18 research outputs found
Incremental branching programs
We propose a new model of restricted branching programs
which we call {em incremental branching programs}.
We show that {em syntactic}
incremental branching programs capture previously studied
structured models of computation for the problem GEN, namely marking
machines [Cook74].
and Poon\u27s extension [Poon93] of jumping automata
on graphs [CookRackoff80].
We then prove
exponential size lower bounds for our syntactic incremental model,
and for some other restricted branching program models as well.
We further show that nondeterministic syntactic incremental
branching programs are
provably stronger than their deterministic counterpart when solving a
natural NL-complete GEN subproblem.
It remains open if syntactic incremental branching programs are as powerful
as unrestricted branching programs for GEN problems.
Joint work with Anna GĂÂĄl and Michal KouckĂ
Bounds on monotone switching networks for directed connectivity
We separate monotone analogues of L and NL by proving that any monotone
switching network solving directed connectivity on vertices must have size
at least .Comment: 49 pages, 12 figure
New Time-Space Upperbounds for Directed Reachability in High-genus and H-minor-free Graphs
We obtain the following new simultaneous time-space upper bounds for the directed reachability problem. (1) A polynomial-time, O(n^{2/3} * g^{1/3})-space algorithm for directed graphs embedded on orientable surfaces of genus g. (2) A polynomial-time, O(n^{2/3})-space algorithm for all H-minor-free graphs given the tree decomposition, and (3) for K_{3,3}-free and K_5-free graphs, a polynomial-time, O(n^{1/2 + epsilon})-space algorithm, for every epsilon > 0.
For the general directed reachability problem, the best known simultaneous time-space upper bound is the BBRS bound, due to Barnes, Buss, Ruzzo, and Schieber, which achieves a space bound of O(n/2^{k * sqrt(log(n))}) with polynomial running time, for any constant k. It is a significant open question to improve this bound for reachability over general directed graphs. Our algorithms beat the BBRS bound for graphs embedded on surfaces of genus n/2^{omega(sqrt(log(n))}, and for all H-minor-free graphs. This significantly broadens the class of directed graphs for which the BBRS bound can be improved
Formulas vs. Circuits for Small Distance Connectivity
We give the first super-polynomial separation in the power of bounded-depth
boolean formulas vs. circuits. Specifically, we consider the problem Distance
Connectivity, which asks whether two specified nodes in a graph of size
are connected by a path of length at most . This problem is solvable
(by the recursive doubling technique) on {\bf circuits} of depth
and size . In contrast, we show that solving this problem on {\bf
formulas} of depth requires size for all . As corollaries:
(i) It follows that polynomial-size circuits for Distance Connectivity
require depth for all . This matches the
upper bound from recursive doubling and improves a previous lower bound of Beame, Pitassi and Impagliazzo [BIP98].
(ii) We get a tight lower bound of on the size required to
simulate size- depth- circuits by depth- formulas for all and . No lower bound better than
was previously known for any .
Our proof technique is centered on a new notion of pathset complexity, which
roughly speaking measures the minimum cost of constructing a set of (partial)
paths in a universe of size via the operations of union and relational
join, subject to certain density constraints. Half of our proof shows that
bounded-depth formulas solving Distance Connectivity imply upper bounds
on pathset complexity. The other half is a combinatorial lower bound on pathset
complexity
Space Complexity of the Directed Reachability Problem over Surface-Embedded Graphs
The graph reachability problem, the computational task of deciding whether there is a path between two given nodes in a graph is the canonical problem for studying space bounded computations. Three central open questions regarding the space complexity of the reachabil-ity problem over directed graphs are: (1) improving Savitchâs O(log2 n) space bound, (2) designing a polynomial-time algorithm with O(n1â) space bound, and (3) designing an unambiguous non-deterministic log-space (UL) algorithm. These are well-known open questions in complex-ity theory, and solving any one of them will be a major breakthrough. We will discuss some of the recent progress reported on these questions for certain subclasses of surface-embedded directed graphs
An O(n 1 2 +É)-Space and Polynomial-Time Algorithm for Directed Planar Reachability
AbstractâWe show that the reachability problem over directed planar graphs can be solved simultaneously in polynomial time and approximately O ( â n) space. In contrast, the best space bound known for the reachability problem on general directed graphs with polynomial running time is O(n/2 â log n Keywords-reachability, directed planar graph, sublinear space, polynomial time I
Sublinear-Space Lexicographic Depth-First Search for Bounded Treewidth Graphs and Planar Graphs
The lexicographic depth-first search (Lex-DFS) is one of the first basic graph problems studied in the context of space-efficient algorithms. It is shown independently by Asano et al. [ISAAC 2014] and Elmasry et al. [STACS 2015] that Lex-DFS admits polynomial-time algorithms that run with O(n)-bit working memory, where n is the number of vertices in the graph. Lex-DFS is known to be P-complete under logspace reduction, and giving or ruling out polynomial-time sublinear-space algorithms for Lex-DFS on general graphs is quite challenging. In this paper, we study Lex-DFS on graphs of bounded treewidth. We first show that given a tree decomposition of width O(n^(1-?)) with ? > 0, Lex-DFS can be solved in sublinear space. We then complement this result by presenting a space-efficient algorithm that can compute, for w ? ?n, a tree decomposition of width O(w ?nlog n) or correctly decide that the graph has a treewidth more than w. This algorithm itself would be of independent interest as the first space-efficient algorithm for computing a tree decomposition of moderate (small but non-constant) width. By combining these results, we can show in particular that graphs of treewidth O(n^(1/2 - ?)) for some ? > 0 admits a polynomial-time sublinear-space algorithm for Lex-DFS. We can also show that planar graphs admit a polynomial-time algorithm with O(n^(1/2+?))-bit working memory for Lex-DFS
Space-efficient Basic Graph Algorithms
We reconsider basic algorithmic graph problems in a setting where an n-vertex input graph is read-only and the computation must take place in a working memory of O(n) bits or little more than that. For computing connected components and performing breadth-first search, we match
the running times of standard algorithms that have no memory restrictions, for depth-first search and related problems we come within a factor of Theta(loglog n), and for computing minimum spanning forests and single-source shortest-paths trees we come close for sparse input graphs
Pebbling, Entropy and Branching Program Size Lower Bounds
We contribute to the program of proving lower bounds on the size of branching
programs solving the Tree Evaluation Problem introduced by Cook et. al. (2012).
Proving a super-polynomial lower bound for the size of nondeterministic thrifty
branching programs (NTBP) would separate from for thrifty models
solving the tree evaluation problem. First, we show that {\em Read-Once NTBPs}
are equivalent to whole black-white pebbling algorithms thus showing a tight
lower bound (ignoring polynomial factors) for this model.
We then introduce a weaker restriction of NTBPs called {\em Bitwise
Independence}. The best known NTBPs (of size ) for the tree
evaluation problem given by Cook et. al. (2012) are Bitwise Independent. As our
main result, we show that any Bitwise Independent NTBP solving
must have at least states. Prior to this work, lower
bounds were known for NTBPs only for fixed heights (See Cook et. al.
(2012)). We prove our results by associating a fractional black-white pebbling
strategy with any bitwise independent NTBP solving the Tree Evaluation Problem.
Such a connection was not known previously even for fixed heights.
Our main technique is the entropy method introduced by Jukna and Z{\'a}k
(2001) originally in the context of proving lower bounds for read-once
branching programs. We also show that the previous lower bounds given by Cook
et. al. (2012) for deterministic branching programs for Tree Evaluation Problem
can be obtained using this approach. Using this method, we also show tight
lower bounds for any -way deterministic branching program solving Tree
Evaluation Problem when the instances are restricted to have the same group
operation in all internal nodes.Comment: 25 Pages, Manuscript submitted to Journal in June 2013 This version
includes a proof for tight size bounds for (syntactic) read-once NTBPs. The
proof is in the same spirit as the proof for size bounds for bitwise
independent NTBPs present in the earlier version of the paper and is included
in the journal version of the paper submitted in June 201