I/O-optimal algorithms on grid graphs

Given a graph of which the n vertices form a regular two-dimensional grid, and in which each (possibly weighted and/or directed) edge connects a vertex to one of its eight neighbours, the following can be done in O(scan(n)) I/Os, provided M = Omega(B^2): computation of shortest paths with non-negative edge weights from a single source, breadth-first traversal, computation of a minimum spanning tree, topological sorting, time-forward processing (if the input is a plane graph), and an Euler tour (if the input graph is a tree). The minimum-spanning tree algorithm is cache-oblivious. The best previously published algorithms for these problems need Theta(sort(n)) I/Os. Estimates of the actual I/O volume show that the new algorithms may often be very efficient in practice.Comment: 12 pages' extended abstract plus 12 pages' appendix with details, proofs and calculations. Has not been published in and is currently not under review of any conference or journa

Approximation Algorithms for the Asymmetric Traveling Salesman Problem : Describing two recent methods

The paper provides a description of the two recent approximation algorithms for the Asymmetric Traveling Salesman Problem, giving the intuitive description of the works of Feige-Singh[1] and Asadpour et.al\ [2].\newline [1] improves the previous $O(\log n)$ approximation algorithm, by improving the constant from 0.84 to 0.66 and modifying the work of Kaplan et. al\ [3] and also shows an efficient reduction from ATSPP to ATSP. Combining both the results, they finally establish an approximation ratio of $\left(\frac{4}{3}+\epsilon \right)\log n$ for ATSPP,\ considering a small $\epsilon>0$,\ improving the work of Chekuri and Pal.[4]\newline Asadpour et.al, in their seminal work\ [2], gives an $O\left(\frac{\log n}{\log \log n}\right)$ randomized algorithm for the ATSP, by symmetrizing and modifying the solution of the Held-Karp relaxation problem and then proving an exponential family distribution for probabilistically constructing a maximum entropy spanning tree from a spanning tree polytope and then finally defining the thin-ness property and transforming a thin spanning tree into an Eulerian walk.\ The optimization methods used in\ [2] are quite elegant and the approximation ratio could further be improved, by manipulating the thin-ness of the cuts.Comment: 12 page

Balancing Minimum Spanning and Shortest Path Trees

This paper give a simple linear-time algorithm that, given a weighted digraph, finds a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous trade-off: given the two trees and epsilon > 0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+epsilon times the shortest-path distance, and yet the total weight of the tree is at most 1+2/epsilon times the weight of a minimum spanning tree. This is the best tradeoff possible. The paper also describes a fast parallel implementation.Comment: conference version: ACM-SIAM Symposium on Discrete Algorithms (1993