1,166 research outputs found
Polynomial expansion and sublinear separators
Let be a class of graphs that is closed under taking subgraphs.
We prove that if for some fixed , every -vertex graph of
has a balanced separator of order , then any
depth- minor (i.e. minor obtained by contracting disjoint subgraphs of
radius at most ) of a graph in has average degree . This confirms a conjecture of Dvo\v{r}\'ak
and Norin.Comment: 6 pages, no figur
Route Planning in Transportation Networks
We survey recent advances in algorithms for route planning in transportation
networks. For road networks, we show that one can compute driving directions in
milliseconds or less even at continental scale. A variety of techniques provide
different trade-offs between preprocessing effort, space requirements, and
query time. Some algorithms can answer queries in a fraction of a microsecond,
while others can deal efficiently with real-time traffic. Journey planning on
public transportation systems, although conceptually similar, is a
significantly harder problem due to its inherent time-dependent and
multicriteria nature. Although exact algorithms are fast enough for interactive
queries on metropolitan transit systems, dealing with continent-sized instances
requires simplifications or heavy preprocessing. The multimodal route planning
problem, which seeks journeys combining schedule-based transportation (buses,
trains) with unrestricted modes (walking, driving), is even harder, relying on
approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4,
previously published by Microsoft Research. This work was mostly done while
the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at
Microsoft Research Silicon Valle
Search for the end of a path in the d-dimensional grid and in other graphs
We consider the worst-case query complexity of some variants of certain
\cl{PPAD}-complete search problems. Suppose we are given a graph and a
vertex . We denote the directed graph obtained from by
directing all edges in both directions by . is a directed subgraph of
which is unknown to us, except that it consists of vertex-disjoint
directed paths and cycles and one of the paths originates in . Our goal is
to find an endvertex of a path by using as few queries as possible. A query
specifies a vertex , and the answer is the set of the edges of
incident to , together with their directions. We also show lower bounds for
the special case when consists of a single path. Our proofs use the theory
of graph separators. Finally, we consider the case when the graph is a grid
graph. In this case, using the connection with separators, we give
asymptotically tight bounds as a function of the size of the grid, if the
dimension of the grid is considered as fixed. In order to do this, we prove a
separator theorem about grid graphs, which is interesting on its own right
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