On some conditioning results in the probabilistic analysis of algorithms

A key step in the probabilistic analysis of combinatorial algorithms is often that of establishing that certain conditioning introduced by the operation so far of the algorithm either helps or at least does not hurt too much. In this note we consider a conditioning problem that arises from searching adjacency lists, and which occurs for example in the analysis of algorithms for finding Hamiltonian cycles, perfect matchings, connected components or blocks [l], [2]. We introduce a natural method of generating the random input which reduces the problem to a simple combinatorial one. We are interested in the probabilistic analysis of algorithms which operate on certain random subgraphs R of some fixed (finite, simple) graph G. This graph G may correspond to a part of another graph, for example a complete graph K,,, not explored by a previous phase of the algorithm. We suppose that the random graph R is presented as a family of ordered adjacency lists. At each step, depending on what has happened so far, the algorithm either will terminate or will choose some vertex u and look at the next entry in the adjacency list for u. If it finds vertex w then the edge {u, w} has been selected. For each possible random graph R we suppose that each possible ordering of the adjacency lists is equally likely. Also we suppose that each subgraph of G with the same number of edges is equally likely (see Remark (ii) below). For each vertex u of G let D, denote its degree in the random graph R. Our main result is the following.