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 w ( N ) or linear space Theta( 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 ( k O (1) ). In our case studies, as examples, we present (1) a randomized greedy algorithm of time O ( N ) and space O ( k 2 ) for a parameterized version of the M AXIMAL M ATCHING problem; and (2) randomized kernelization algorithms of time O ( N ) and space O ( k O (1) ) for a number of well-known NP -hard problems. Our kernelization algorithms have their kernel sizes match the best kernel sizes by known polynomialtime 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 V ERTEX C OVER problem that has an optimal update time complexity while matches the best known space complexity of streaming algorithms for the problem.