Abstract
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 in 2004 that any partially retroactive data structure with operation time T_{op}(n,m) can be transformed into a fully retroactive data structure with operation time O(sqrt{m} * T_{op}(n,m)), where n is the size of the data structure and m is the number of operations in the timeline [Demaine et al., 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 << 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}^{1o(1)} assuming any of the following conjectures:
 Conjecture I: Circuit SAT requires 2^{n  o(n)} time on ninput circuits of size 2^{o(n)}. This conjecture is far weaker than the wellbelieved SETH conjecture from complexity theory, which asserts that CNF SAT with n variables and O(n) clauses already requires 2^{no(n)} time.
 Conjecture II: Online (min,+) product between an integer n x n matrix and n vectors requires n^{3  o(1)} time. This conjecture is weaker than the APSP conjectures widely used in finegrained complexity.
 Conjecture III (3SUM 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. This 1995 conjecture [Anka Gajentaan and Mark H. Overmars, 1995] was the first conjecture in finegrained complexity.
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.
BibTeX  Entry
@InProceedings{chen_et_al:LIPIcs:2018:8859,
author = {Lijie Chen and Erik D. Demaine and Yuzhou Gu and Virginia Vassilevska Williams and Yinzhan Xu and Yuancheng Yu},
title = {{Nearly Optimal Separation Between Partially and Fully Retroactive Data Structures}},
booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)},
pages = {33:133:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770682},
ISSN = {18688969},
year = {2018},
volume = {101},
editor = {David Eppstein},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8859},
URN = {urn:nbn:de:0030drops88593},
doi = {10.4230/LIPIcs.SWAT.2018.33},
annote = {Keywords: retroactive data structure, conditional lower bound}
}
Keywords: 

retroactive data structure, conditional lower bound 
Collection: 

16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018) 
Issue Date: 

2018 
Date of publication: 

04.06.2018 