62,488 research outputs found
Shorter paths to graph algorithms
AbstractWe illustrate the use of formal languages and relations in compact formal derivations of some graph algorithms
Shorter paths to graph algorithms
We illustrate the use of formal languages and relations in compact formal derivations of some graph algorithms
On Hamiltonicity of {claw, net}-free graphs
An st-path is a path with the end-vertices s and t. An s-path is a path with
an end-vertex s. The results of this paper include necessary and sufficient
conditions for a {claw, net}-free graph G with given two different vertices s,
t and an edge e to have (1)a Hamiltonian s-path, (2) a Hamiltonian st-path, (3)
a Hamiltonian s- and st-paths containing edge e when G has connectivity one,
and (4) a Hamiltonian cycle containing e when G is 2-connected. These results
imply that a connected {claw, net}-free graph has a Hamiltonian path and a
2-connected {claw, net}-free graph has a Hamiltonian cycle [D. Duffus, R.J.
Gould, M.S. Jacobson, Forbidden Subgraphs and the Hamiltonian Theme, in The
Theory and Application of Graphs (Kalamazoo, Mich., 1980$), Wiley, New York
(1981) 297--316.] Our proofs of (1)-(4) are shorter than the proofs of their
corollaries in [D. Duffus, R.J. Gould, M.S. Jacobson] and provide
polynomial-time algorithms for solving the corresponding Hamiltonicity
problems.
Keywords: graph, claw, net, {claw, net}-free graph, Hamiltonian path,
Hamiltonian cycle, polynomial-time algorithm.Comment: 9 page
Repairing Wireless Sensor Network connectivity with mobility and hop-count constraints
Wireless Sensor Networks can become partitioned due to node failure or damage, and must be repaired by deploying new sensors, relays or sink nodes to restore some quality of service. We formulate the task as a multi-objective problem over two graphs. The solution specifies additional nodes to reconnect a connectivity graph subject to network path-length constraints, and a path through a mobility graph to visit those locations. The objectives are to minimise both the cost of the additional nodes and the length of the mobility path. We propose two heuristic algorithms which prioritise the different objectives. We evaluate the two algorithms on randomly generated graphs, and compare their solutions to the optimal solutions for the individual objectives. Finally, we assess the total restoration time for different classes of agent, i.e. small robots and larger vehicles, which allows us to trade-off longer computation times for shorter mobility paths
Speeding up shortest path algorithms
Given an arbitrary, non-negatively weighted, directed graph we
present an algorithm that computes all pairs shortest paths in time
, where is the number of
different edges contained in shortest paths and is a running
time of an algorithm to solve a single-source shortest path problem (SSSP).
This is a substantial improvement over a trivial times application of
that runs in . In our algorithm we use
as a black box and hence any improvement on results also in improvement
of our algorithm.
Furthermore, a combination of our method, Johnson's reweighting technique and
topological sorting results in an all-pairs
shortest path algorithm for arbitrarily-weighted directed acyclic graphs.
In addition, we also point out a connection between the complexity of a
certain sorting problem defined on shortest paths and SSSP.Comment: 10 page
Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm
Given an undirected graph and two disjoint vertex pairs and
, the Shortest two disjoint paths problem (S2DP) asks for the minimum
total length of two vertex disjoint paths connecting with , and
with , respectively.
We show that for cubic planar graphs there are NC algorithms, uniform
circuits of polynomial size and polylogarithmic depth, that compute the S2DP
and moreover also output the number of such minimum length path pairs.
Previously, to the best of our knowledge, no deterministic polynomial time
algorithm was known for S2DP in cubic planar graphs with arbitrary placement of
the terminals. In contrast, the randomized polynomial time algorithm by
Bj\"orklund and Husfeldt, ICALP 2014, for general graphs is much slower, is
serial in nature, and cannot count the solutions.
Our results are built on an approach by Hirai and Namba, Algorithmica 2017,
for a generalisation of S2DP, and fast algorithms for counting perfect
matchings in planar graphs
Physiology-Aware Rural Ambulance Routing
In emergency patient transport from rural medical facility to center tertiary
hospital, real-time monitoring of the patient in the ambulance by a physician
expert at the tertiary center is crucial. While telemetry healthcare services
using mobile networks may enable remote real-time monitoring of transported
patients, physiologic measures and tracking are at least as important and
requires the existence of high-fidelity communication coverage. However, the
wireless networks along the roads especially in rural areas can range from 4G
to low-speed 2G, some parts with communication breakage. From a patient care
perspective, transport during critical illness can make route selection patient
state dependent. Prompt decisions with the relative advantage of a longer more
secure bandwidth route versus a shorter, more rapid transport route but with
less secure bandwidth must be made. The trade-off between route selection and
the quality of wireless communication is an important optimization problem
which unfortunately has remained unaddressed by prior work.
In this paper, we propose a novel physiology-aware route scheduling approach
for emergency ambulance transport of rural patients with acute, high risk
diseases in need of continuous remote monitoring. We mathematically model the
problem into an NP-hard graph theory problem, and approximate a solution based
on a trade-off between communication coverage and shortest path. We profile
communication along two major routes in a large rural hospital settings in
Illinois, and use the traces to manifest the concept. Further, we design our
algorithms and run preliminary experiments for scalability analysis. We believe
that our scheduling techniques can become a compelling aid that enables an
always-connected remote monitoring system in emergency patient transfer
scenarios aimed to prevent morbidity and mortality with early diagnosis
treatment.Comment: 6 pages, The Fifth IEEE International Conference on Healthcare
Informatics (ICHI 2017), Park City, Utah, 201
Counting approximately-shortest paths in directed acyclic graphs
Given a directed acyclic graph with positive edge-weights, two vertices s and
t, and a threshold-weight L, we present a fully-polynomial time
approximation-scheme for the problem of counting the s-t paths of length at
most L. We extend the algorithm for the case of two (or more) instances of the
same problem. That is, given two graphs that have the same vertices and edges
and differ only in edge-weights, and given two threshold-weights L_1 and L_2,
we show how to approximately count the s-t paths that have length at most L_1
in the first graph and length at most L_2 in the second graph. We believe that
our algorithms should find application in counting approximate solutions of
related optimization problems, where finding an (optimum) solution can be
reduced to the computation of a shortest path in a purpose-built auxiliary
graph
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