Xray diffraction algorithm solves Sudoku puzzles
29 March 2006
Cornell physicist Veit Elser and colleagues have discovered an algorithm
critical for Xray diffraction microscopy that gives researchers a new tool
for imaging the tiniest and most delicate of biological specimens. They
discovered that the same algorithm also solves the internationally popular
numbers puzzle Sudoku — not just one puzzle, all of them.

A Sudoku puzzle designed by Elser's
difference map algorithm. The challenge is to fill in the
missing cells of the table with letters AY so that there are no
duplicates in any row, column or fivebyfive block (outlined in
red). 
The Sudoku discovery appeals to Elser's whimsical side. But the
algorithm, he notes, has potential for all kinds of other applications.
"There are a lot of problems that you can represent in terms of this
language," he says. The socalled differencemap algorithm, which Elser says
could have applications from productivity optimization to nanofabrication,
tackles problems for which the solution must meet two independent
constraints. In the case of Sudoku, the constraints are simple: Each of nine
numbers, considered alone, appears nine times in the grid so that there is
only one per row and column. And all nine numbers appear within each of the
nine blocks.
In Xray diffraction microscopy, the constraints are more complex. But
the beauty of the algorithm, as Elser demonstrates, is that complexity
doesn't matter. By applying the algorithm to the jumble of raw data from
such an experiment, researchers can now reconstruct from it a clear,
detailed image.
The result, shown in images published with a recent article in the
Proceedings of the National Academy of Sciences (PNAS) with lead author
David Shapiro of the State University of New York at Stony Brook and other
colleagues, is a richly detailed image of a specimen as small as a single
yeast cell — taken without staining, sectioning or otherwise damaging the
specimen.
Unlike optical microscopes, which use a lens to focus light on a target
(and are therefore only useful for specimens larger than the wavelength of
visible light), imaging methods based on Xrays and electrons take advantage
of the finer wavelengths provided by these forms of illumination.
But some of these methods can damage the specimen with harmful radiation,
require that a specimen be stained or otherwise altered or lack the
penetrating power necessary for threedimensional reconstructions. Xray
diffraction microscopy, which uses "soft" Xrays and measures the resulting
diffraction pattern, is often the method of choice because it gives a
detailrich image and leaves the specimen relatively unscathed.
The tricky part comes in using the diffraction pattern to reconstruct an
image of the specimen. Instead of directly measuring the pixelbypixel
contrast within the specimen, researchers are left with data that represents
the object broken down into its constituent waves: a vast number of them, of
different frequencies and amplitudes, waiting to be added up in a process
called a Fourier synthesis to reconstruct the image.
"But if it were just combining waves with a definite oscillation and
adding them up, that would be a piece of cake for a computer to do," says
Elser.
The challenge is in the waves' phases, which are critical in
reconstructing an image from the diffraction data. Without attention to that
piece of the puzzle, the resulting image is reduced to noise.
"People used to say that's an impossible problem," Elser says. "Then
people got to thinking about the fact that there are going to be constraints
coming from some rather mundane facts — that in principle will make the
problem of deducing the phase like the solution of a solvable puzzle."
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