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.
A celebrated theorem of Savitch [Sav70] states that N SP ACE(S) ⊆ DSP ACE(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 ⊆ DSP ACE(log 2 n). Savitch's algorithm runs in time 2 O(log 2 n) . It has been a longstanding open problem to improve Savitch's theorem i.e., to prove (i) NL ⊆ DSP ACE(o(log 2 n)) or (ii) NL ⊆ SC 2 , i.e., ST-CONNECTIVITY can be solved by a deterministic algorithm in polynomial time and O(log 2 n) space.
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. Allender’s survey [All07] gives an update of progress related to several special cases of ST-CONNECTIVITY. Recently ST-CONNECTIVITY in planar DAGs with O(logn) sources is shown to be in L [SBV10]. Stolee All the connectivity problems considered in the literature so far are essentially special cases of ST-CONNECTIVITY. In the first half of 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.
In the second half of this paper, we study the space complexity of SGSLogCFL (see Section 4.1 for definition). We define generalizations of graph squaring and transitive closure, present efficient parallel algorithms for SGSLogCFL and use the techniques of Trifonov [Tri08] to show that SGSLogCFL is contained in DSP ACE(lognloglogn). This implies that BALANCED ST-CONNECTIVITY, a natural graph connectivity problem which lies between L and NL, is contained in DSP ACE(lognloglogn).
Auxiliary Pushdown Automata : A language is accepted by a non-deterministic pushdown automaton (PDA) if and only if it is a context-free language. Deterministic context-free languages are those accepted by the deterministic PDAs. LogCFL is the set of all languages that are log-space reducible to a context-free language. Similarly, LogDCFL is the set of all languages that are log-space reducible to a deterministic context-free language. There are many equivalent characterizations of LogCFL. Sudborough [Sud78] gave the machine class equivalence. Ruzzo [Ruz80] gave an alternating Turing machine (ATM) class equivalent to LogCFL. Venkateswaran [Ven91] gave a circuit characterization and showed that LogCFL = SAC 1 . For a survey of parallel complexity classes and LogCFL see Limaye’s thesis [Lim05].
An Auxiliary Pushdown Automaton (NAuxPDA or simply AuxPDA), introduced by Cook [Coo71], is a two-way PDA augmented with an S(n)-space bounded work tape. If a deterministic two-way PDA is augmented with an S(n)-space bounded work tape then we get a Deterministic Auxiliary Pushdown Automaton (DAuxPDA). We present the formal definitions in the appendix (see Section A). Let NAuxPDA-SpaceTime (S(n),T (n)) be the class of languages accepted by an AuxPDA with S(n)-space bounded work tapes and the running time bounded by T (n). Let the corresponding deterministic class be DAuxPDA-SpaceTime (S(n),T (n)). It is easy to see that NL ⊆ NAuxPDA-SpaceTime (O(logn), poly(n)). It is shown by Sudborough that NAuxPDA-SpaceTime (O(logn), poly(n)) = LogCFL and DAuxPDA-SpaceTime (O(logn),poly(n))
= LogDCFL [Sud78]. Using ATM simulations, Ruzzo showed that LogCFL ⊆ NC 2 [Ruz80]. Simpler proofs of DAuxPDA-SpaceTime (O(logn),poly(n)) = LogDCFL and LogCFL = SAC 1 are given in [MRV99].
Many proof techniques and results obtained in the context of NL, are generalized to obtain the corresponding results for LogCFL. For example : (i) Borodin [Bor77] proved that NL ⊆ NC 2 . Ruzzo [Ruz80] introduced tree-size-bounded alternating Turing machines, gave a new characterization of LogCFL, and proved that LogCFL ⊆ NC 2 . (ii) Immerman [Imm88] and Szelepcsényi [Sze87] proved that NL = co-NL. Borodin et. al. [BCD + 89] generalized their inductive counting technique and proved that LogCFL = co-LogCFL. In fact, they proved a stronger result showing that SAC i is closed under complementation for i > 0. (iii) Wigderson [Wig94] proved that NL ≤ r ⊕NL. Gál and Wigderson [GW96] proved that LogCFL ≤ r ⊕LogCFL. (iv) Nisan [Nis94] proved that BPL ⊆ SC 2 . Venkateswaran [Ven06,Ven09] proved that BPLogCFL ⊆ SC 2 and BPLogCFL ⊆ NC 2 . Here BPLogCFL (resp. RLogCFL and ZPLogCFL) is the bounded error (resp. one-sided error and zero error) probabilistic version of LogCFL. All the above results are elegant and non-trivial generalizations of the corresponding results in t
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