Depth-First Search performance in a random digraph with geometric outdegree distribution

TL;DR

This paper analyzes the performance of depth-first search in a random directed graph with geometric outdegree distribution, providing asymptotic results for various graph properties and revealing a normal distribution for height.

Contribution

It offers the first comprehensive asymptotic analysis of DFS in a specific random digraph model, including distributional results for height.

Findings

Asymptotic results for depth profile, height, and average depth

Height follows an asymptotic normal distribution

Number of trees and their sizes characterized asymptotically

Abstract

We present an analysis of the depth-first search algorithm in a random digraph model with independent outdegrees having a geometric distribution. The results include asymptotic results for the depth profile of vertices, the height (maximum depth) and average depth, the number of trees in the forest, the size of the largest and second-largest trees, and the numbers of arcs of different types in the depth-first jungle. Most results are first order. For the height we show an asymptotic normal distribution. This analysis proposed by Donald Knuth in his next to appear volume of The Art of Computer Programming gives interesting insight in one of the most elegant and efficient algorithm for graph analysis due to Tarjan.