Several fast and space-optimal sequential and parallel algorithms for solving the satisfaction problem of functional and multivalued dependencies (FDs and MVDs) are presented. Two frameworks to verify an MVD for a relation and their implementation by exploring the existing fast space-optimal sorting techniques are described. The space optimality means that only a constant amount of extra memory space is needed for the sequential implementations, and O(M) amount of extra memory space for parallel algorithms that use M processors. This feature makes the algorithms attractive whenever space is a critical resource and I/O transfers should be reduced to the minimal, as is often the case for relational database systems. The time requirements for in-place FD and MVD verification are given in terms of M and of N, which is the number of tuples in a relation. The effect of relation modification on FD and MVD verification is examined. >