Incremental Graph Computations: Doable and Undoable

Abstract

The incremental problem for a class \({\mathcal {Q}} \) of graph queries aims to compute, given a query \(Q \in {\mathcal {Q}} \) , graph G , answers Q ( G ) to Q in G and updates ΔG to G as input, changes ΔO to output Q ( G ) such that Q ( G ⊕ ΔG ) = Q ( G )⊕ ΔO . It is called bounded if its cost can be expressed as a polynomial function in the sizes of Q , ΔG and ΔO , which reduces the computations on possibly big G to small ΔG and ΔO . No matter how desirable, however, our first results are negative: for common graph queries such as traversal, connectivity, keyword search, pattern matching, and maximum cardinality matching, their incremental problems are unbounded. In light of the negative results, we propose two characterizations for the effectiveness of incremental graph computation: (a) localizable , if its cost is decided by small neighbors of nodes in ΔG instead of the entire G ; and (b) bounded relative to a batch graph algorithm \({\mathcal {T}} \) , if the cost is determined by the sizes of ΔG and changes to the affected area that is necessarily checked by any algorithms that incrementalize \({\mathcal {T}} \) . We show that the incremental computations above are either localizable or relatively bounded, by providing corresponding incremental algorithms. That is, we can either reduce the incremental computations on big graphs to small data, or incrementalize existing batch graph algorithms by minimizing unnecessary recomputation. Using real-life and synthetic data, we experimentally verify the effectiveness of our incremental algorithms.

Evaluating Complex Queries on Streaming Graphs

“…In particular, the novel PATH operator and its transformation rules enable the integration of existing approaches for RPQ evaluation with standard optimization techniques such as join ordering and pushing down selection in a principled manner. Traditionally, path query evaluation follows two main approaches: graph traversals guided by finite automata, or relational algebra extended with transitive closure, i.e., 𝑎𝑙𝑝ℎ𝑎-RA [12,27,43,62,68,79]. Yakovets et al introduce a hybrid approach (Waveguide) and model the cost factors that impact the efficiency of RPQ evaluation on static graphs [79].…”

“…x z [24,32) z u [25,35) x y [26,33) y w [27,40) z t [28,37) y u [29,41) u v [30,38) u s (w, 1) [26,33) (z, 1) [23,31) (u, 1) [24,31) (t, 1)…”

“…(y, 1) [25,35) (w, 1) [26,33) (u, 1) [28,35) (v, 1) [29,35) (s, 1) [30,35) (z, 1) [23,31) (t, 1) [27,31) (c) 𝑡 = 30…”

Updates-Aware Graph Pattern based Node Matching

“…Sun et al, [21] extended incremental methods to find maximal cliques that contain vertices incident to an edge which has been inserted. Fan et al, [22] further proposed incremental algorithms for four types of typical pattern graphs, which can reduce the computations on big graphs and minimize unnecessary recomputation. Ma et al, [23] proposed a method to find dense subgraphs in temporal networks.…”

Regular Path Query Evaluation on Streaming Graphs

“…Our proposed algorithms have a lower amortized cost, O(n), for insertions at the expense of O(n 2 ) amortized time for deletions -a trade-o justi ed by the insert-heavy nature of real-world streaming graphs. Fan et al [29] characterize the complexity of various graph problems, including RPQ evaluation, in the dynamic model and show that most graph problems are unbounded under edge updates, i.e., the cost of computing changes to query answers cannot be expressed as a polynomial of the size of the changes in the input and output. They prove that RPQ is bounded relative to its batch counterpart; the batch algorithm can be e ciently incrementalized by minimizing unnecessary computation.…”