* [OT] Hints for an algorithm - avoiding O(n^2) @ 2004-03-12 9:15 Jano 2004-03-12 11:06 ` Jean-Pierre Rosen ` (2 more replies) 0 siblings, 3 replies; 16+ messages in thread From: Jano @ 2004-03-12 9:15 UTC (permalink / raw) Sorry for the off-topic, but this is the only group I read frequently that I'm sure is packed with knowledgeable people. The problem is a simple one: I'm planning to write a simulator of sideral-like objects using the Newton laws (precision is not a top priority). The obvious way is to compute the force interactions between each pair of objects, sum them all, and iterate over... but this is O(n^2). To be precise, there are n(n-1)/2 forces to be computed if I'm right (n the number of objects). I was wondering if you know of another approach more efficient. A simple hint is enough, I'll look further as necessary once in the right path. I'm having some thoughts about making a discrete division of the space and compute the potential field, but the idea is not mature enough... I'm not sure if the loss of precision will be too big, or if the results will not be similar in time (I'm planning to manage about 1000-2000 objects simultaneously). Thanks! P.s: It will be programmed in Ada... just to ease the OT a bit :) ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-12 9:15 [OT] Hints for an algorithm - avoiding O(n^2) Jano @ 2004-03-12 11:06 ` Jean-Pierre Rosen 2004-03-12 12:53 ` Stuart Palin ` (3 more replies) 2004-03-13 0:12 ` Wes Groleau 2004-03-17 2:24 ` jtg 2 siblings, 4 replies; 16+ messages in thread From: Jean-Pierre Rosen @ 2004-03-12 11:06 UTC (permalink / raw) [-- Warning: decoded text below may be mangled, UTF-8 assumed --] [-- Attachment #1: Type: text/plain, Size: 876 bytes --] "Jano" <402450@cepsz.unizar.es> a �crit dans le message de news:5d6fdb61.0403120115.7c102e3c@posting.google.com... > The problem is a simple one: I'm planning to write a simulator of > sideral-like objects using the Newton laws (precision is not a top > priority). The obvious way is to compute the force interactions > between each pair of objects, sum them all, and iterate over... but > this is O(n^2). To be precise, there are n(n-1)/2 forces to be > computed if I'm right (n the number of objects). > If I remember my Newton laws correctly, you just need to compute the force between an object and the center of mass of all other objects. Hmmm.... not sure that it won't be O(n^2) still, but it might ease a bit. -- --------------------------------------------------------- J-P. Rosen (rosen@adalog.fr) Visit Adalog's web site at http://www.adalog.fr ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-12 11:06 ` Jean-Pierre Rosen @ 2004-03-12 12:53 ` Stuart Palin 2004-03-12 12:55 ` Stuart Palin ` (2 subsequent siblings) 3 siblings, 0 replies; 16+ messages in thread From: Stuart Palin @ 2004-03-12 12:53 UTC (permalink / raw) Jean-Pierre Rosen wrote: > > "Jano" <402450@cepsz.unizar.es> a �crit dans le message de news:5d6fdb61.0403120115.7c102e3c@posting.google.com... > > The problem is a simple one: I'm planning to write a simulator of > > sideral-like objects using the Newton laws (precision is not a top > > priority). The obvious way is to compute the force interactions > > between each pair of objects, sum them all, and iterate over... but > > this is O(n^2). To be precise, there are n(n-1)/2 forces to be > > computed if I'm right (n the number of objects). > > > If I remember my Newton laws correctly, you just need to compute the force > between an object and the center of mass of all other objects. No! It is quite a straight-forward proof to show that for a point inside a uniformaly dense sphere the mass at a greater distance from the centre (than the point) has no net gravitational effect on the point. Also a counter-example is very easy to construct. Consider two large equal masses separated by a large distance. Their centre of mass (which will be the sum of their masses) is at the mid-point of the shortest line joining them. Another mass placed a little way from this mid-point will in fact feel little gravitational effect because the masses are a long way away. However, a calculation based on the centre of mass would suggest there is a very strong force acting! No help to the OP - they might be better off posting in an astronomical group as this is more likely to be where people with knowledge of any special algorithms might hang out. -- Stuart Palin ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-12 11:06 ` Jean-Pierre Rosen 2004-03-12 12:53 ` Stuart Palin @ 2004-03-12 12:55 ` Stuart Palin 2004-03-12 13:48 ` Dmitry A. Kazakov 2004-03-12 13:30 ` Björn Persson 2004-03-14 17:27 ` Jano 3 siblings, 1 reply; 16+ messages in thread From: Stuart Palin @ 2004-03-12 12:55 UTC (permalink / raw) Jean-Pierre Rosen wrote: > > "Jano" <402450@cepsz.unizar.es> a �crit dans le message de news:5d6fdb61.0403120115.7c102e3c@posting.google.com... > > The problem is a simple one: I'm planning to write a simulator of > > sideral-like objects using the Newton laws (precision is not a top > > priority). The obvious way is to compute the force interactions > > between each pair of objects, sum them all, and iterate over... but > > this is O(n^2). To be precise, there are n(n-1)/2 forces to be > > computed if I'm right (n the number of objects). > > > If I remember my Newton laws correctly, you just need to compute the force > between an object and the center of mass of all other objects. No! It is quite a straight-forward proof to show that for a point inside a uniformly dense sphere the mass at a greater distance from the centre (than the point) has no net gravitational effect on the point. Also a counter-example is very easy to construct. Consider two large equal masses separated by a large distance. Their centre of mass (which will be the sum of their masses) is at the mid-point of the shortest line joining them. Another mass placed a little way from this mid-point will in fact feel little gravitational effect because the masses are a long way away. However, a calculation based on the centre of mass would suggest there is a very strong force acting! No help to the OP - they might be better off posting in an astronomical group as this is more likely to be where people with knowledge of any special algorithms might hang out. -- Stuart Palin ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-12 12:55 ` Stuart Palin @ 2004-03-12 13:48 ` Dmitry A. Kazakov 2004-03-17 1:16 ` jtg 0 siblings, 1 reply; 16+ messages in thread From: Dmitry A. Kazakov @ 2004-03-12 13:48 UTC (permalink / raw) On Fri, 12 Mar 2004 12:55:19 +0000, Stuart Palin <stuart.palin@0.0> wrote: >Jean-Pierre Rosen wrote: >> >> "Jano" <402450@cepsz.unizar.es> a �crit dans le message de >news:5d6fdb61.0403120115.7c102e3c@posting.google.com... >> > The problem is a simple one: I'm planning to write a simulator of >> > sideral-like objects using the Newton laws (precision is not a top >> > priority). The obvious way is to compute the force interactions >> > between each pair of objects, sum them all, and iterate over... but >> > this is O(n^2). To be precise, there are n(n-1)/2 forces to be >> > computed if I'm right (n the number of objects). >> > >> If I remember my Newton laws correctly, you just need to compute the force >> between an object and the center of mass of all other objects. > >No! It is quite a straight-forward proof to show that for a >point inside a uniformly dense sphere the mass at a greater >distance from the centre (than the point) has no net >gravitational effect on the point. > >Also a counter-example is very easy to construct. Consider >two large equal masses separated by a large distance. Their >centre of mass (which will be the sum of their masses) is at >the mid-point of the shortest line joining them. Another >mass placed a little way from this mid-point will in fact >feel little gravitational effect because the masses are a >long way away. However, a calculation based on the centre >of mass would suggest there is a very strong force acting! This is because the third object is within the sphere containing the first two. If I correctly remember, that is a consequence of the Gauss theorem. But if you arrange your objects in spheric clusters centered in their mass centers, so that the interesting object will be outside of any of them, then the result will hold. Whether that will be better than O(n**2) is another question. >No help to the OP - they might be better off posting in an >astronomical group as this is more likely to be where people >with knowledge of any special algorithms might hang out. -- Regards, Dmitry Kazakov www.dmitry-kazakov.de ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-12 13:48 ` Dmitry A. Kazakov @ 2004-03-17 1:16 ` jtg 0 siblings, 0 replies; 16+ messages in thread From: jtg @ 2004-03-17 1:16 UTC (permalink / raw) Dmitry A. Kazakov wrote: > On Fri, 12 Mar 2004 12:55:19 +0000, Stuart Palin <stuart.palin@0.0> > wrote: > > >>Jean-Pierre Rosen wrote: >> >>>"Jano" <402450@cepsz.unizar.es> a �crit dans le message de >> >>news:5d6fdb61.0403120115.7c102e3c@posting.google.com... >> >>>>The problem is a simple one: I'm planning to write a simulator of >>>>sideral-like objects using the Newton laws (precision is not a top >>>>priority). The obvious way is to compute the force interactions >>>>between each pair of objects, sum them all, and iterate over... but >>>>this is O(n^2). To be precise, there are n(n-1)/2 forces to be >>>>computed if I'm right (n the number of objects). >>>> >>> >>>If I remember my Newton laws correctly, you just need to compute the force >>>between an object and the center of mass of all other objects. >> >>No! It is quite a straight-forward proof to show that for a >>point inside a uniformly dense sphere the mass at a greater >>distance from the centre (than the point) has no net >>gravitational effect on the point. >> >>Also a counter-example is very easy to construct. Consider >>two large equal masses separated by a large distance. Their >>centre of mass (which will be the sum of their masses) is at >>the mid-point of the shortest line joining them. Another >>mass placed a little way from this mid-point will in fact >>feel little gravitational effect because the masses are a >>long way away. However, a calculation based on the centre >>of mass would suggest there is a very strong force acting! > > > This is because the third object is within the sphere containing the > first two. If I correctly remember, that is a consequence of the Gauss > theorem. But if you arrange your objects in spheric clusters centered > in their mass centers, so that the interesting object will be outside > of any of them, then the result will hold. Whether that will be better > than O(n**2) is another question. No! According to Newton laws the object can for instance orbit one of the masses, but if you assume you can model the masses with one mass between them, the object could instead only orbit the center (which is actually empty) and could not orbit any of the masses! ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-12 11:06 ` Jean-Pierre Rosen 2004-03-12 12:53 ` Stuart Palin 2004-03-12 12:55 ` Stuart Palin @ 2004-03-12 13:30 ` Björn Persson 2004-03-12 13:42 ` James Rogers 2004-03-12 14:29 ` Robert I. Eachus 2004-03-14 17:27 ` Jano 3 siblings, 2 replies; 16+ messages in thread From: Björn Persson @ 2004-03-12 13:30 UTC (permalink / raw) Jean-Pierre Rosen wrote: > If I remember my Newton laws correctly, you just need to compute the force between an object and the center of mass of all other > objects. If that were true, the moon would orbit the sun and not the Earth (if we ignore the rest of the universe). -- Björn Persson jor ers @sv ge. b n_p son eri nu ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-12 13:30 ` Björn Persson @ 2004-03-12 13:42 ` James Rogers 2004-03-12 14:29 ` Robert I. Eachus 1 sibling, 0 replies; 16+ messages in thread From: James Rogers @ 2004-03-12 13:42 UTC (permalink / raw) [-- Warning: decoded text below may be mangled, UTF-8 assumed --] [-- Attachment #1: Type: text/plain, Size: 280 bytes --] Bj�rn Persson <spam-away@nowhere.nil> wrote in news:1Yi4c.85891$dP1.243229@newsc.telia.net: > If that were true, the moon would orbit the sun and not the Earth (if we > ignore the rest of the universe). The moon does orbit the Sun. It also orbits the Earth. Jim Rogers ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-12 13:30 ` Björn Persson 2004-03-12 13:42 ` James Rogers @ 2004-03-12 14:29 ` Robert I. Eachus 2004-03-12 23:44 ` Björn Persson 1 sibling, 1 reply; 16+ messages in thread From: Robert I. Eachus @ 2004-03-12 14:29 UTC (permalink / raw) Bj�rn Persson wrote: > If that were true, the moon would orbit the sun and not the Earth (if we > ignore the rest of the universe). Actually it does. If you compute the gravitational force of the Earth on the Moon, and compare it to the gravitational force of the Sun, the Sun has a much greater effect. In fact, the orbit of the Moon around the Sun is everywhere convex. However, back to the OP's question. The first thing you want to do is not use brute force, but use Runga-Kutta for each pair of objects. This makes the calculation about ten times as complex, but gives a much better fit. Second when two objects are "close" you should do the Runga-Kutta interpolation much more often. This usually means calculating the effects of all gravitational attractions on the objects that are in close proximity more often as well. (You don't have to do the reverse, because the effect of the two nearby objects on remote objects will be that of their center of mass.) Of course you also have to allow for the possibility that there are several objects that are gravitationally close. The normal way to do this is to, at the end of every major cycle look at the positions of all the objects, and decide which ones get the intermediate steps. It is not worth the effort to work with two pairs instead of four 'close' objects. -- Robert I. Eachus "The only thing necessary for the triumph of evil is for good men to do nothing." --Edmund Burke ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-12 14:29 ` Robert I. Eachus @ 2004-03-12 23:44 ` Björn Persson 2004-03-13 15:21 ` Robert I. Eachus 0 siblings, 1 reply; 16+ messages in thread From: Björn Persson @ 2004-03-12 23:44 UTC (permalink / raw) Robert I. Eachus wrote: > Björn Persson wrote: > >> If that were true, the moon would orbit the sun and not the Earth (if >> we ignore the rest of the universe). > > > Actually it does. If you compute the gravitational force of the Earth > on the Moon, and compare it to the gravitational force of the Sun, the > Sun has a much greater effect. In fact, the orbit of the Moon around > the Sun is everywhere convex. I didn't mean to say that the Sun doesn't affect the Moon. "If what J-P wrote were true, the Moon would not orbit both the Sun and the Earth, as it does, but only orbit the solar system's center of gravity (if we ignore the rest of the universe)." Is that better? You can't choose your words too carefully when there are Ada language lawyers in the audience. :-) -- Björn Persson jor ers @sv ge. b n_p son eri nu ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-12 23:44 ` Björn Persson @ 2004-03-13 15:21 ` Robert I. Eachus 2004-03-14 17:27 ` Jano 0 siblings, 1 reply; 16+ messages in thread From: Robert I. Eachus @ 2004-03-13 15:21 UTC (permalink / raw) Bj�rn Persson wrote: > I didn't mean to say that the Sun doesn't affect the Moon. "If what J-P > wrote were true, the Moon would not orbit both the Sun and the Earth, as > it does, but only orbit the solar system's center of gravity (if we > ignore the rest of the universe)." Is that better? > > You can't choose your words too carefully when there are Ada language > lawyers in the audience. :-) I think you missed my point. Earth's moon is unique in the solar system. It is the only moon whose path around the Sun doesn't cross itself. In fact, the effect of the Earth's gravity is so slight that if you are calculating the Moon's orbit, it is better to compute it the same way as other planets. (Treat it as orbiting the sun, with perturbations by the gravity from other planets. For all other moons, you are better off calculating them as orbiting some planet, with the Sun's gravitational effect as a perturbation.) In a discussion of how to calculate the positions in a complex gravitational environment, that is a point worth knowing. -- Robert I. Eachus "The only thing necessary for the triumph of evil is for good men to do nothing." --Edmund Burke ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-13 15:21 ` Robert I. Eachus @ 2004-03-14 17:27 ` Jano 2004-03-15 5:34 ` Robert I. Eachus 0 siblings, 1 reply; 16+ messages in thread From: Jano @ 2004-03-14 17:27 UTC (permalink / raw) [-- Warning: decoded text below may be mangled, UTF-8 assumed --] [-- Attachment #1: Type: text/plain, Size: 1204 bytes --] Robert I. Eachus dice... > Bj�rn Persson wrote: > > > I didn't mean to say that the Sun doesn't affect the Moon. "If what J-P > > wrote were true, the Moon would not orbit both the Sun and the Earth, as > > it does, but only orbit the solar system's center of gravity (if we > > ignore the rest of the universe)." Is that better? > > > > You can't choose your words too carefully when there are Ada language > > lawyers in the audience. :-) > > I think you missed my point. Earth's moon is unique in the solar > system. It is the only moon whose path around the Sun doesn't cross > itself. In fact, the effect of the Earth's gravity is so slight that if > you are calculating the Moon's orbit, it is better to compute it the > same way as other planets. (Treat it as orbiting the sun, with > perturbations by the gravity from other planets. For all other moons, > you are better off calculating them as orbiting some planet, with the > Sun's gravitational effect as a perturbation.) I had never stopped to think about it, but if I understand you correctly, that means that the Moon's orbit has a zig-zag path. Nice :) I'll look further into your Runge-Kutta suggestion, thanks. ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-14 17:27 ` Jano @ 2004-03-15 5:34 ` Robert I. Eachus 0 siblings, 0 replies; 16+ messages in thread From: Robert I. Eachus @ 2004-03-15 5:34 UTC (permalink / raw) Jano wrote: > I had never stopped to think about it, but if I understand you > correctly, that means that the Moon's orbit has a zig-zag path. Nice :) Not even that. No zigs, no zags. If you draw a set of lines perpendicular to the Moon's path around the Sun (from a heliocentric view), at any point in its orbit, the points where the lines cross will be inside the Sun or very near. It is only from a geocentric point of view that the Moon and Earth seem to orbit a common point. (And that point is actually about one Earth radius from the Earth...) > I'll look further into your Runge-Kutta suggestion, thanks. I wish I could recommend a good book on how to implement Runge-Kutta, but your choice is probably a book about how the algorithm works, or one about how to use some math library that implements it. It isn't that hard to implement, but be sure to use double precision unless you are really careful. -- Robert I. Eachus "The only thing necessary for the triumph of evil is for good men to do nothing." --Edmund Burke ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-12 11:06 ` Jean-Pierre Rosen ` (2 preceding siblings ...) 2004-03-12 13:30 ` Björn Persson @ 2004-03-14 17:27 ` Jano 3 siblings, 0 replies; 16+ messages in thread From: Jano @ 2004-03-14 17:27 UTC (permalink / raw) [-- Warning: decoded text below may be mangled, UTF-8 assumed --] [-- Attachment #1: Type: text/plain, Size: 828 bytes --] Jean-Pierre Rosen dice... > > "Jano" <402450@cepsz.unizar.es> a �crit dans le message de news:5d6fdb61.0403120115.7c102e3c@posting.google.com... > > The problem is a simple one: I'm planning to write a simulator of > > sideral-like objects using the Newton laws (precision is not a top > > priority). The obvious way is to compute the force interactions > > between each pair of objects, sum them all, and iterate over... but > > this is O(n^2). To be precise, there are n(n-1)/2 forces to be > > computed if I'm right (n the number of objects). > > > If I remember my Newton laws correctly, you just need to compute the force between an object and the center of mass of all other > objects. I discussed this too with a colleague, but it's equivalent (apart from the subtleties noted by other posters) in computing time. ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-12 9:15 [OT] Hints for an algorithm - avoiding O(n^2) Jano 2004-03-12 11:06 ` Jean-Pierre Rosen @ 2004-03-13 0:12 ` Wes Groleau 2004-03-17 2:24 ` jtg 2 siblings, 0 replies; 16+ messages in thread From: Wes Groleau @ 2004-03-13 0:12 UTC (permalink / raw) Jano wrote: > I was wondering if you know of another approach more efficient. A > simple hint is enough, I'll look further as necessary once in the > right path. I would think that one could locate software already being used to compute/predict trajectories of real astronomical bodies (or unmanned spacecraft that fly near them) and if you want Ada (who wouldn't?) just translate. -- Wes Groleau Alive and Well http://freepages.religions.rootsweb.com/~wgroleau/ ^ permalink raw reply [flat|nested] 16+ messages in thread
* Re: [OT] Hints for an algorithm - avoiding O(n^2) 2004-03-12 9:15 [OT] Hints for an algorithm - avoiding O(n^2) Jano 2004-03-12 11:06 ` Jean-Pierre Rosen 2004-03-13 0:12 ` Wes Groleau @ 2004-03-17 2:24 ` jtg 2 siblings, 0 replies; 16+ messages in thread From: jtg @ 2004-03-17 2:24 UTC (permalink / raw) Jano wrote: > Sorry for the off-topic, but this is the only group I read frequently > that I'm sure is packed with knowledgeable people. > > The problem is a simple one: I'm planning to write a simulator of > sideral-like objects using the Newton laws (precision is not a top > priority). The obvious way is to compute the force interactions > between each pair of objects, sum them all, and iterate over... but > this is O(n^2). To be precise, there are n(n-1)/2 forces to be > computed if I'm right (n the number of objects). > > I was wondering if you know of another approach more efficient. A > simple hint is enough, I'll look further as necessary once in the > right path. > > I'm having some thoughts about making a discrete division of the space > and compute the potential field, but the idea is not mature enough... > I'm not sure if the loss of precision will be too big, or if the > results will not be similar in time (I'm planning to manage about > 1000-2000 objects simultaneously). > Two ideas 1. You can assign groups of objects. For every group you can calculate its mass and the center of the mass. Then you calculate acceleration between all the objects in the same group, and then acceleration between all the groups. Of course you apply group acceleration to every object in that group. For instance you can assign sqrt(N) groups of sqrt(N) objects, and then you have O(sqrt(N)^2) + sqrt(N)*O(sqrt(N)^2) = O(N*sqrt(N)) But there are still some problems to solve for instance group boundary can divide two masses which are close to each other. You can use for instance more intelligent clustering algs, skip group interaction between adjacent groups (calculating object interactions instead) or even apply more levels of complexity (objects, small groups, bigger groups etc.) 2. You are going to make a simulation, but you think only about static calculation of gravity forces. Wrong! It is possible to do the calculations between objects much less frequently, while mantaining accuracy and fluent object movement, if you use some tricks. For instance you can adaptively change time step, making it small when needed (if two masses come very close, you calculate their paths more precisely). This will protect your simulation when two masses go very close. You can also make some asynchronous calculations (not in time steps), i.e. calculate gravitational pull for every object, in every time step just integrate it, and from time to time update it - more frequently for close objects, less frequently for distant objects. However it is memory intensive. Connecting these two ideas would be VERY effective. Of course you can calculate group interactions less frequently than object-in-group interactions. And since you can treat each group as single object, asynchronous calculations do not demand much memory in this case. If you find these ideas helpful, please e-mail me, I'm curious about the results. :-) ^ permalink raw reply [flat|nested] 16+ messages in thread
end of thread, other threads:[~2004-03-17 2:24 UTC | newest] Thread overview: 16+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 2004-03-12 9:15 [OT] Hints for an algorithm - avoiding O(n^2) Jano 2004-03-12 11:06 ` Jean-Pierre Rosen 2004-03-12 12:53 ` Stuart Palin 2004-03-12 12:55 ` Stuart Palin 2004-03-12 13:48 ` Dmitry A. Kazakov 2004-03-17 1:16 ` jtg 2004-03-12 13:30 ` Björn Persson 2004-03-12 13:42 ` James Rogers 2004-03-12 14:29 ` Robert I. Eachus 2004-03-12 23:44 ` Björn Persson 2004-03-13 15:21 ` Robert I. Eachus 2004-03-14 17:27 ` Jano 2004-03-15 5:34 ` Robert I. Eachus 2004-03-14 17:27 ` Jano 2004-03-13 0:12 ` Wes Groleau 2004-03-17 2:24 ` jtg
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