A celebrated theorem of Savitch states that NSPACE(S) is contained in DSPACE(S^2). In particular, Savitch gave a deterministic algorithm to solve ST-CONNECTIVITY (an NL-complete problem) using O(log^2 n ) space, implying NL is in DSPACE(log^2 n ). While Savitch's theorem itself has not been improved in the last four decades, studying the space complexity of several special cases of ST-CONNECTIVITY has provided new insights into the space-bounded complexity classes. In this paper, we introduce new kind of graph connectivity problems which we call graph realizability problems. All of our graph realizability problems are generalizations of UNDIRECTED ST-CONNECTIVITY. ST-REALIZABILITY, the most general graph realizability problem, is LogCFL-complete. We define the corresponding complexity classes that lie between L and LogCFL and study their relationships. As special cases of our graph realizability problems we define two natural problems, BALANCED ST-CONNECTIVITY and POSITIVE BALANCED ST-CONNECTIVITY, that lie between L and NL. We present a deterministic O(lognloglogn) space algorithm for BALANCED ST-CONNECTIVITY. More generally we prove that SGSLogCFL, a generalization of BALANCED ST-CONNECTIVITY, is contained in DSPACE(lognloglogn). To achieve this goal we generalize several concepts (such as graph squaring and transitive closure) and algorithms (such as parallel algorithms) known in the context of UNDIRECTED ST-CONNECTIVITY.
Symmetric AuxPDAs : In sec:ustconn , we define , a symmetric" version of . To study the space complexity of we define symmetric auxiliary pushdown automata, a natural generalization of symmetric Turing machines introduced by Lewis and Papadimitriou @cite_17 . We introduce a new complexity class called , a generalization of and show that @math @math .
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Abstract A symmetric Turing machine is one such that the “yields” relation between configurations is symmetric. The space complexity classes for such machines are found to be intermediate between the corresponding deterministic and nondeterministic space complexity classes. Certain natural problems are shown to be complete for symmetric space complexity classes, and the relationship of symmetry to determinism and nondeterminism is investigated.