This paper presents an interesting class of algorithms for finding the minimal nontrivial congruences, subalgebras, and ideals of a finite algebra specified by its table of operations. This algorithm is based on a clever modification of Tarjan’s algorithm for finding strongly corrected components of a graph [1]. The running times of the given algorithm are O(n^1+r) for the case of congruences and O(n^r) for subalgebras and ideals where n is the number of elements in the algebra and r is the maximum arity of its operations. Another interesting feature of the algorithm is that when the given algebra is, in fact, a group or ring, the complexity of finding minimal congruences reduces to O(n²).