194 research outputs found
Efficient Parallel Implementation of the Ramalingam Decremental Algorithm for Updating the Shortest Paths Subgraph
We propose an efficient parallel implementation of the Ramalingam algorithm for dynamic updating the shortest paths subgraph of a directed weighted graph with a sink after deletion of an edge. To this end, a model of associative (content addressable) parallel systems with vertical processing (the STAR-machine) is used. On the STAR-machine, the Ramalingam decremental algorithm for dynamic updating the shortest paths subgraph is represented as the main procedure DeleteArc that uses a group of auxiliary procedures. We provide the DeleteArc procedure along with the auxiliary procedures, prove correctness of these procedures and evaluate the time complexity. We also consider an example of implementing the DeleteArc procedure on the STAR-machine
Fine-Grained Complexity Analysis of Two Classic TSP Variants
We analyze two classic variants of the Traveling Salesman Problem using the
toolkit of fine-grained complexity. Our first set of results is motivated by
the Bitonic TSP problem: given a set of points in the plane, compute a
shortest tour consisting of two monotone chains. It is a classic
dynamic-programming exercise to solve this problem in time. While the
near-quadratic dependency of similar dynamic programs for Longest Common
Subsequence and Discrete Frechet Distance has recently been proven to be
essentially optimal under the Strong Exponential Time Hypothesis, we show that
bitonic tours can be found in subquadratic time. More precisely, we present an
algorithm that solves bitonic TSP in time and its bottleneck
version in time. Our second set of results concerns the popular
-OPT heuristic for TSP in the graph setting. More precisely, we study the
-OPT decision problem, which asks whether a given tour can be improved by a
-OPT move that replaces edges in the tour by new edges. A simple
algorithm solves -OPT in time for fixed . For 2-OPT, this is
easily seen to be optimal. For we prove that an algorithm with a runtime
of the form exists if and only if All-Pairs
Shortest Paths in weighted digraphs has such an algorithm. The results for
may suggest that the actual time complexity of -OPT is
. We show that this is not the case, by presenting an algorithm
that finds the best -move in time for
fixed . This implies that 4-OPT can be solved in time,
matching the best-known algorithm for 3-OPT. Finally, we show how to beat the
quadratic barrier for in two important settings, namely for points in the
plane and when we want to solve 2-OPT repeatedly.Comment: Extended abstract appears in the Proceedings of the 43rd
International Colloquium on Automata, Languages, and Programming (ICALP 2016
Distance Oracles for Time-Dependent Networks
We present the first approximate distance oracle for sparse directed networks
with time-dependent arc-travel-times determined by continuous, piecewise
linear, positive functions possessing the FIFO property.
Our approach precomputes approximate distance summaries from
selected landmark vertices to all other vertices in the network. Our oracle
uses subquadratic space and time preprocessing, and provides two sublinear-time
query algorithms that deliver constant and approximate
shortest-travel-times, respectively, for arbitrary origin-destination pairs in
the network, for any constant . Our oracle is based only on
the sparsity of the network, along with two quite natural assumptions about
travel-time functions which allow the smooth transition towards asymmetric and
time-dependent distance metrics.Comment: A preliminary version appeared as Technical Report ECOMPASS-TR-025 of
EU funded research project eCOMPASS (http://www.ecompass-project.eu/). An
extended abstract also appeared in the 41st International Colloquium on
Automata, Languages, and Programming (ICALP 2014, track-A
A single-source shortest path algorithm for dynamic graphs
Graphs are mathematical structures used in many applications. In recent years, many applications emerged that require the processing of large dynamic graphs where the graph’s structure and properties change constantly over time. Examples include social networks, communication networks, transportation networks, etc. One of the most challenging problems in large scale dynamic graphs is the single-source shortest path (SSSP) problem. Traditional solutions (based on Dijkstra’s algorithms) to the SSSP problem do not scale to large dynamic graphs with a high change frequency. In this paper, we propose an efficient SSSP algorithm for large dynamic graphs. We first present our algorithm and give a formal proof of its correctness. Then, we give an analytical evaluation of the proposed solution
- …