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# Unambiguous logarithmic space bounded computations

#### Abstract

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. ^

#### Subject Area

Mathematics|Computer Science

#### Recommended Citation

Tewari, Raghunath, "Unambiguous logarithmic space bounded computations" (2011). *ETD collection for University of Nebraska - Lincoln*. AAI3449405.

http://digitalcommons.unl.edu/dissertations/AAI3449405