Average-Case Analysis of Graph-Searching Algorithms (Thesis)

Abstract:

We estimate the expected value of various search quantities for a variety of
graph-searching methods, for example, depth-first search and breadth-first
search. Our analysis applies to both directed and undirected random graphs,
and it covers the range of interesting graph densities, including densities
at which a random graph is disconnected with a giant connected component.
We estimate the number of edges examined during the search, since this number
is proportional to the running time of the algorithm. We find that for hardly
connected graphs, all of the edges might be examined, but for denser graphs
many fewer edges are generally required. We prove that any searching algorithm
examines O(n log n) edges, if present, on all random graphs with n nodes but
not necessarily on the complete graphs.
One property of some searching algorithms is the maximum depth of the search.
In depth-first search, this depth can be used to estimate the space needed for
the recursion stack. For random graphs of any density, even for disconnected
graphs, we prove that this space is O(n). On the other hand, the depth of
breadth-first search is O(log n/log(pn)), where p is the probability of the
existence of an edge. The size of the data structure needed by any searching
algorithm is proved to be O(n). If the search terminates at a particular node,
any searching algorithm needs a data structure of size O(n), and examines
only O(n) edges.
Finally we derive similar results for variants of the above searching
algorithms, more general classes of searching algorithms, and for random
graphs with multiple edges.
These results are verified through simulations. The techniques employed to
permit simulations of big graphs (few millions of nodes) and the results
obtained are of general interest, especially to those performing similar
experiments.