# Recent Advances in Efficient Dynamic Graph Processing
## 4 Algorithms
### Graph Separation
#### 4.5.1 Graph Partitioning  
Effective problem decomposition plays a pivotal role in both distributed computing systems and the optimization of sequential processing. Among various decomposition strategies, graph partitioning has emerged as a particularly prominent approach. At its core, this technique involves splitting a graph into multiple subgraphs of approximately equal size while simultaneously reducing the number of connections between these partitions. This is a well-studied challenge that has attracted significant research attention [5].  
**Definition 15** (**Graph Partitioning**).: _Given a graph \(G=(V,E)\) and an integer \(k\), graph partitioning is to divide \(V\) into \(k\) blocks of vertices \(V_{1},\ldots,V_{k}\) such that \(V_{1}\cup\cdots\cup V_{k}=V\) and \(V_{i}\cap V_{j}=\emptyset\) for \(i,j\in[1,k]\), \(i\
eq j\), while simultaneously optimizing an objective function and maintaining balance: \(\forall i\in\{1,\ldots,k\}:|V_{i}|\leq(1+\epsilon)\lceil|V|/k\rceil\) for some \(\epsilon\geq 0\). The most common objective is to minimize the total cut \(\Sigma_{i<j}|E_{ij}|\), where \(E_{ij}=\{\{u,v\}\in e:u\in V_{i},v\in V_{j}\}\)._  
Recent research has increasingly focused on dynamic graph partitioning, recognizing its crucial role in the preprocessing stage of distributed graph processing systems. While traditional approaches primarily addressed static graphs, the growing need to handle evolving graph structures in distributed analytics has spurred significant interest in dynamic partitioning methods.  
In response to this challenge, Ref. [111] introduces an adaptive partitioning algorithm specifically designed for dynamic graphs. This decentralized approach employs an iterative vertex migration strategy that operates from any initial partition configuration. LogGP [117] introduces an innovative partition refinement approach that leverages both historical partitioning data and real-time statistical log information. Hermes [24] presents an efficient local-view approach for dynamic graph repartitioning. The system focuses on progressively enhancing existing partitions by reducing edge-cuts while preserving near-perfect load balance. Building upon existing streaming techniques [116], Leopard [112] presents an innovative approach to dynamic graph partitioning. The algorithm models evolving graphs as transitional states within a single-pass partitioning framework, effectively adapting traditional one-pass heuristics to dynamic scenarios. [PERSON]. [113] proposes an alternative to conventional single-vertex migration approaches.  
Different from the above researches that focus on the practical performance of the dynamic graph partitioning algorithms in distributed graph analytic systems, [PERSON]. [115] theoretically proves that the dynamic graph partitioning problem is NP-complete and unbounded. Due to the intractability and unboundedness of the problem, [PERSON]. [115] proposes a notion of heuristic boundedness to characterize the effectiveness of incremental heuristic algorithms and devises corresponding heuristically bounded dynamic graph partitioning algorithms based on the existing batch graph partition algorithms, such as DNE [167], KGGGP [168], FENNEL [116], and HDRF [169].  
Recently, starting from the focus and characteristics of different dynamic graph partitioning algorithms, [PERSON]. [170] systematically introduces various algorithms that can be used to solve the dynamic graph partitioning problem.