MOD-CHAR: an implementation of Char's spanning tree enumeration algorithm and its complexity analysis

Abstract

Two complexity analyses of MOD-CHAR are presented. It is shown that MOD-CHAR leads to better complexity results for J.P. Char's algorithm than what could be obtained using the straightforward implementation implied in Char's original presentation (see IEEE Trans. Circuit Theory, Vol.15, p.228-38, 1968). The class of graphs for which MOD-CHAR and, hence, Char's algorithm, has linear time complexity per spanning tree generated is identified. This class is more general than the corresponding one identified in R. Jayakumar et al. (see ibid., vol.31, no.10, p.853-60, 1984). Using a result on random graphs, it is proved that for almost all graphs MOD-CHAR has linear worst-case time complexity per spanning tree generated. It is also shown that for any complete graph MOD-CHAR requires, on the average, at most seven computational steps to generate a spanning tree. This result and computational experiences provide evidence to believe that for dense graphs of any order the time complexity of MOD-CHAR is O(t), where t is the number of spanning trees generated. On the other hand, there is enough evidence to conclude that for sparse graphs, Char's original implementation is superior to MOD-CHAR.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>