency list]]) is efficient in shared memory.

In the case of a [[shared memory]] model, the graph representations used for parallel processing are the same as in the sequential case,<ref name=":0">{{Cite book|last=Sanders|first=Peter|url=https://www.springer.com/gp/book/9783030252083|title=Sequential and Parallel Algorithms and Data Structures: The Basic Toolbox|last2=Mehlhorn|first2=Kurt|last3=Dietzfelbinger|first3=Martin|last4=Dementiev|first4=Roman|date=2019|publisher=Springer International Publishing|isbn=978-3-030-25208-3|language=en}}</ref> since parallel read-only access to the graph representation (e.g. an [[adjacency list]]) is efficient in shared memory.

In the case of a shared memory model, the graph representations used for parallel processing are the same as in the sequential case, since parallel read-only access to the graph representation (e.g. an adjacency list) is efficient in shared memory.

In the case of a shared memory model, the graph representations used for parallel processing are the same as in the sequential case, since parallel read-only access to the graph representation (e.g. an adjacency list) is efficient in shared memory.

在共享内存模型的情况下，用于并行处理的图表示与顺序处理的方式相同，因为对图表示的并行只读访问(例如:邻接表)是共享内存的有效方法。

在共享内存模型的情况下，之所以用于并行处理的图表示与顺序处理的方式相同，是因为对图表示的并行只读访问(例如:邻接表)是共享内存的有效方法。

   --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）通读一遍  注意多余符号的问题(例如:。邻接表)

   --[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）通读一遍  注意多余符号的问题(例如:。邻接表)

=== Distributed Memory ===

=== Distributed Memory 分布式存储===

分布式存储<br>

<br>

In the [[distributed memory]] model, the usual approach is to [[Graph partition|partition]] the vertex set <math>V</math> of the graph into <math>p</math> sets <math>V_0, \dots, V_{p-1}</math>. Here, <math>p</math> is the amount of available processing elements (PE). The vertex set partitions are then distributed to the PEs with matching index, additionally to the corresponding edges. Every PE has its own [[Subgraph (graph theory)|subgraph]] representation, where edges with an endpoint in another partition require special attention. For standard communication interfaces like [[Message Passing Interface|MPI]], the ID of the PE owning the other endpoint has to be identifiable. During computation in a distributed graph algorithms, passing information along these edges implies communication.<ref name=":0" />

In the [[distributed memory]] model, the usual approach is to [[Graph partition|partition]] the vertex set <math>V</math> of the graph into <math>p</math> sets <math>V_0, \dots, V_{p-1}</math>. Here, <math>p</math> is the amount of available processing elements (PE). The vertex set partitions are then distributed to the PEs with matching index, additionally to the corresponding edges. Every PE has its own [[Subgraph (graph theory)|subgraph]] representation, where edges with an endpoint in another partition require special attention. For standard communication interfaces like [[Message Passing Interface|MPI]], the ID of the PE owning the other endpoint has to be identifiable. During computation in a distributed graph algorithms, passing information along these edges implies communication.<ref name=":0" />

在分布式存储模型中，常用的方法是将图的顶点集合<math>V</math> 分解为<math>P</math> 集合 <math> Vo,…,V{ p-1}</math> 。这里，<math>p</math> 是可用处理元素(PE)的数量。然后，顶点集合分区被分配到具有匹配索引的 PE 中，并附加到相应的边上。每个 PE 都有自己的子图表示法，其中带有另一个分区中端点的边需要特别注意。对于像 MPI 这样的标准通信接口，拥有其他端点的 PE 的 ID 必须是可识别的。在分布式图算法的计算过程中，沿着这些边传递信息意味着通信。

在分布式存储模型中，常用的方法是将图的顶点集合<math>V</math> 分解为<math>P</math> 集合 <math> Vo,…,V{ p-1}</math> 。这里，<math>p</math> 是可用处理元素(PE)的数量。然后，顶点集合分区被分配到具有匹配索引的 PE 中，并附加到相应的边上。每个 PE 都有自己的子图表示法，其中带有另一个分区中端点的边需要特别注意。对于像 MPI 这样的标准通信接口，拥有其他端点的 PE 的 ID 必须是可识别的。在分布式图算法的计算过程中，沿着这些边传递信息意味着通信。

--信白该句存疑Vo到Vp-1没写好

--信白该句存疑Vo到Vp-1没写好