On the Performance of the Depth First Search Algorithm in Supercritical Random Graphs

Abstract

We consider the performance of the Depth First Search (DFS) algorithm on the random graph $G\left(n,\frac{1+\epsilon}{n}\right)$, $\epsilon>0$ a small constant. Recently, Enriquez, Faraud and Ménard proved that the stack $U$ of the DFS follows a specific scaling limit, reaching the maximal height of $\left(1+o_{\epsilon}(1)\right)\epsilon^2n$. Here we provide a simple analysis for the typical length of a maximum path discovered by the DFS.

Site percolation on pseudo‐random graphs

“…We note that in the application of the DFS algorithm to the case of bond percolation (see [20]), there is a random variable corresponding to the number of queries between the stack $$$ U $$$ and the set $$$ T $$$, corresponding to edges whose coin flip was answered in the negative. Therefore, in order to obtain the asymptotic order of the giant in bond percolation utilizing the DFS algorithm, one needs to estimate this random variable (see [9] for an estimation of this random variable and a careful yet relatively simple analysis of the performance of the DFS in $$$ G\left(n,p\right) $$$). Here, on the other hand, all the coin flips answered in the negative correspond to a vertex moving to $$$ W $$$.…”