Global Rebuilding, Coroutines, and Defunctionalization
In 1983, Mark Overmars described global rebuilding in The Design of Dynamic Data Structures. The problem it was aimed at solving was turning the amortized time complexity bounds of batched rebuilding into worst-case bounds. In batched rebuilding we perform a series of updates to a data structure which may cause the performance of operations to degrade, but occasionally we expensively rebuild the data structure back into an optimal arrangement. If the updates don’t degrade performance too much before we rebuild, then we can achieve our target time complexity bounds in an amortized sense. An update that doesn’t degrade performance too much is called a weak update.
Taking an example from Okasaki’s Purely Functional Data Structures, we can consider a binary search tree where deletions occur by simply marking the deleted nodes as deleted. Then, once about half the tree is marked as deleted, we rebuild the tree into a balanced binary search tree and clean out the nodes marked as deleted at that time. In this case, the deletions count as weak updates because leaving the deleted nodes in the tree even when it corresponds to up to half the tree can only mildly impact the time complexity of other operations. Specifically, assuming the tree was balanced at the start, then deleting half the nodes could only reduce the tree’s depth by about 1. On the other hand, naive inserts are not weak updates as they can quickly increase the tree’s depth.
The idea of global rebuilding is relatively straightforward, though how you would actually realize it in any particular example is not. The overall idea is simply that instead of waiting until the last moment and then rebuilding the data structure all at once, we’ll start the rebuild sooner and work at it incrementally as we perform other operations. If we update the new version faster than we update the original version, we’ll finish it by the time we would have wanted to perform a batch rebuild, and we can just switch to this new version.
