Janos Simon

Space-bounded hierarchies and probabilistic computations

Abstract We study three aspects of the power of space-bounded probabilistic Turing machines. First, we give a simple alternative proof of Simons result that space-bounded probabilistic complexity classes are closed under complement. Second, we demonstrate that any language recognizable by an alternating Turing machine in log n space with a constant number of alternations (the log n space “alternation hierarch”) also can be recognized by a log n spacebounded probabilistic Turing machine with small error probability; this is a generalization of Gills result that any language in NSPACE (log n) can be recognized by such a machine. Third, we give a new definition of space-bounded oracle machines, and use it to define a space-bounded “oracle hierarchy” analogous to the original definition of the polynomial time hierarchy. Unlike its polynomial time analogue, the entire log n space “alternation hierarchy” is contained in the second level of the log n space “oracle hierarchy.” However, the entire log n space “oracle hierarchy”is still contained in bounded-error probabilistic space log n.

An information-theoretic approach to time bounds for on-line computation

Abstract Static, descriptional complexity (program size) can be used to obtain lower bounds on dynamic, computational complexity (such as running time). We discuss the approach and use it to obtain lower time bounds for on-line simulation of one abstract storage unit by another. Our main results show that more points of access into multidimensional or tree-shaped storage can save significant time.

Probabilistic Communication Complexity

We study (unbounded error) probabilistic communication complexity. Our new results include -one way and two complexities differ by at most 1 - certain functions like equality and the verification of Hamming distance have upper bounds that are considerably better than their counterparts in deterministic, nondeterministic, or bounded error probabilistic model - there exists a function which requires /spl Omega/(logn) information transfer. As an application, we prove that a certain language requires /spl Omega/(nlogn) time to be recognized by a 1-tape (unbounded error) probabilistic Turing machine. This bound is optimal. (Previous lower bound results [Yao 1] require acceptance by bounded error computation. We believe that this is the first nontrivial lower bound on the time required by unrestricted probabilistic Turing machines.

Lower bounds on graph threading by probabilistic machines

It is likely that reliable and fast space-bounded probabilistic acceptors are less powerful than nondeterministic ones. We consider a restricted model of space-bounded probabilistic computation, the random analog of a model studied in [CR]. We show that maze traversal (a complete problem for nondeterministic space log n) requires space Ω(log2n/loglogn) by random machines, even if fast is relaxed to mean only subexponential. In particular, the lower bound on space holds for the time complexity of Savitchs algorithm (which can be simulated in the model).

Division in idealized unit cost RAMs

Abstract We study the power of RAM acceptors with several instruction sets. We exhibit several instances where the availability of the division operator increases the power of the acceptors. We also show that in certain situations parallelism and stochastic features (“distributed random choices”) are provably more powerful than either parallelism or randomness alone. We relate the class of probabilistic Turing machine computations to random access machines with multiplication (but without boolean vector operations). Again, the availability of integer division seems to play a crucial role in these results.

Lower bounds on the time of probabilistic on-line simulations

We study probabilistic on-line simulators for several machine models (or memory structures). The simulators have a more constrained access to data than the virtual machines, but are allowed to use probabilistic means to improve average access time. We show that in many cases coin tosses can not make up for inadequate access.

Space-bounded probabilistic turing machine complexity classes are closed under complement (Preliminary Version)

For tape constructible functions S(n)≥log n, if a language L is accepted by an S(n) tape bounded probabilistic Turing machine, then there is an S(n) tape bounded probabilistic Turing machine that accepts &Lmarc;, the complement of L.

Space efficient algorithms for some graph theoretical problems

SummaryWe present space-efficient-O(log2n)-deterministic algorithms for some graph theoretical problems such as planarity testing, producing a plane embedding, finding minimum cost spanning trees, obtaining the connected, biconnected and triconnected components of a graph. Previous planarity algorithms used Ω(n) space. Several algorithms are based on a space-efficient matrix inversion method. The same bounds hold for uniform circuit depth.

Parallel Algorithms in Graph Theory: Planarity Testing (preliminary version)

We present 0(log2n) step parallel algorithms for planarity testing and for finding the triply connected components of a graph. The algorithms use a polynomial number of synchronous processors with shared memory.

Lower Bounds for Solving Undirected Graph Problems on VLSI

We study VLSI solutions to the connected component problem on networks that have area too small to store all the edges of the graph for the entire computation. We give lower bounds on the time needed to solve this problem on such networks. The lower bounds use a new proof technique combining adversary strategy, information flow, and Kolmogorov complexity arguments. The lower bounds obtained for the connected components problem hold for a number of other undirected graph problems.