# Space Efficient Algorithms for Breadth-Depth Search

## TL;DR

This paper introduces space-efficient algorithms for the graph search method known as breadth-depth search, reducing space complexity significantly while maintaining near-linear time performance.

## Contribution

The paper presents novel algorithms for breadth-depth search that operate in o(n log n) space, improving upon classical space bounds without substantial time penalties.

## Key findings

- Achieved space complexity below n log n bits
- Maintained linear time complexity for graph traversal
- Successfully visited all vertices in correct order

## Abstract

Continuing the recent trend, in this article we design several space-efficient algorithms for two well-known graph search methods. Both these search methods share the same name {\it breadth-depth search} (henceforth {\sf BDS}), although they work entirely in different fashion. The classical implementation for these graph search methods takes $O(m+n)$ time and $O(n \lg n)$ bits of space in the standard word RAM model (with word size being $\Theta(\lg n)$ bits), where $m$ and $n$ denotes the number of edges and vertices of the input graph respectively. Our goal here is to beat the space bound of the classical implementations, and design $o(n \lg n)$ space algorithms for these search methods by paying little to no penalty in the running time. Note that our space bounds (i.e., with $o(n \lg n)$ bits of space) do not even allow us to explicitly store the required information to implement the classical algorithms, yet our algorithms visits and reports all the vertices of the input graph in correct order.