The performance of the Rapidly-exploring Random Tree (RRT) algorithm to find low-stretch spanning trees is analyzed on a special family of graphs. The construction of this graph family is described in the report, and we show that it is possible to force a group of edges to be removed with high probability. On this graph family, running the RRT algorithm gives a spanning tree with one group of edges that has expected stretch O(nc), c < 1. Adding multi-edges to the graph will lead to a counter example for the RRT algorithm because there exists at least one graph family such that running the RRT algorithm on this graph family gives a spanning tree of average stretch O(nc). Several ways to modify the algorithm are proposed. The idea of using random walk in the algorithm was proposed in Fall 2017 and a generalization of a random-walk step is investigated in this project. However, experimental evidence shows that a generalized random walk version of Rapidly-exploring Random Tree algorithm does not give trees with average stretch lower than O(log n). Another idea for changing the RRT algorithm by sampling vertices based on weighted degree also does not give trees with low average stretch. We also present another modified version of the RRT algorithm that may lead to a promising approach to find low-stretch trees. The new algorithm relies on the interpretation of graphs as systems of resistors or as flow networks, and the Laplacian of the graph is used to determine the sampling distribution on vertices.
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