Finding symmetries is a way to highlight shortcuts to answers that, for example, verify the safety of train schedules, identify bugs in software and hardware designs, or speed up common search tasks.
The algorithm is an update to software called "saucy" that the researchers developed in 2004 and shared with colleagues. Paul Darga, a graduate student in the Department of Electrical Engineering and Computer Science, will present the algorithm on June 10 at the Design Automation Conference in Anaheim, Calif. Darga's co-authors are Igor Markov, associate professor in the Department of Electrical Engineering and Computer Science, and Karem Sakallah, a professor in the same department.
The software's applications extend to artificial intelligence and logistics.It speeds up solutions to fundamental computer science problems and quickly solves what's called the graph automorphism problem. "Our new algorithm solves the graph automorphism problem so quickly in real-life applications that the problem is starting to look easy," Markov said.
Symmetries are, in a sense, interchangeable options that lead to the same outcome. In complicated equations, symmetries point to repeated branches of the search for solutions that only need to be figured out once. Current programs that look for symmetries can take days to give results even when they find no instances, Darga said. The new method finishes in seconds even when there are millions of variables.
To illustrate how finding symmetries can simplify equations, Markov pointed to the pigeonhole principle. This says you can't, for example, fit 10 birds in nine pigeonholes (unless they share.) The particular problem has a nine-fold symmetry because it doesn't matter which hole each bird occupies. One will always end up homeless. It also has a 10-fold symmetry because the birds are considered interchangeable.
"If you ask a computer to put 20 trains on 19 tracks, this computation may take forever," Markov said. "But if you use an approach with symmetry breaking, these cases can be solved in seconds."
Symmetry breaking in train scheduling and logistics can also help figure the shortest itineraries. In artificial intelligence, the ability to recognize symmetries quickly could help a computer generate a plan or an optimal schedule. The computer would know when the order of tasks was interchangeable.
The new algorithm starts working in the same way as existing symmetry breaking software. It converts the complicated equation into a graph and looks for similarities in the arrangement of the vertices. Like the original version of saucy, it narrows the search while exploiting what Darga calls "sparsity"---the fact that almost every node on the graph is only connected to a few other nodes.
The saucy update recognizes that it's not just the node connections that are sparse. It turns out that most important symmetries themselves are sparse too, in that they involve only several nodes at a time. Other symmetries can be derived from sparse symmetries, and the number of distinct symmetries can grow exponentially with the size of the system.
"Just like snowflakes, many interconnected systems in technology and nature are sparse and exhibit structural symmetries," Sakallah said. "The internet connectivity graph we worked with reminds me of a giant snowflake. It has a quarter million vertices and half a million edges, but it exhibits more symmetries than there are electrons in the universe."
In less than a half-second, the new software captured 1083,687 different symmetries in an Internet connectivity graph of routers around the world. A symmetry in this graph signifies a way the routers could be shuffled that wouldn't change the operation.
Previous methods timed out in the 30 minutes they were given to generate results in these experiments. Darga said it would take these older programs days to solve such a complicated problem. In searching for symmetries in the road networks between cities and towns in Illinois, the new algorithm captured the 104,843 symmetries in less than a half-second, whereas the most robust previous algorithm took 16 minutes.
The paper is called "Faster Symmetry Discovery Using Sparsity of Symmetries." It is available at http://www.eecs.umich.edu/~imarkov/pubs/conf/dac08-sym.pdf. Information about how to obtain the software is at http://vlsicad.eecs.umich.edu/BK/SAUCY/.For more information:
Nicole Casal Moore | alfa
UT professor develops algorithm to improve online mapping of disaster areas
29.11.2016 | University of Tennessee at Knoxville
New standard helps optical trackers follow moving objects precisely
23.11.2016 | National Institute of Standards and Technology (NIST)
A multi-institutional research collaboration has created a novel approach for fabricating three-dimensional micro-optics through the shape-defined formation of porous silicon (PSi), with broad impacts in integrated optoelectronics, imaging, and photovoltaics.
Working with colleagues at Stanford and The Dow Chemical Company, researchers at the University of Illinois at Urbana-Champaign fabricated 3-D birefringent...
In experiments with magnetic atoms conducted at extremely low temperatures, scientists have demonstrated a unique phase of matter: The atoms form a new type of quantum liquid or quantum droplet state. These so called quantum droplets may preserve their form in absence of external confinement because of quantum effects. The joint team of experimental physicists from Innsbruck and theoretical physicists from Hannover report on their findings in the journal Physical Review X.
“Our Quantum droplets are in the gas phase but they still drop like a rock,” explains experimental physicist Francesca Ferlaino when talking about the...
The Max Planck Institute for Physics (MPP) is opening up a new research field. A workshop from November 21 - 22, 2016 will mark the start of activities for an innovative axion experiment. Axions are still only purely hypothetical particles. Their detection could solve two fundamental problems in particle physics: What dark matter consists of and why it has not yet been possible to directly observe a CP violation for the strong interaction.
The “MADMAX” project is the MPP’s commitment to axion research. Axions are so far only a theoretical prediction and are difficult to detect: on the one hand,...
Broadband rotational spectroscopy unravels structural reshaping of isolated molecules in the gas phase to accommodate water
In two recent publications in the Journal of Chemical Physics and in the Journal of Physical Chemistry Letters, researchers around Melanie Schnell from the Max...
The efficiency of power electronic systems is not solely dependent on electrical efficiency but also on weight, for example, in mobile systems. When the weight of relevant components and devices in airplanes, for instance, is reduced, fuel savings can be achieved and correspondingly greenhouse gas emissions decreased. New materials and components based on gallium nitride (GaN) can help to reduce weight and increase the efficiency. With these new materials, power electronic switches can be operated at higher switching frequency, resulting in higher power density and lower material costs.
Researchers at the Fraunhofer Institute for Solar Energy Systems ISE together with partners have investigated how these materials can be used to make power...
16.11.2016 | Event News
01.11.2016 | Event News
14.10.2016 | Event News
02.12.2016 | Medical Engineering
02.12.2016 | Agricultural and Forestry Science
02.12.2016 | Physics and Astronomy