# An Optimal Vector Clock Algorithm for Multithreaded Systems

## TL;DR

This paper introduces an optimal hybrid vector clock algorithm for multithreaded systems that minimizes clock size by leveraging graph theory, improving efficiency in causality tracking.

## Contribution

It presents a novel algorithm combining thread and object components, using bipartite graph matching to determine the minimal vector clock size.

## Key findings

- Offline algorithm reduces vector clock size below min(n, m)
- Online mechanisms are effective for sparse bipartite graphs
- Algorithm outperforms traditional thread-only or object-only clocks

## Abstract

Tracking causality (or happened-before relation) between events is useful for many applications such as debugging and recovery from failures. Consider a concurrent system with $n$ threads and $m$ objects. For such systems, either a vector clock of size $n$ is used with one component per thread or a vector clock of size $m$ is used with one component per object. A natural question is whether one can use a vector clock of size strictly less than the minimum of $m$ and $n$ to timestamp events. We give an algorithm in this paper that uses a hybrid of thread and object components. Our algorithm is guaranteed to return the minimum number of components necessary for vector clocks. We first consider the case when the interaction between objects and threads is statically known. This interaction is modeled by a thread-object bipartite graph. Our algorithm is based on finding the maximum bipartite matching of such a graph and then applying K\"{o}nig-Egerv\'{a}ry Theorem to compute the minimum vertex cover to determine the optimal number of components necessary for the vector clock. We also propose two mechanisms to compute such an vector clock when computation is revealed in an online fashion. Evaluation on different types of graphs indicates that our offline algorithm generates a size vector clock which is significantly less than the minimum of $m$ and $n$. These mechanisms are more effective when the underlying bipartite graph is not dense.