Space limited linear-time graph algorithms on big data

We study algorithms for graph problems in which the graphs are of extremely large size N so that super-linear time ω(N) or linear space Θ(N) would become impractical. We use a parameter k to characterize the computational power of a normal computer that can provide additional time and space bounded by polynomials of k in dealing with the large graphs. In particular, we are interested in strict linear-time algorithms using space O(kO(1)). In our case studies, as examples, we present (1) a randomized greedy algorithm of time O(N) and space O(k2) for a parameterized version of the Maximal Matching problem; and (2) randomized kernelization algorithms of time O(N) and space O(kO(1)) for a number of well-known NP-hard problems. Our kernelization algorithms have their kernel sizes match the best kernel sizes by known polynomial-time kernelization algorithms with no space constraints for the problems. We also study the relationship between our proposed model and the streaming model. This study motivates a new streaming kernelization algorithm for the famous Vertex Cover problem that has an optimal update time complexity while matches the best known space complexity of streaming algorithms for the problem.