Constructing Generalized Universal Traversing Sequences of Polynomial Size for Graphs with Small Diameter (Extended Abstract)

The paper constructs a generalized version of universal traversing sequences. The generalization preserves the features of the universal traversing sequences that make them attractive for applications to derandomizations and space-bounded computation. For every n, a sequence is constructed that is used by a finite-automaton with 0(1) states in order to traverse all the n-vertex labeled undirected graphs. The automaton walks on the graph; when it is at a certain vertex, it uses the edge labels and the sequence in order to decide which edge to follow. When walking on an edge, the automaton can see the edge labeling. The generalized sequences have size 2°(8(")) and traverse all the n-vertex undirected graphs G satisfying Diam(G) * log(A(G)) 5 a(n), where Diam(G) is the diameter of G, and A(G) is the maximum degree of G. As a corollary we obtain polynomial size generalized universal traversing sequences construcible in DSpace(1ogn) for the following classes of graphs, where a ( n ) = O(1ogn): expanders of constant degree, random graphs, butterfly networks, shuffle-exchange networks, cubeconnected-cycles networks, de Bruijn networks, cliques. For other classes of graphs, the construction gives better traversing bounds than the universal traversing sequences constructed by Nisan [ll] for arbitrary undirected graphs; for example in the case of the hypercubes, our sequences have size n0(lo9'ogn). 'Supported by NSF Grant No. CCR-8810074 The only known class of graphs where universal traversing sequences of polynomial size can be constructed is the class of graphs of maximum degree 2 [9]. The construction of universal traversing sequences (in their standard or generalized form) for arbitrary undirected graphs in DSpace(1ogn) will have strong consequences in complexity theory. What we may call The Undirected Graph Connectivity Conjecture: UNDIRECTED CONNECTIVITY is in DSpace(1ogn) will be established by such a construction. As UNDIRECTED CONNECTIVITY is complete for the complexity class SymmetricSpace(logn), it will follow that DSpace(logn) = SymmetricSpace(1ogn) and therefore a variety of fundamental computational problems such as planarity testing, minimum spanning forests will be solvable in DSpace(logn). 1 History of the problem and previous construct ions S. Cook introduced the concept of universal traversing sequence and asked the question of the existence of short, i.e., polynomial size, such sequences. R. Aleliunas (1978) (11 showed that such sequences of size O(nS) exist for 2-regular graphs. (The result was re-obtained CH2925-6/90/0000/0439$01 .OO