Dynamic Algorithms Against an Adaptive Adversary: Generic Constructions and Lower Bounds

TL;DR

This paper introduces faster dynamic algorithms resilient to adaptive adversaries using techniques from privacy and cryptography, and establishes fundamental lower bounds showing some problems inherently require more time against adaptive adversaries.

Contribution

It provides a general reduction from oblivious to adaptive adversary models and demonstrates improved algorithms and lower bounds for dynamic problems.

Findings

Faster dynamic algorithms for global minimum cut, all pairs distances, and effective resistance.

A reduction technique using differential privacy to protect randomness in algorithms.

Lower bounds showing polynomial slowdown is necessary for certain problems against adaptive adversaries.

Abstract

A dynamic algorithm against an adaptive adversary is required to be correct when the adversary chooses the next update after seeing the previous outputs of the algorithm. We obtain faster dynamic algorithms against an adaptive adversary and separation results between what is achievable in the oblivious vs. adaptive settings. To get these results we exploit techniques from differential privacy, cryptography, and adaptive data analysis. We give a general reduction transforming a dynamic algorithm against an oblivious adversary to a dynamic algorithm robust against an adaptive adversary. This reduction maintains several copies of the oblivious algorithm and uses differential privacy to protect their random bits. Using this reduction we obtain dynamic algorithms against an adaptive adversary with improved update and query times for global minimum cut, all pairs distances, and all pairs effective resistance. We further improve our update and query times by showing how to maintain a sparsifier over an expander decomposition that can be refreshed fast. This fast refresh enables it to be robust against what we call a blinking adversary that can observe the output of the algorithm only following refreshes. We believe that these techniques will prove useful for additional problems. On the flip side, we specify dynamic problems that, assuming a random oracle, every dynamic algorithm that solves them against an adaptive adversary must be polynomially slower than a rather straightforward dynamic algorithm that solves them against an oblivious adversary. We first show a separation result for a search problem and then show a separation result for an estimation problem. In the latter case our separation result draws from lower bounds in adaptive data analysis.