Relative Subboundedness of Contraction Hierarchy and Hierarchical 2-Hop Index in Dynamic Road Networks

Computing the shortest path for any two given vertices is an important problem in road networks. Since real road networks are dynamically updated due to real-time traffic conditions and it is costly to recompute the oracle O in use from scratch, O needs to be updated to reflect the changes in the network using incremental algorithms. An incremental algorithm is said to be bounded if its cost is polynomial in |CHANGED|, where CHANGED comprises both the changes to the graph and the resulting changes to O. An incremental problem is bounded if it has a bounded algorithm and is unbounded otherwise. We study the boundedness of the incremental counterparts of two state-of-the-art oracles, namely contraction hierarchy (CH) and hierarchical 2-hop index (H2H). We prove that under specific computational models, both CH and H2H are unbounded to maintain. Despite this fact, we introduce relative subboundedness as an alternative to boundedness. We prove that the state-of-the-art incremental algorithm for CH is relatively subbounded, and moreover, we propose a relatively subbounded algorithm for H2H. Our experimental study on real road networks shows that the algorithms studied are faster than recomputing from scratch even when 10% of the index needs to be updated, thereby verifying the effectiveness of relative subboundedness.