Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Many practical computing problems concern large graphs. Standard examples include the Web graph and various social networks. The scale of these graphs - in some cases billions of vertices, trillions of edges - poses challenges to their efficient processing. In this paper we present a computational model suitable for this task. Programs are expressed as a sequence of iterations, in each of which a vertex can receive messages sent in the previous iteration, send messages to other vertices, and modify its own state and that of its outgoing edges or mutate graph topology. This vertex-centric approach is flexible enough to express a broad set of algorithms. The model has been designed for efficient, scalable and fault-tolerant implementation on clusters of thousands of commodity computers, and its implied synchronicity makes reasoning about programs easier. Distribution-related details are hidden behind an abstract API. The result is a framework for processing large graphs that is expressive and easy to program.

/pdf/pregel-a-system-for-large-scale-graph-processing-30lgjenxxg.pdf

Pregel: a system for large-scale graph processing

where A represents the magnetic vector potential, is an integral of the hydromagnetic equations. This -integral made it possible to formulate a variational principle for the force-free magnetic fields. The integral expresses the fact that motions cannot transform a given field in an entirely arbitrary different field, if the conductivity of the medium isconsidered infinite. In this paper we shall show that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent. These integrals, as we shall presently verify, are I2 =fbHvdV, (2)

Proceedings of the National Academy of Sciences

Computational geometry

We study the average-case running-time of single-source shortest-path (SSSP) algorithms assuming arbitrary directed graphs with n nodes, m edges, and independent random edge weights uniformly distributed in [0,1]. We give the first label-setting and label-correcting algorithms that run in linear time O(n + m) on the average. In fact, the result for the label-setting version is even obtained for dependent edge weights. In case of independence, however, the linear-time bound holds with high probability, too.Furthermore, we propose a general method to construct graphs with random edge weights that cause large expected running times when input to many traditional SSSP algorithms. We use our method to prove lower bounds on the average-case complexity of the following algorithms: the "Bellman-Ford algorithm" [R. Bellman, Quart. Appl. Math. 16 (1958) 87-90, L.R. Ford, D.R. Fulkerson, 1963], "Pallottino's Incremental Graph algorithm" [S. Pallottino, Networks 14 (1984) 257-267], the "Threshold approach" [F. Glover, R. Glover, D. Klingman, Networks 14 (1984) 23-37, F. Glover, D. Klingman, N. Phillips, Oper. Res. 33 (1985) 65-73, F. Glover, D. Klingman, N. Phillips, R.F. Schneider, Management Sci. 31 (1985) 1106-1128], the "Topological Ordering SSSP algorithm" [A.V. Goldberg, T. Radzik, Appl. Math. Lett. 6 (1993) 3-6], the "Approximate Bucket implementation" of Dijkstra's algorithm [B.V. Cherkassky, A.V. Goldberg, T. Radzik, Math. Programming 73 (1996) 129-174], and the "Δ-Stepping algorithm" [U. Meyer, P. Sanders, 1998].

Average-case complexity of single-source shortest-paths algorithms: lower and upper bounds

We present improved cache-oblivious data structures and algorithms for breadth-first search and the single-source shortest path problem on undirected graphs with non-negative edge weights. Our results removes the performance gap between the currently best cache-aware algorithms for these problems and their cache-oblivious counterparts. Our shortest-path algorithm relies on a new data structure, called bucket heap, which is the first cache-oblivious priority queue to efficiently support a weak DecreaseKey operation.

http://www.cs.au.dk/~gerth/papers/brics-rs-04-2.pdf

Cache-Oblivious Data Structures and Algorithms for Undirected Breadth-First Search and Shortest Paths

We present an I/O-efficient algorithm for the single-source shortest path problem on undirected graphs G=(V,E). Our algorithm performs $\mathcal{O}$($\sqrt{(VE/B){\rm log_2}(W/w)}$sort(V + E)loglog(VB/E)) I/Os, where w ∈ ℝ + and W ∈ ℝ + are the minimal and maximal edge weights in G, respectively. For uniform random edge weights in (0,1], the expected I/O-complexity of our algorithm is $\mathcal{O}$($\sqrt{VE/B} + ((V + E)/B){log_2}B +$sort(V+E)).

http://web.cs.dal.ca/~nzeh/Publications/sssp_esa03.pdf

I/O-Efficient Undirected Shortest Paths

Breadth first search (BFS) traversal on massive graphs in external memory was considered non-viable until recently, because of the large number of I/Os it incurs. Ajwani et al. [3] showed that the randomized variant of the o(n) I/O algorithm of Mehlhorn and Meyer [24] (MM_BFS) can compute the BFS level decomposition for large graphs (around a billion edges) in a few hours for small diameter graphs and a few days for large diameter graphs. We improve upon their implementation of this algorithm by reducing the overhead associated with each BFS level, thereby improving the results for large diameter graphs which are more difficult for BFS traversal in external memory. Also, we present the implementation of the deterministic variant of MM_BFS and show that in most cases, it outperforms the randomized variant. The running time for BFS traversal is further improved with a heuristic that preserves the worst case guarantees of MM_BFS. Together, they reduce the time for BFS on large diameter graphs from days shown in [3] to hours. In particular, on line graphs with random layout on disks, our implementation of the deterministic variant of MM_BFS with the proposed heuristic is more than 75 times faster than the previous best result for the randomized variant of MM_BFS in [3].

/pdf/improved-external-memory-bfs-implementations-1yuy8m43eb.pdf

Improved external memory BFS implementations

We show that the well-known random incremental constructlon of Clarkson and Shor [14] can be adapted via gradations to provide efficient external-memory algorithms for some geomctric problems. In particular, as the main result, we obtain an optimal randomized algorithm for the problem of computing the trapezoidal decomposition determined by a set of N line scgmcnts in the plane with K pairwise intersections, that requires G($$ logMjB Q + 5) expected disk accesses (I/OS), where M is the size of the available internal memory and B is the size of the block transfer. The approach is sufficiently general to obtain algorithms for the problems of 2-d and 3-d convex hulls, 2-d abstract Voronoi diagrams and batched point location in a planar subdivision, which require an optimal expected number of I/OS and are olmplcr than the ones previously known. The results extend to a external-memory model with multiple disks.

/pdf/randomized-external-memory-algorithms-for-some-geometric-3gxwqulym4.pdf

Ulrich Meyer

Papers

Average-case complexity of single-source shortest-paths algorithms: lower and upper bounds

Cache-Oblivious Data Structures and Algorithms for Undirected Breadth-First Search and Shortest Paths

I/O-Efficient Undirected Shortest Paths

Improved external memory BFS implementations

Randomized external-memory algorithms for some geometric problems