Log In Sign Up

Coarse-Grained Complexity for Dynamic Algorithms

by   Sayan Bhattacharya, et al.

To date, the only way to argue polynomial lower bounds for dynamic algorithms is via fine-grained complexity arguments. These arguments rely on strong assumptions about specific problems such as the Strong Exponential Time Hypothesis (SETH) and the Online Matrix-Vector Multiplication Conjecture (OMv). While they have led to many exciting discoveries, dynamic algorithms still miss out some benefits and lessons from the traditional “coarse-grained” approach that relates together classes of problems such as P and NP. In this paper we initiate the study of coarse-grained complexity theory for dynamic algorithms. Below are among questions that this theory can answer. What if dynamic Orthogonal Vector (OV) is easy in the cell-probe model? A research program for proving polynomial unconditional lower bounds for dynamic OV in the cell-probe model is motivated by the fact that many conditional lower bounds can be shown via reductions from the dynamic OV problem. Since the cell-probe model is more powerful than word RAM and has historically allowed smaller upper bounds, it might turn out that dynamic OV is easy in the cell-probe model, making this research direction infeasible. Our theory implies that if this is the case, there will be very interesting algorithmic consequences: If dynamic OV can be maintained in polylogarithmic worst-case update time in the cell-probe model, then so are several important dynamic problems such as k-edge connectivity, (1+ϵ)-approximate mincut, (1+ϵ)-approximate matching, planar nearest neighbors, Chan's subset union and 3-vs-4 diameter. The same conclusion can be made when we replace dynamic OV by, e.g., subgraph connectivity, single source reachability, Chan's subset union, and 3-vs-4 diameter. Lower bounds for k-edge connectivity via dynamic OV? (see the full abstract in the pdf file).


page 1

page 2

page 3

page 4


New Amortized Cell-Probe Lower Bounds for Dynamic Problems

We build upon the recent papers by Weinstein and Yu (FOCS'16), Larsen (F...

Fine-Grained Complexity Theory: Conditional Lower Bounds for Computational Geometry

Fine-grained complexity theory is the area of theoretical computer scien...

Tight Cell Probe Bounds for Succinct Boolean Matrix-Vector Multiplication

The conjectured hardness of Boolean matrix-vector multiplication has bee...

Fine-Grained I/O Complexity via Reductions: New lower bounds, faster algorithms, and a time hierarchy

This paper initiates the study of I/O algorithms (minimizing cache misse...

Tight Dynamic Problem Lower Bounds from Generalized BMM and OMv

The main theme of this paper is using k-dimensional generalizations of t...

Fully Dynamic Single-Source Reachability in Practice: An Experimental Study

Given a directed graph and a source vertex, the fully dynamic single-sou...

Fine-Grained Complexity of Safety Verification

We study the fine-grained complexity of Leader Contributor Reachability ...