An (O)over-tilde(mn) Gomory-Hu Tree Construction Algorithm for Unweighted Graphs

We present a fast algorithm for computing a Gomory-Hu tree or cut tree for an unweighted undirected graph G = (V, E). The expected running time of our algorithm is (O) over tilde (mc) where vertical bar E vertical bar = m and c is the maximum u-v edge connectivity, where u, v is an element of V. When the input graph is also simple (i.e., it has no parallel edges), then the u-v edge connectivity for each pair of vertices u and v is at most n - 1; so the expected run-ning time of our algorithm for simple unweighted graphs is (O) over tilde (mn). All the algorithms currently known for constructing a Gomory-Hu tree [8, 9] use n - 1 minimum s-t cut (i.e., max flow) subroutines. This in conjunction with the current fastest (O) over tilde (n(20/9)) max flow algorithm due to Karger and Levine[11] yields the current best running time of (O) over tilde (n(20/9)n) for Gomory-Hu tree construction on simple unweighted graphs with m edges and n vertices. Thus we present the first (O) over tilde (mn) algorithm for constructing a Gomory-Hu tree for simple unweighted graphs. We do not use a max flow subroutine here; we present an efficient tree packing algorithm for computing Steiner edge connectivity and use this algorithm as our main subroutine. The advantage in using a tree packing algorithm for constructing a Gomory-Hu tree is that the work done in computing a minimum Steiner cut for a Steiner set S subset of V can be reused for computing a minimum Steiner cut for certain Steiner sets S' subset of S.

Guest-Assisted Self-assembly of Organostannoxane Nanotubules on a Mica Surface

The reaction of n-BuSn(O)OH](n), and 9-hydroxy-9-fluorenecarboxylic acid in the presence of p-X-C6H4-OH (X = F, Br) afforded hydroxyl-rich hexameric organostannoxane prismanes. The crystal structures of these prismanes reveal guest-assisted supramolecular structures. Self-assembly of these compounds on a mica surface affords organooxotin nanotubules.

Crystallization of SrCO3 on a self-assembled monolayer substrate: an in-situ synchrotron X-ray study

Self-assembled monolayers (SAMs) of alkanethiols on gold surfaces show great promise in controlling the nucleation and growth of inorganic minerals from solution. In doing so, they mimic the role of some biogenic macromolecules in natural biomineralisation processes. Crystallization on SAM surfaces is usually monitored ex-situ; by allowing the process to commence and to evolve for some time, removing the substrate from the mother solution, and then examining it using microscopy, diffraction etc. We present here for the first time, the use of high energy monochromatic synchrotron X-radiation in conjunction with a two dimensional detector to monitor in situ, in a time resolved fashion, the growth of SrCO3 (strontianite) crystals on a SAM substrate.

Fast edge splitting and Edmonds' arborescence construction for unweighted graphs

Given an unweighted undirected or directed graph with n vertices, m edges and edge connectivity c, we present a new deterministic algorithm for edge splitting. Our algorithm splits-off any specified subset S of vertices satisfying standard conditions (even degree for the undirected case and in-degree ≥ out-degree for the directed case) while maintaining connectivity c for vertices outside S in Õ(m+nc2) time for an undirected graph and Õ(mc) time for a directed graph. This improves the current best deterministic time bounds due to Gabow [8], who splits-off a single vertex in Õ(nc2+m) time for an undirected graph and Õ(mc) time for a directed graph. Further, for appropriate ranges of n, c, |S| it improves the current best randomized bounds due to Benczúr and Karger [2], who split-off a single vertex in an undirected graph in Õ(n2) Monte Carlo time. We give two applications of our edge splitting algorithms. Our first application is a sub-quadratic (in n) algorithm to construct Edmonds' arborescences. A classical result of Edmonds [5] shows that an unweighted directed graph with c edge-disjoint paths from any particular vertex r to every other vertex has exactly c edge-disjoint arborescences rooted at r. For a c edge connected unweighted undirected graph, the same theorem holds on the digraph obtained by replacing each undirected edge by two directed edges, one in each direction. The current fastest construction of these arborescences by Gabow [7] takes Õ(n2c2) time. Our algorithm takes Õ(nc3+m) time for the undirected case and Õ(nc4+mc) time for the directed case. The second application of our splitting algorithm is a new Steiner edge connectivity algorithm for undirected graphs which matches the best known bound of Õ(nc2 + m) time due to Bhalgat et al [3]. Finally, our algorithm can also be viewed as an alternative proof for existential edge splitting theorems due to Lovász [9] and Mader [11].

An O(mn) Gomory-Hu Tree Construction Algorithm for Unweighted Graphs

We present a fast algorithm for computing a Gomory-Hu tree or cut tree for an unweighted undirected graph G = (V,E). The expected running time of our algorithm is Õ(mc) where |E| = m and c is the maximum u-vedge connectivity, where u,v ∈ V. When the input graph is also simple (i.e., it has no parallel edges), then the u-v edge connectivity for each pair of vertices u and v is at most n-1; so the expected running time of our algorithm for simple unweighted graphs is Õ(mn).All the algorithms currently known for constructing a Gomory-Hu tree [8,9] use n-1 minimum s-t cut (i.e., max flow) subroutines. This in conjunction with the current fastest Õ(n20/9) max flow algorithm due to Karger and Levine [11] yields the current best running time of Õ(n20/9n) for Gomory-Hu tree construction on simpleunweighted graphs with m edges and n vertices. Thus we present the first Õ(mn) algorithm for constructing a Gomory-Hu tree for simple unweighted graphs.We do not use a max flow subroutine here; we present an efficient tree packing algorithm for computing Steiner edge connectivity and use this algorithm as our main subroutine. The advantage in using a tree packing algorithm for constructing a Gomory-Hu tree is that the work done in computing a minimum Steiner cut for a Steiner set S ⊆ V can be reused for computing a minimum Steiner cut for certain Steiner sets S' ⊆ S.

A provably tight delay-driven concurrently congestion mitigating global routing algorithm

Routing is a very important step in VLSI physical design. A set of nets are routed under delay and resource constraints in multi-net global routing. In this paper a delay-driven congestion-aware global routing algorithm is developed, which is a heuristic based method to solve a multi-objective NP-hard optimization problem. The proposed delay-driven Steiner tree construction method is of O(n(2) log n) complexity, where n is the number of terminal points and it provides n-approximation solution of the critical time minimization problem for a certain class of grid graphs. The existing timing-driven method (Hu and Sapatnekar, 2002) has a complexity O(n(4)) and is implemented on nets with small number of sinks. Next we propose a FPTAS Gradient algorithm for minimizing the total overflow. This is a concurrent approach considering all the nets simultaneously contrary to the existing approaches of sequential rip-up and reroute. The algorithms are implemented on ISPD98 derived benchmarks and the drastic reduction of overflow is observed. (C) 2014 Elsevier Inc. All rights reserved.