elements to remove from the tree. The algorithms for bulk operations aren’t just applicable to the red–black tree, but can be adapted to other sorted sequence data structures also, like the 2–3 tree, 2–3–4 tree and (a,b)-tree. In the following different algorithms for bulk insert will be explained, but the same algorithms can also be applied to removal and update. Bulk insert is an operation that inserts each element of a sequence into a tree . Join-based This approach can be applied to every sorted sequence data structure that supports efficient join- and split-operations. The general idea is to split and in multiple parts and perform the insertions on these parts in parallel. First the bulk of elements to insert must be sorted. After that, the algorithm splits into parts of about equal sizes. Next the tree must be split into parts in a way, so that for every following constraints hold: Now the algorithm inserts each element of into sequentially. This step must be performed for every , which can be done by up to processors in parallel. Finally, the resulting trees will be joined to form the final result of the entire operation. Note that in Step 3 the constraints for splitting assure that in Step 5 the trees can be joined again and the resulting sequence is sorted. The pseudo code shows a simple divide-and-conquer implementation of the join-based algorithm for bulk-insert. Both recursive calls can be executed in parallel. The join operation used here differs from the version explained in this article, instead join2 is used, which misses the second parameter k. bulkInsert(T, I, k): I.sort() bulklInsertRec(T, I, k) bulkInsertRec(T, I, k): if k = 1: forall e in I: T.insert(e) else m := ⌊size(I) / 2⌋ (T1, _, T2) := split(T, I[m]) bulkInsertRec(T1, I[0 .. m], ⌈k / 2⌉) || bulkInsertRec(T2, I[m + 1 .. size(I) - 1], ⌊k / 2⌋) T ← join2(T1, T2) Execution time Sorting is not considered in this analysis. This can be improved by using parallel algorithms for splitting and joining. In this case the execution time is . Work Pipelining Another method of parallelizing bulk operations is to use a pipelining approach. This can be done by breaking the task of processing a basic operation up into a sequence of subtasks. For multiple basic operations the subtasks can be processed in parallel by assigning each subtask to a separate