3-HOP: A High-Compression Indexing Scheme for Reachability Query

title: 3-HOP: A High-Compression Indexing Scheme for Reachability Query

venue: SIGMOD'09

author: Ruoming Jin, Yang Xiang, Ning Ruan, and David Fuhry

abstract: Reachability queries on large directed graphs have attracted much attention recently. The existing work either uses spanning structures, such as chains or trees, to compress the complete transitive closure, or utilizes the 2-hop strategy to describe the reachability. Almost all of these approaches work well for very sparse graphs. However, the challenging problem is that as the ratio of the number of edges to the number of vertices increases, the size of the compressed transitive closure grows very large. In this paper, we propose a new 3-hop indexing scheme for directed graphs with higher density. The basic idea of 3-hop indexing is to use chain structures in combination with hops to minimize the number of structures that must be indexed. Technically, our goal is to find a 3-hop scheme over dense DAGs (directed acyclic graphs) with minimum index size. We develop an efficient algorithm to discover a transitive closure contour, which yields near optimal index size. Empirical studies show that our 3-hop scheme has much smaller index size than state-of-the-art reachability query schemes such as 2-hop and pathtree when DAGs are not very sparse, while our query time is close to path-tree, which is considered to be one of the best reachability query schemes.

主要思想：

1. 扩展2-hop至3-hop
2. 解构graph和vertex为一系列的chain
3. 3-hop: 第一条chain1分为起点段(incoming segments），第二条chain2为高速（highway chain有entry points和exit points)，第三条chain3分为终点段（outgoing segments）

技术细节：

1. transitive closure binary matrix(V行V列，V为vetex数，1为连接，0为未连接）
2. transitive closure contour（为dominating diagonal，其实为entry points和exit points分别与highway chain的连接）
3. Computing transitive closure contour givin a chain decomposition
4. Compress thransitive closure contour: NP-hard, generilized factorization, set-cover prob0lem(how to map to these problems)
5. How to compute the smallest labels
6. chain-center bipartite graph

总结

将2-hop拓展为3-hop，利用chain decomposition和chain-center bipartite graph预处理然后构建3-hop（Section 3）。其后将最小3-hop等价于最小2-hop问题（densest subgraph selection Section 4）。设计两种query方式（判断contour或segment）。