# Optimal In-place Algorithms for Basic Graph Problems

## TL;DR

This paper introduces linear-time in-place algorithms for fundamental graph problems, significantly improving previous results and exploiting input graph sortedness for efficiency.

## Contribution

It presents the first linear-time in-place algorithms for various core graph problems, surpassing prior polynomial-time solutions.

## Key findings

- Algorithms run in linear time, optimal for the problems addressed.
- Exploiting input sortedness is key to achieving in-place efficiency.
- The techniques may apply to other graph problems in the future.

## Abstract

We present linear time {\it in-place} algorithms for several basic and fundamental graph problems including the well-known graph search methods (like depth-first search, breadth-first search, maximum cardinality search), connectivity problems (like biconnectivity, $2$-edge connectivity), decomposition problem (like chain decomposition) among various others, improving the running time (by polynomial multiplicative factor) of the recent results of Chakraborty et al. [ESA, 2018] who designed $O(n^3 \lg n)$ time in-place algorithms for a strict subset of the above mentioned problems. The running times of all our algorithms are essentially optimal as they run in linear time. One of the main ideas behind obtaining these algorithms is the detection and careful exploitation of sortedness present in the input representation for any graph without loss of generality. This observation alone is powerful enough to design some basic linear time in-place algorithms, but more non-trivial graph problems require extra techniques which, we believe, may find other applications while designing in-place algorithms for different graph problems in the future.