533 research outputs found
Time-Optimal Tree Computations on Sparse Meshes
The main goal of this work is to fathom the suitability of the mesh with multiple broadcasting architecture (MMB) for some tree-related computations. We view our contribution at two levels: on the one hand, we exhibit time lower bounds for a number of tree-related problems on the MMB. On the other hand, we show that these lower bounds are tight by exhibiting time-optimal tree algorithms on the MMB. Specifically, we show that the task of encoding and/or decoding n-node binary and ordered trees cannot be solved faster than Ω(log n) time even if the MMB has an infinite number of processors. We then go on to show that this lower bound is tight. We also show that the task of reconstructing n-node binary trees and ordered trees from their traversais can be performed in O(1) time on the same architecture. Our algorithms rely on novel time-optimal algorithms on sequences of parentheses that we also develop
Fast Parallel Algorithms for Basic Problems
Parallel processing is one of the most active research areas these days. We are interested in one aspect of parallel processing, i.e. the design and analysis of parallel algorithms. Here, we focus on non-numerical parallel algorithms for basic combinatorial problems, such as data structures, selection, searching, merging and sorting. The purposes of studying these types of problems are to obtain basic building blocks which will be useful in solving complex problems, and to develop fundamental algorithmic techniques.
In this thesis, we study the following problems: priority queues, multiple search and multiple selection, and reconstruction of a binary tree from its traversals. The research on priority queue was motivated by its various applications. The purpose of studying multiple search and multiple selection is to explore the relationships between four of the most fundamental problems in algorithm design, that is, selection, searching, merging and sorting; while our parallel solutions can be used as subroutines in algorithms for other problems. The research on the last problem, reconstruction of a binary tree from its traversals, was stimulated by a challenge proposed in a recent paper by Berkman et al. ( Highly Parallelizable Problems, STOC 89) to design doubly logarithmic time optimal parallel algorithms because a remarkably small number of such parallel algorithms exist
Time-Optimal Algorithms on Meshes With Multiple Broadcasting
The mesh-connected computer architecture has emerged as a natural choice for solving a large number of computational tasks in image processing, computational geometry, and computer vision. However, due to its large communication diameter, the mesh tends to be slow when it comes to handling data transfer operations over long distances. In an attempt to overcome this problem, mesh-connected computers have recently been augmented by the addition of various types of bus systems. One such system known as the mesh with multiple broadcasting involves enhancing the mesh architecture by the addition of row and column buses. The mesh with multiple broadcasting has proven to be feasible to implement in VLSI, and is used in the DAP family of computers. In recent years, efficient algorithms to solve a number of computational problems on meshes with multiple broadcasting have been proposed in the literature.
The problems considered in this thesis are semigroup computations, sorting, multiple search, various convexity-related problems, and some tree problems. Based on the size of the input data for the problem under consideration, existing results can be broadly classified into sparse and dense. Specifically, for a given √n x √n mesh with multiple broadcasting, we refer to problems involving ) items as sparse, while the case £ O(n) will be referred to as dense. Finally, the case corresponding to 2 ≤ m ≤ n is be termed general. The motivation behind the current work is twofold. First, time-optimal solutions are proposed for the problems listed above. Secondly, an attempt is made to remove the artificial limitation of problems studied to sparse and dense cases.
To establish the time-optimality of the algorithms presented in this work, we use some existing lower bound techniques along with new ones that we develop. We solve the semigroup computation problem for the general case and present a novel lower bound argument. We solve the multiple search problem in the general case and present some surprising applications to computational geometry. In the case of sorting, the general case is defined to be slightly different. For the specified range of the size of input, we present a time and VLSI-optimal algorithm. We also present time lower bound results and matching algorithms for a number of convexity related and tree problems in the sparse case
Efficient pebbling for list traversal synopses
We show how to support efficient back traversal in a unidirectional list,
using small memory and with essentially no slowdown in forward steps. Using
memory for a list of size , the 'th back-step from the
farthest point reached so far takes time in the worst case, while
the overhead per forward step is at most for arbitrary small
constant . An arbitrary sequence of forward and back steps is
allowed. A full trade-off between memory usage and time per back-step is
presented: vs. and vice versa. Our algorithms are based on a
novel pebbling technique which moves pebbles on a virtual binary, or -ary,
tree that can only be traversed in a pre-order fashion. The compact data
structures used by the pebbling algorithms, called list traversal synopses,
extend to general directed graphs, and have other interesting applications,
including memory efficient hash-chain implementation. Perhaps the most
surprising application is in showing that for any program, arbitrary rollback
steps can be efficiently supported with small overhead in memory, and marginal
overhead in its ordinary execution. More concretely: Let be a program that
runs for at most steps, using memory of size . Then, at the cost of
recording the input used by the program, and increasing the memory by a factor
of to , the program can be extended to support an
arbitrary sequence of forward execution and rollback steps: the 'th rollback
step takes time in the worst case, while forward steps take O(1)
time in the worst case, and amortized time per step.Comment: 27 page
Inference of Many-Taxon Phylogenies
Phylogenetic trees are tree topologies that represent the evolutionary history of a set of organisms. In this thesis, we address computational challenges related to the analysis of large-scale datasets with Maximum Likelihood based phylogenetic inference. We have approached this using different strategies: reduction of memory requirements, reduction of running time, and reduction of man-hours
Leader Election for Anonymous Asynchronous Agents in Arbitrary Networks
We study the problem of leader election among mobile agents operating in an
arbitrary network modeled as an undirected graph. Nodes of the network are
unlabeled and all agents are identical. Hence the only way to elect a leader
among agents is by exploiting asymmetries in their initial positions in the
graph. Agents do not know the graph or their positions in it, hence they must
gain this knowledge by navigating in the graph and share it with other agents
to accomplish leader election. This can be done using meetings of agents, which
is difficult because of their asynchronous nature: an adversary has total
control over the speed of agents. When can a leader be elected in this
adversarial scenario and how to do it? We give a complete answer to this
question by characterizing all initial configurations for which leader election
is possible and by constructing an algorithm that accomplishes leader election
for all configurations for which this can be done
Learning mutational graphs of individual tumour evolution from single-cell and multi-region sequencing data
Background. A large number of algorithms is being developed to reconstruct
evolutionary models of individual tumours from genome sequencing data. Most
methods can analyze multiple samples collected either through bulk multi-region
sequencing experiments or the sequencing of individual cancer cells. However,
rarely the same method can support both data types.
Results. We introduce TRaIT, a computational framework to infer mutational
graphs that model the accumulation of multiple types of somatic alterations
driving tumour evolution. Compared to other tools, TRaIT supports multi-region
and single-cell sequencing data within the same statistical framework, and
delivers expressive models that capture many complex evolutionary phenomena.
TRaIT improves accuracy, robustness to data-specific errors and computational
complexity compared to competing methods.
Conclusions. We show that the application of TRaIT to single-cell and
multi-region cancer datasets can produce accurate and reliable models of
single-tumour evolution, quantify the extent of intra-tumour heterogeneity and
generate new testable experimental hypotheses
- …