Upper and Lower Bounds for Fully Retroactive Graph Problems

Classic dynamic data structure problems maintain a data structure subject to a sequence S of updates and they answer queries using the latest version of the data structure, i.e., the data structure after processing the whole sequence. To handle operations that change the sequence S of updates, Demaine et al. (TALG 2007) introduced retroactive data structures. A retroactive operation modifies the update sequence S in a given position t, called time, and either creates or cancels an update in S at time t. A partially retroactive data structure restricts queries to be executed exclusively in the latest version of the data structure. A fully retroactive data structure supports queries at any time t: a query at time t is answered using only the updates of S up to time t. If the sequence S only consists of insertions, the resulting data structure is an incremental retroactive data structure. While efficient retroactive data structures have been proposed for classic data structures, e.g., stack, priority queue and binary search tree, the retroactive version of graph problems are rarely studied. In this paper we study retroactive graph problems including connectivity, minimum spanning forest (MSF), maximum degree, etc. We provide fully retroactive data structures for maintaining the maximum degree, connectivity and MSF in $\tilde{O}(n)$ time per operation. We also give an algorithm for the incremental fully retroactive connectivity with $\tilde{O}(1)$ time per operation. We compliment our algorithms with almost tight hardness results. We show that under the OMv conjecture (proposed by Henzinger et al. (STOC 2015)), there does not exist fully retroactive data structures maintaining connectivity or MSF, or incremental fully retroactive data structure maintaining the maximum degree with $O(n^{1-\epsilon})$ time per operation, for any constant $\epsilon > 0$.