04.03.2013

A new technique for solving ‘graph Laplacians’ is drastically simpler than its predecessors, with implications for a huge range of practical problems.

In the last decade, theoretical computer science has seen remarkable progress on the problem of solving graph Laplacians — the esoteric name for a calculation with hordes of familiar applications in scheduling, image processing, online product recommendation, network analysis, and scientific computing, to name just a few.

At this year’s ACM Symposium on the Theory of Computing, MIT researchers will present a new algorithm for solving graph Laplacians that is not only faster than its predecessors, but also drastically simpler. “The 2004 paper required fundamental innovations in multiple branches of mathematics and computer science, but it ended up being split into three papers that I think were 130 pages in aggregate,” says Jonathan Kelner, an associate professor of applied mathematics at MIT who led the new research. “We were able to replace it with something that would fit on a blackboard.”

The MIT researchers — Kelner; Lorenzo Orecchia, an instructor in applied mathematics; and Kelner’s students Aaron Sidford and Zeyuan Zhu — believe that the simplicity of their algorithm should make it both faster and easier to implement in software than its predecessors. But just as important is the simplicity of their conceptual analysis, which, they argue, should make their result much easier to generalize to other contexts.

... more about:

»Carnegie »Problem Solving »algorithm »computer science »conceptual analysis »familiar applications »graph Laplacian »image processing

»Carnegie »Problem Solving »algorithm »computer science »conceptual analysis »familiar applications »graph Laplacian »image processing

Overcoming resistance

A graph Laplacian is a matrix — a big grid of numbers — that describes a graph, a mathematical abstraction common in computer science. A graph is any collection of nodes, usually depicted as circles, and edges, depicted as lines that connect the nodes. In a logistics problem, the nodes might represent tasks to be performed, while in an online recommendation engine, they might represent titles of movies.

In many graphs, the edges are “weighted,” meaning that they have different numbers associated with them. Those numbers could represent the cost — in time, money or energy — of moving from one step to another in a complex logistical operation, or they could represent the strength of the correlations between the movie preferences of customers of an online video service.

The Laplacian of a graph describes the weights between all the edges, but it can also be interpreted as a series of linear equations. Solving those equations is crucial to many techniques for analyzing graphs.

One intuitive way to think about graph Laplacians is to imagine the graph as a big electrical circuit and the edges as resistors. The weights of the edges describe the resistance of the resistors; solving the Laplacian tells you how much current would flow between any two points in the graph.

Earlier approaches to solving graph Laplacians considered a series of ever-simpler approximations of the graph of interest. Solving the simplest provided a good approximation of the next simplest, which provided a good approximation of the next simplest, and so on. But the rules for constructing the sequence of graphs could get very complex, and proving that the solution of the simplest was a good approximation of the most complex required considerable mathematical ingenuity.

Looping back

The MIT researchers’ approach is much more straightforward. The first thing they do is find a “spanning tree” for the graph. A tree is a particular kind of graph that has no closed loops. A family tree is a familiar example; there, a loop might mean that someone was both parent and sibling to the same person. A spanning tree of a graph is a tree that touches all of the graph’s nodes but dispenses with the edges that create loops. Efficient algorithms for constructing spanning trees are well established.

The spanning tree in hand, the MIT algorithm then adds back just one of the missing edges, creating a loop. A loop means that two nodes are connected by two different paths; on the circuit analogy, the voltage would have to be the same across both paths. So the algorithm sticks in values for current flow that balance the loop. Then it adds back another missing edge and rebalances.

In even a simple graph, values that balance one loop could imbalance another one. But the MIT researchers showed that, remarkably, this simple, repetitive process of adding edges and rebalancing will converge on the solution of the graph Laplacian. Nor did the demonstration of that convergence require sophisticated mathematics: “Once you find the right way of thinking about the problem, everything just falls into place,” Kelner explains.

Paradigm shift

Daniel Spielman, a professor of applied mathematics and computer science at Yale University, was Kelner’s thesis advisor and one of two co-authors of the 2004 paper. According to Spielman, his algorithm solved Laplacians in nearly linear time “on problems of astronomical size that you will never ever encounter unless it’s a much bigger universe than we know. Jon and colleagues’ algorithm is actually a practical one.”

Spielman points out that in 2010, researchers at Carnegie Mellon University also presented a practical algorithm for solving Laplacians. Theoretical analysis shows that the MIT algorithm should be somewhat faster, but “the strange reality of all these things is, you do a lot of analysis to make sure that everything works, but you sometimes get unusually lucky, or unusually unlucky, when you implement them. So we’ll have to wait to see which really is the case.”

The real value of the MIT paper, Spielman says, is in its innovative theoretical approach. “My work and the work of the folks at Carnegie Mellon, we’re solving a problem in numeric linear algebra using techniques from the field of numerical linear algebra,” he says. “Jon’s paper is completely ignoring all of those techniques and really solving this problem using ideas from data structures and algorithm design. It’s substituting one whole set of ideas for another set of ideas, and I think that’s going to be a bit of a game-changer for the field. Because people will see there’s this set of ideas out there that might have application no one had ever imagined.”

Sarah McDonnell | EurekAlert!

Further information:

http://www.mit.edu

http://web.mit.edu/newsoffice/2013/short-algorithm-long-range-consequences-0301.html

**Further reports about:**
> Carnegie
> Problem Solving
> algorithm
> computer science
> conceptual analysis
> familiar applications
> graph Laplacian
> image processing

Five developments for improved data exploitation

19.04.2017 | Deutsches Forschungszentrum für Künstliche Intelligenz GmbH, DFKI

Smart Manual Workstations Deliver More Flexible Production

04.04.2017 | Deutsches Forschungszentrum für Künstliche Intelligenz GmbH, DFKI

The nearby, giant radio galaxy M87 hosts a supermassive black hole (BH) and is well-known for its bright jet dominating the spectrum over ten orders of magnitude in frequency. Due to its proximity, jet prominence, and the large black hole mass, M87 is the best laboratory for investigating the formation, acceleration, and collimation of relativistic jets. A research team led by Silke Britzen from the Max Planck Institute for Radio Astronomy in Bonn, Germany, has found strong indication for turbulent processes connecting the accretion disk and the jet of that galaxy providing insights into the longstanding problem of the origin of astrophysical jets.

Supermassive black holes form some of the most enigmatic phenomena in astrophysics. Their enormous energy output is supposed to be generated by the...

The probability to find a certain number of photons inside a laser pulse usually corresponds to a classical distribution of independent events, the so-called...

Microprocessors based on atomically thin materials hold the promise of the evolution of traditional processors as well as new applications in the field of flexible electronics. Now, a TU Wien research team led by Thomas Müller has made a breakthrough in this field as part of an ongoing research project.

Two-dimensional materials, or 2D materials for short, are extremely versatile, although – or often more precisely because – they are made up of just one or a...

Two researchers at Heidelberg University have developed a model system that enables a better understanding of the processes in a quantum-physical experiment...

Glaciers might seem rather inhospitable environments. However, they are home to a diverse and vibrant microbial community. It’s becoming increasingly clear that they play a bigger role in the carbon cycle than previously thought.

A new study, now published in the journal Nature Geoscience, shows how microbial communities in melting glaciers contribute to the Earth’s carbon cycle, a...

Anzeige

Anzeige

Event News

Expert meeting “Health Business Connect” will connect international medical technology companies

20.04.2017 | Event News

Wenn der Computer das Gehirn austrickst

18.04.2017 | Event News

7th International Conference on Crystalline Silicon Photovoltaics in Freiburg on April 3-5, 2017

03.04.2017 | Event News

Latest News

New quantum liquid crystals may play role in future of computers

21.04.2017 | Physics and Astronomy

A promising target for kidney fibrosis

21.04.2017 | Health and Medicine

Light rays from a supernova bent by the curvature of space-time around a galaxy

21.04.2017 | Physics and Astronomy

VideoLinks

NASA | A Year in the Life of Earth's CO2

NASA Computer Model Provides a New Portrait of Carbon Dioxide

Black Holes Come to the Big Screen

The new movie "Interstellar" explores a longstanding fascination, but UA astrophysicists are using cutting-edge technology to go one better.

NASA's Swift Mission Observes Mega Flares from a Mini Star

NASA's Swift satellite detected the strongest, hottest, and longest-lasting sequence of stellar flares ever seen from a nearby red dwarf star.

NASA | Global Hawks Soar into Storms

NASA's airborne Hurricane and Severe Storm Sentinel or HS3 mission, will revisit the Atlantic Ocean for the third year in a row.

Baffin Island - Disappearing ice caps

Giff Miller, geologist and paleoclima-tologist, is walking the margins of melting glaciers on Baffin Island, Nunavut, Canada.

The Infrasound Network and how it works

The CTBTO uses infrasound stations to monitor the Earth mainly for atmospheric explosions.

B2B-VideoLinks

Efficient reduction of odour and grease with Heraeus UV solutions

Kitchen exhaust air cleaning with UV in gastronomy

Drying and curing of paints on glass and ceramics

Bright and brilliant paints on glass and ceramics require safe solutions for drying and curing.

JULABO World of Temperature

Explore the World of Temperature with JULABO - Superior Temperature Technology for a Better Life.

Acoustic Wave Separation: How It Works

In this animated video, see how Acoustic Wave Separation technology works in full detail.

Infrared Heat for printed electronics

Drying and sintering of printed electronics by specialty light sources from Heraeus

All about Data Logger, how to use

Wolfgang Rudolph explains: all information worth knowing about the data logger and the practical test by means of a drone