Making Sudoku puzzles less puzzling

ScienceDaily (Oct. 11, 2012) — For anyone who has ever struggled while attempting to solve a Sudoku puzzle, University of Notre Dame complex networks researcher Zoltan Toroczkai and Notre Dame postdoctoral researcher Maria Ercsey-Ravasz are coming to the rescue. They can not only explain why some Sudoku puzzles are harder than others, they have also developed a mathematical algorithm that solves Sudoku puzzles very quickly, without any guessing or backtracking.

Toroczkai and Ercsey-Ravasz, of Romania's Babes-Bolyai University, began studying Sudoku as part of their research into the theory of optimization and computational complexity. They note that most Sudoku enthusiasts use what is known as a "brute force" system to solve problems, combined with a good deal of guessing. Brute force systems essentially deploy all possible combinations of numbers in a Sudoku puzzle until the correct answer is found. While the method is successful, it is also time consuming.

Instead, Toroczkai and Ercsey-Ravasz have proposed a universal analog algorithm that is completely deterministic (no guessing or exhaustive searching) and always arrives at the correct solution to a problem, and does so much more quickly.

The researchers also discovered that the time it took to solve a problem with their analog algorithm correlated with the difficulty of the problem as rated by human solvers. This led them to develop a ranking scale for problem or puzzle difficulty. The scale runs from 1 through 4, and it matches up nicely with the "easy" through "hard" to "ultra-hard" classification currently applied to Sudoku puzzles. A puzzle with a rating of 2 takes, on average, 10 times as long to solve than one with rating of 1. According to this system, the hardest known puzzle so far has a rating of 3.6, and it is not known if there are even harder puzzles out there.

"I had not been interested in Sudoku until we started working on the much more general class of Boolean satisfiability problems," Toroczkai said. "Since Sudoku is a part of this class, it seemed like a good testbed for our solver, so I familiarized myself with it. To me, and to a number of researchers studying such problems, a fascinating question is how far can us humans go in solving Sudoku puzzles deterministically, without backtracking -- that is without making a choice at random, then seeing where that leads to and if it fails, restarting. Our analog solver is deterministic -- there are no random choices or backtracks made during the dynamics."

Toroczkai and Ercsey-Ravasz believe their analog algorithm potentially can be applied to a wide variety of problems in industry, computer science and computational biology.

The research experience has also made Toroczkai a devotee of Sudoku puzzles.

"Both my wife and I have several Sudoku apps on our iPhones, and we must have played thousands of times, racing to get the shortest completion times on all levels," he said. "She often sees combinations of patterns that I completely miss. I have to deduce them. Without paper and pencil to jot down possibilities, it becomes impossible for me to solve many of the puzzles that our solver categorizes as hard or ultra-hard."

Toroczkai and Ercsey-Ravasz's methodology was first published in the journal Nature Physics, and its application to Sudoku, appears in the Oct. 11 edition of the journal Nature Scientific Reports.

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The above story is reprinted from materials provided by University of Notre Dame. The original article was written by William G. Gilroy.

Note: Materials may be edited for content and length. For further information, please contact the source cited above.

Journal References:

Mária Ercsey-Ravasz, Zoltán Toroczkai. Optimization hardness as transient chaos in an analog approach to constraint satisfaction. Nature Physics, 2011; 7 (12): 966 DOI: 10.1038/nphys2105Mária Ercsey-Ravasz, Zoltán Toroczkai. The Chaos Within Sudoku. Scientific Reports, 2012; 2 DOI: 10.1038/srep00725

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Elusive capacity of networks: Calculating data network's total capacity notoriously difficult, but theorists making some headway

ScienceDaily (May 15, 2012) — In its early years, information theory -- which grew out of a landmark 1948 paper by MIT alumnus and future professor Claude Shannon -- was dominated by research on error-correcting codes: How do you encode information so as to guarantee its faithful transmission, even in the presence of the corrupting influences engineers call "noise"?

Recently, one of the most intriguing developments in information theory has been a different kind of coding, called network coding, in which the question is how to encode information in order to maximize the capacity of a network as a whole. For information theorists, it was natural to ask how these two types of coding might be combined: If you want to both minimize error and maximize capacity, which kind of coding do you apply where, and when do you do the decoding?

What makes that question particularly hard to answer is that no one knows how to calculate the data capacity of a network as a whole -- or even whether it can be calculated. Nonetheless, in the first half of a two-part paper, which was published recently in IEEE Transactions on Information Theory, MIT's Muriel Médard, California Institute of Technology's Michelle Effros and the late Ralf Koetter of the University of Technology in Munich show that in a wired network, network coding and error-correcting coding can be handled separately, without reduction in the network's capacity. In the forthcoming second half of the paper, the same researchers demonstrate some bounds on the capacities of wireless networks, which could help guide future research in both industry and academia.

A typical data network consists of an array of nodes -- which could be routers on the Internet, wireless base stations or even processing units on a single chip -- each of which can directly communicate with a handful of its neighbors. When a packet of data arrives at a node, the node inspects its addressing information and decides which of several pathways to send it along.

Calculated confusion

With network coding, on the other hand, a node scrambles together the packets it receives and sends the hybrid packets down multiple paths; at each subsequent node they're scrambled again in different ways. Counterintuitively, this can significantly increase the capacity of the network as a whole: Hybrid packets arrive at their destination along multiple paths. If one of those paths is congested, or if one of its links fails outright, the packets arriving via the other paths will probably contain enough information that the recipient can piece together the original message.

But each link between nodes could be noisy, so the information in the packets also needs to be encoded to correct for errors. "Suppose that I'm a node in a network, and I see a communication coming in, and it is corrupted by noise," says Médard, a professor of electrical engineering and computer science. "I could try to remove the noise, but by doing that, I'm in effect making a decision right now that maybe would have been better taken by someone downstream from me who might have had more observations of the same source."

On the other hand, Médard says, if a node simply forwards the data it receives without performing any error correction, it could end up squandering bandwidth. "If the node takes all the signal it has and does not whittle down his representation, then it might be using a lot of energy to transmit noise," she says. "The question is, how much of the noise do I remove, and how much do I leave in?"

In their first paper, Médard and her colleagues analyze the case in which the noise in a given link is unrelated to the signals traveling over other links, as is true of most wired networks. In that case, the researchers show, the problems of error correction and network coding can be separated without limiting the capacity of the network as a whole.

Noisy neighbors

In the second paper, the researchers tackle the case in which the noise on a given link is related to the signals on other links, as is true of most wireless networks, since the transmissions of neighboring base stations can interfere with each other. This complicates things enormously: Indeed, Médard points out, information theorists still don't know how to quantify the capacity of a simple three-node wireless network, in which two nodes relay messages to each other via a third node.

Nonetheless, Médard and her colleagues show how to calculate upper and lower bounds on the capacity of a given wireless network. While the gap between the bounds can be very large in practice, knowing the bounds could still help network operators evaluate the benefits of further research on network coding. If the observed bit rate on a real-world network is below the lower bound, the operator knows the minimum improvement that the ideal code would provide; if the observed rate is above the lower bound but below the upper, then the operator knows the maximum improvement that the ideal code might provide. If even the maximum improvement would afford only a small savings in operational expenses, the operator may decide that further research on improved coding isn't worth the money.

"The separation theorem they proved is of fundamental interest," says Raymond Yeung, a professor of information engineering and co-director of the Institute of Network Coding at the Chinese University of Hong Kong. "While the result itself is not surprising, it is somewhat unexpected that they were able to prove the result in such a general setting."

Yeung cautions, however, that while the researchers have "decomposed a very difficult problem into two," one of those problems "remains very difficult. … The bound is in terms of the solution to another problem which is difficult to solve," he says. "It is not clear how tight this bound is; that needs further research."

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Story Source:

The above story is reprinted from materials provided by Massachusetts Institute of Technology. The original article was written by Larry Hardesty, MIT News Office.

Note: Materials may be edited for content and length. For further information, please contact the source cited above.

Journal Reference:

Ralf Koetter, Michelle Effros, Muriel Medard. A Theory of Network Equivalence -- Part I: Point-to-Point Channels. IEEE Transactions on Information Theory, 2011; 57 (2): 972 DOI: 10.1109/TIT.2010.2102110

Note: If no author is given, the source is cited instead.

Disclaimer: Views expressed in this article do not necessarily reflect those of ScienceDaily or its staff.

View the original article here

Elusive capacity of networks: Calculating data network's total capacity notoriously difficult, but theorists making some headway

ScienceDaily (May 15, 2012) — In its early years, information theory -- which grew out of a landmark 1948 paper by MIT alumnus and future professor Claude Shannon -- was dominated by research on error-correcting codes: How do you encode information so as to guarantee its faithful transmission, even in the presence of the corrupting influences engineers call "noise"?

Recently, one of the most intriguing developments in information theory has been a different kind of coding, called network coding, in which the question is how to encode information in order to maximize the capacity of a network as a whole. For information theorists, it was natural to ask how these two types of coding might be combined: If you want to both minimize error and maximize capacity, which kind of coding do you apply where, and when do you do the decoding?

What makes that question particularly hard to answer is that no one knows how to calculate the data capacity of a network as a whole -- or even whether it can be calculated. Nonetheless, in the first half of a two-part paper, which was published recently in IEEE Transactions on Information Theory, MIT's Muriel Médard, California Institute of Technology's Michelle Effros and the late Ralf Koetter of the University of Technology in Munich show that in a wired network, network coding and error-correcting coding can be handled separately, without reduction in the network's capacity. In the forthcoming second half of the paper, the same researchers demonstrate some bounds on the capacities of wireless networks, which could help guide future research in both industry and academia.

A typical data network consists of an array of nodes -- which could be routers on the Internet, wireless base stations or even processing units on a single chip -- each of which can directly communicate with a handful of its neighbors. When a packet of data arrives at a node, the node inspects its addressing information and decides which of several pathways to send it along.

Calculated confusion

With network coding, on the other hand, a node scrambles together the packets it receives and sends the hybrid packets down multiple paths; at each subsequent node they're scrambled again in different ways. Counterintuitively, this can significantly increase the capacity of the network as a whole: Hybrid packets arrive at their destination along multiple paths. If one of those paths is congested, or if one of its links fails outright, the packets arriving via the other paths will probably contain enough information that the recipient can piece together the original message.

But each link between nodes could be noisy, so the information in the packets also needs to be encoded to correct for errors. "Suppose that I'm a node in a network, and I see a communication coming in, and it is corrupted by noise," says Médard, a professor of electrical engineering and computer science. "I could try to remove the noise, but by doing that, I'm in effect making a decision right now that maybe would have been better taken by someone downstream from me who might have had more observations of the same source."

On the other hand, Médard says, if a node simply forwards the data it receives without performing any error correction, it could end up squandering bandwidth. "If the node takes all the signal it has and does not whittle down his representation, then it might be using a lot of energy to transmit noise," she says. "The question is, how much of the noise do I remove, and how much do I leave in?"

In their first paper, Médard and her colleagues analyze the case in which the noise in a given link is unrelated to the signals traveling over other links, as is true of most wired networks. In that case, the researchers show, the problems of error correction and network coding can be separated without limiting the capacity of the network as a whole.

Noisy neighbors

In the second paper, the researchers tackle the case in which the noise on a given link is related to the signals on other links, as is true of most wireless networks, since the transmissions of neighboring base stations can interfere with each other. This complicates things enormously: Indeed, Médard points out, information theorists still don't know how to quantify the capacity of a simple three-node wireless network, in which two nodes relay messages to each other via a third node.

Nonetheless, Médard and her colleagues show how to calculate upper and lower bounds on the capacity of a given wireless network. While the gap between the bounds can be very large in practice, knowing the bounds could still help network operators evaluate the benefits of further research on network coding. If the observed bit rate on a real-world network is below the lower bound, the operator knows the minimum improvement that the ideal code would provide; if the observed rate is above the lower bound but below the upper, then the operator knows the maximum improvement that the ideal code might provide. If even the maximum improvement would afford only a small savings in operational expenses, the operator may decide that further research on improved coding isn't worth the money.

"The separation theorem they proved is of fundamental interest," says Raymond Yeung, a professor of information engineering and co-director of the Institute of Network Coding at the Chinese University of Hong Kong. "While the result itself is not surprising, it is somewhat unexpected that they were able to prove the result in such a general setting."

Yeung cautions, however, that while the researchers have "decomposed a very difficult problem into two," one of those problems "remains very difficult. … The bound is in terms of the solution to another problem which is difficult to solve," he says. "It is not clear how tight this bound is; that needs further research."

Share this story on Facebook, Twitter, and Google:

Other social bookmarking and sharing tools:

Story Source:

The above story is reprinted from materials provided by Massachusetts Institute of Technology. The original article was written by Larry Hardesty, MIT News Office.

Note: Materials may be edited for content and length. For further information, please contact the source cited above.

Journal Reference:

Ralf Koetter, Michelle Effros, Muriel Medard. A Theory of Network Equivalence -- Part I: Point-to-Point Channels. IEEE Transactions on Information Theory, 2011; 57 (2): 972 DOI: 10.1109/TIT.2010.2102110

Note: If no author is given, the source is cited instead.

Disclaimer: Views expressed in this article do not necessarily reflect those of ScienceDaily or its staff.

View the original article here