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Unambiguous logarithmic space bounded computations
Space complexity investigates the power and limitations of a computational model (e.g. a Turing machine) which has a limited amount of workspace to perform its computation. Particularly interesting is the case when the space is only logarithmic in the input size. Unambiguous computation is a natural restriction of nondeterministic computation, where there is a unique accepting path on a ‘Yes’ instance, and no accepting paths on a ‘No’ instance. In this dissertation we study the power of unambiguous log-space computations (denoted as UL) and whether it is general enough to contain all of nondeterministic log-space (denoted as NL). This leads us to the study of the graph reachability problem, which is known to exactly capture the complexity of NL, and thus exhibiting a UL algorithm for reachability is sufficient to show that NL = UL.^ We prove that UL contains certain important restrictions of directed graph reachability. In particular, we show that reachability in planar graphs and certain non-planar graphs are in UL. We give a proof that planar reachability is in UL, by using a result from multi-variable calculus, known as Green’s Theorem. From another viewpoint, we show that deciding reachability in graphs where the number of paths from the start vertex to any other vertex is bounded by a polynomial, is in UL (this result shows that the complexity class ReachFewL is in UL). We also study and prove an upper bound on the UL hierarchy.^ The NL versus UL question led us to another important problem in complexity theory - space complexity of deciding if a graph has a perfect matching. We prove that perfect matching in bipartite bounded genus graphs is in SPL (a class which is a generalization of UL and not known to be comparable with NL). We also show that over bipartite planar graphs, the perfect matching problem is in UL. ^ Embeddings algorithms for graphs on surfaces is well studied in the context of time complexity. Here we give log-space algorithms that provide us certain useful embeddings of planar and bounded genus graphs, on a corresponding surface. ^
Tewari, Raghunath, "Unambiguous logarithmic space bounded computations" (2011). ETD collection for University of Nebraska - Lincoln. AAI3449405.