Machine learning addresses the question of how to build computers that improve automatically through experience. It is one of today’s most rapidly growing technical fields, lying at the intersection of computer science and statistics, and at the core of artificial intelligence and data science. Recent progress in machine learning has been driven both by the development of new learning algorithms and theory and by the ongoing explosion in the availability of online data and low-cost computation. The adoption of data-intensive machine-learning methods can be found throughout science, technology and commerce, leading to more evidence-based decision-making across many walks of life, including health care, manufacturing, education, financial modeling, policing, and marketing.

Machine learning: Trends, perspectives, and prospects

Computational geometry

2005년 대한 생화학·분자생물학회 하계학술대회를 다녀와서

The ability to analyze multiple single-cell parameters is critical for understanding cellular heterogeneity. Despite recent advances in measurement technology, methods for analyzing high-dimensional single-cell data are often subjective, labor intensive and require prior knowledge of the biological system. To objectively uncover cellular heterogeneity from single-cell measurements, we present a versatile computational approach, spanning-tree progression analysis of density-normalized events (SPADE). We applied SPADE to flow cytometry data of mouse bone marrow and to mass cytometry data of human bone marrow. In both cases, SPADE organized cells in a hierarchy of related phenotypes that partially recapitulated well-described patterns of hematopoiesis. We demonstrate that SPADE is robust to measurement noise and to the choice of cellular markers. SPADE facilitates the analysis of cellular heterogeneity, the identification of cell types and comparison of functional markers in response to perturbations.

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Extracting a cellular hierarchy from high-dimensional cytometry data with SPADE

In this paper, we present a minimum spanning tree (MST) based topology control algorithm, called local minimum spanning tree (LMST), for wireless multi-hop networks. In this algorithm, each node builds its local minimum spanning tree independently and only keeps on-tree nodes that are one-hop away as its neighbors in the final topology. We analytically prove several important properties of LMST: (1) the topology derived under LMST preserves the network connectivity; (2) the node degree of any node in the resulting topology is bounded by 6; and (3) the topology can be transformed into one with bidirectional links (without impairing the network connectivity) after removal of all uni-directional links. These results are corroborated in the simulation study.

/pdf/design-and-analysis-of-an-mst-based-topology-control-4n6elov2l3.pdf

Design and analysis of an MST-based topology control algorithm

We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decision-tree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning tree of a graph with n vertices and m edges that runs in time O(T*(m,n)) where T* is the minimum number of edge-weight comparisons needed to determine the solution. The algorithm is quite simple and can be implemented on a pointer machine.Although our time bound is optimal, the exact function describing it is not known at present. The current best bounds known for T* are T*(m,n) = Ω(m) and T*(m,n) = O(m ∙ α(m,n)), where α is a certain natural inverse of Ackermann's function.Even under the assumption that T* is superlinear, we show that if the input graph is selected from Gn,m, our algorithm runs in linear time with high probability, regardless of n, m, or the permutation of edge weights. The analysis uses a new martingale for Gn,m similar to the edge-exposure martingale for Gn,p.

/pdf/an-optimal-minimum-spanning-tree-algorithm-164wevnj5w.pdf

An optimal minimum spanning tree algorithm

The maximum cardinality and maximum weight matching problems can be solved in O(m√n) time, a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article, we demonstrate that this “m√n barrier” can be bypassed by approximation. For any e > 0, we give an algorithm that computes a (1 − e)-approximate maximum weight matching in O(me−1 log e−1) time, that is, optimal linear time for any fixed e. Our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error.

/pdf/linear-time-approximation-for-maximum-weight-matching-3apthpttr1.pdf

Linear-Time Approximation for Maximum Weight Matching

Symmetry-breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open. In this article we work in the LOCAL model (where the input graph and underlying distributed network are identical) and study the randomized complexity of four fundamental symmetry-breaking problems on graphs: computing MISs (maximal independent sets), maximal matchings, vertex colorings, and ruling sets. A small sample of our results includes the following: —An MIS algorithm running in O(log2Δ + 2o(√log log n)) time, where Δ is the maximum degree. This is the first MIS algorithm to improve on the 1986 algorithms of Luby and Alon, Babai, and Itai, when log n L Δ L 2√log n, and comes close to the Ω(log Δ / log log Δ lower bound of Kuhn, Moscibroda, and Wattenhofer. —A maximal matching algorithm running in O(log Δ + log 4log n) time. This is the first significant improvement to the 1986 algorithm of Israeli and Itai. Moreover, its dependence on Δ is nearly optimal. —A (Δ + 1)-coloring algorithm requiring O(log Δ + 2o(√log log n) time, improving on an O(log Δ + √log n)-time algorithm of Schneider and Wattenhofer. —A method for reducing symmetry-breaking problems in low arboricity/degeneracy graphs to low-degree graphs. (Roughly speaking, the arboricity or degeneracy of a graph bounds the density of any subgraph.) Corollaries of this reduction include an O(√log n)-time maximal matching algorithm for graphs with arboricity up to 2√log n and an O(log 2/3n)-time MIS algorithm for graphs with arboricity up to 2(log n)1/3. Each of our algorithms is based on a simple but powerful technique for reducing a randomized symmetry-breaking task to a corresponding deterministic one on a poly(log n)-size graph.

https://www.cs.bgu.ac.il/~elkinm/symmetry_pettie.pdf

The Locality of Distributed Symmetry Breaking

We present a new all-pairs shortest path algorithm that works with real-weighted graphs in the traditional comparison-addition model. It runs in O(mn + n2 log log n) time, improving on the long-standing bound of O(mn + n2 log n) derived from an implementation of Dijkstra's algorithm with Fibonacci heaps. Here m and n are the number of edges and vertices, respectively.Our algorithm is rooted in the so-called component hierarchy approach to shortest paths invented by Thorup for integer-weighted undirected graphs, and generalized by Hagerup to integer-weighted directed graphs. The technical contributions of this paper include a method for approximating shortest path distances and a method for leveraging approximate distances in the computation of exact ones. We also provide a simple, one line characterization of the class of hierarchy-type shortest path algorithms. This characterization leads to some pessimistic lower bounds on computing single-source shortest paths with a hierarchy-type algorithm.

A new approach to all-pairs shortest paths on real-weighted graphs

An (α, β)-spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k−1, 0)-spanner of size O(n1+1/k) and an (additive) (1,2)-spanner of size O(n3/2). However no other additive spanners are known to exist.In this article we develop a couple of new techniques for constructing (α, β)-spanners. Our first result is an additive (1,6)-spanner of size O(n4/3). The construction algorithm can be understood as an economical agent that assigns costs and values to paths in the graph, purchasing affordable paths and ignoring expensive ones, which are intuitively well approximated by paths already purchased. We show that this path buying algorithm can be parameterized in different ways to yield other sparseness-distortion tradeoffs. Our second result addresses the problem of which (α, β)-spanners can be computed efficiently, ideally in linear time. We show that, for any k, a (k,k−1)-spanner with size O(kn1+1/k) can be found in linear time, and, further, that in a distributed network the algorithm terminates in a constant number of rounds. Previous spanner constructions with similar performance had roughly twice the multiplicative distortion.

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Seth Pettie

Papers

An optimal minimum spanning tree algorithm

Linear-Time Approximation for Maximum Weight Matching

The Locality of Distributed Symmetry Breaking

A new approach to all-pairs shortest paths on real-weighted graphs

Additive spanners and (α, β)-spanners