6,700 research outputs found
Finding Simple Shortest Paths and Cycles
The problem of finding multiple simple shortest paths in a weighted directed
graph has many applications, and is considerably more difficult than
the corresponding problem when cycles are allowed in the paths. Even for a
single source-sink pair, it is known that two simple shortest paths cannot be
found in time polynomially smaller than (where ) unless the
All-Pairs Shortest Paths problem can be solved in a similar time bound. The
latter is a well-known open problem in algorithm design. We consider the
all-pairs version of the problem, and we give a new algorithm to find
simple shortest paths for all pairs of vertices. For , our algorithm runs
in time (where ), which is almost the same bound as
for the single pair case, and for we improve earlier bounds. Our approach
is based on forming suitable path extensions to find simple shortest paths;
this method is different from the `detour finding' technique used in most of
the prior work on simple shortest paths, replacement paths, and distance
sensitivity oracles.
Enumerating simple cycles is a well-studied classical problem. We present new
algorithms for generating simple cycles and simple paths in in
non-decreasing order of their weights; the algorithm for generating simple
paths is much faster, and uses another variant of path extensions. We also give
hardness results for sparse graphs, relative to the complexity of computing a
minimum weight cycle in a graph, for several variants of problems related to
finding simple paths and cycles.Comment: The current version includes new results for undirected graphs. In
Section 4, the notion of an (m,n) reduction is generalized to an f(m,n)
reductio
Fast network configuration in Software Defined Networking
Software Defined Networking (SDN) provides a framework to dynamically adjust and re-program the data plane with the use of flow rules. The realization of highly adaptive SDNs with the ability to respond to changing demands or recover after a network failure in a short period of time, hinges on efficient updates of flow rules. We model the time to deploy a set of flow rules by the update time at the bottleneck switch, and formulate the problem of selecting paths to minimize the deployment time under feasibility constraints as a mixed integer linear program (MILP). To reduce the computation time of determining flow rules, we propose efficient heuristics designed to approximate the minimum-deployment-time solution by relaxing the MILP or selecting the paths sequentially. Through extensive simulations we show that our algorithms outperform current, shortest path based solutions by reducing the total network configuration time up to 55% while having similar packet loss, in the considered scenarios. We also demonstrate that in a networked environment with a certain fraction of failed links, our algorithms are able to reduce the average time to reestablish disrupted flows by 40%
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