Depth-First Search Performance in a Random Digraph with Geometric Outdegree Distribution

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.