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. 擴展2-hop至3-hop
2. 解構graph和vertex爲一系列的chain
3. 3-hop: 第一條chain1分爲起點段(incoming segments），第二條chain2爲高速（highway chain有entry points和exit points)，第三條chain3分爲終點段（outgoing segments）

技術細節：

1. transitive closure binary matrix(V行V列，V爲vetex數，1爲鏈接，0爲未鏈接）
2. transitive closure contour（爲dominating diagonal，其實爲entry points和exit points分別與highway chain的鏈接）
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

總結

將2-hop拓展爲3-hop，利用chain decomposition和chain-center bipartite graph預處理而後構建3-hop（Section 3）。其後將最小3-hop等價於最小2-hop問題（densest subgraph selection Section 4）。設計兩種query方式（判斷contour或segment）。