Nearly Optimal Separation Between Partially And Fully Retroactive Data Structures.

Since the introduction of retroactive data structures at SODA 2004, a major unsolved problem has been to bound the gap between the best partially retroactive data structure (where changes can be made to the past, but only the present can be queried) and the best fully retroactive data structure (where the past can also be queried) for any problem. It was proved 2004 that any partially retroactive data structure with operation time $T(n,m)$ can be transformed into a fully retroactive data structure with operation time $O(sqrt{m} cdot T(n,m))$, where $n$ is the size of the data structure and $m$ is the number of operations the timeline [Demaine 2004], but it has been open for 14 years whether such a gap is necessary. In this paper, we prove nearly matching upper and lower bounds on this gap for all $n$ and $m$. We improve the upper bound for $n ll sqrt m$ by showing a new transformation with multiplicative overhead $n log m$. We then prove a lower bound of $min{n log m, sqrt m}^{1-o(1)}$ assuming any of the following conjectures: Conjecture I: Circuit SAT requires $2^{n - o(n)}$ time on $n$-input circuits of size $2^{o(n)}$. (Far weaker than the well-believed SETH conjecture, which asserts that CNF SAT with $n$ variables and $O(n)$ clauses already requires $2^{n-o(n)}$ time.) Conjecture II: Online $(min,+)$ product between an integer $ntimes n$ matrix and $n$ vectors requires $n^{3 - o(1)}$ time. Conjecture III (3-SUM Conjecture): Given three sets $A,B,C$ of integers, each of size $n$, deciding whether there exist $a in A, b in B, c in C$ such that $a + b + c = 0$ requires $n^{2 - o(1)}$ time. Our lower bound construction illustrates an interesting power of fully retroactive queries: they can be used to quickly solve batched pair evaluation. We believe this technique can prove useful for other data structure lower bounds, especially dynamic ones.