Before revision:  More precisely, we provide O(1) round algorithms that can process a batch of updated of size O (S) for the Minimum Spanning Forest problem and a batch of updates of size O (S^{1-\varepsilon}) for the Maximal Matchingproblem, where S is the limit on the local memory of a single machine.
After revision:  More precisely, we provide O(1) round algorithms that can process a batch of updated of size O (S) for the Minimum Spanning Forest problem and a batch of updates of size O (S^{1-\varepsilon}) , for Maximal Matching, in O(1) rounds, where S is the limit on the local memory of a single machine.