9 research outputs found
An O(n 1 2 +ɛ)-Space and Polynomial-Time Algorithm for Directed Planar Reachability
Abstract—We show that the reachability problem over directed planar graphs can be solved simultaneously in polynomial time and approximately O ( √ 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 √ log n Keywords-reachability, directed planar graph, sublinear space, polynomial time I
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
Tight bounds for undirected graph exploration with pebbles and multiple agents
We study the problem of deterministically exploring an undirected and
initially unknown graph with vertices either by a single agent equipped
with a set of pebbles, or by a set of collaborating agents. The vertices of the
graph are unlabeled and cannot be distinguished by the agents, but the edges
incident to a vertex have locally distinct labels. The graph is explored when
all vertices have been visited by at least one agent. In this setting, it is
known that for a single agent without pebbles bits of memory
are necessary and sufficient to explore any graph with at most vertices. We
are interested in how the memory requirement decreases as the agent may mark
vertices by dropping and retrieving distinguishable pebbles, or when multiple
agents jointly explore the graph. We give tight results for both questions
showing that for a single agent with constant memory
pebbles are necessary and sufficient for exploration. We further prove that the
same bound holds for the number of collaborating agents needed for exploration.
For the upper bound, we devise an algorithm for a single agent with constant
memory that explores any -vertex graph using
pebbles, even when is unknown. The algorithm terminates after polynomial
time and returns to the starting vertex. Since an additional agent is at least
as powerful as a pebble, this implies that agents
with constant memory can explore any -vertex graph. For the lower bound, we
show that the number of agents needed for exploring any graph of size is
already when we allow each agent to have at most
bits of memory for any .
This also implies that a single agent with sublogarithmic memory needs
pebbles to explore any -vertex graph
A Type-Based Complexity Analysis of Object Oriented Programs
A type system is introduced for a generic Object Oriented programming
language in order to infer resource upper bounds. A sound andcomplete
characterization of the set of polynomial time computable functions is
obtained. As a consequence, the heap-space and thestack-space requirements of
typed programs are also bounded polynomially. This type system is inspired by
previous works on ImplicitComputational Complexity, using tiering and
non-interference techniques. The presented methodology has several advantages.
First, itprovides explicit big polynomial upper bounds to the programmer,
hence its use could allow the programmer to avoid memory errors.Second, type
checking is decidable in polynomial time. Last, it has a good expressivity
since it analyzes most object oriented featureslike inheritance, overload,
override and recursion. Moreover it can deal with loops guarded by objects and
can also be extended tostatements that alter the control flow like break or
return.Comment: Information and Computation, Elsevier, A Para\^itre, pp.6