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Home -> Community -> Usenet -> comp.databases.theory -> Re: Bags vs. Sets
vc wrote:
> Rght, but imagine a real life relation with on average 10 attributes
> and say cardinality of 50,000. Naively, one would need to end up with
> 10^50,000-1 equations, or perform 50,000 Buchberger reductions to ten
> equations at each step.
Relations should always be reduced, as 10^50,000 is ridiculously large. I'm not able to get any estimate on the polynomial system size however. Here is a radical ideal for the relation with 10 tuples in 3 variables:
{x = 5, y = 5, z = 5}, {x = 1, y = 1, z = 1}, {y = 2, x = 3, z = 1},
{y = 4, x = 3, z = 1}, {z = 2, x = 1, y = 1}, {z = 2, x = 1, y = 2}, {y = 1, z = 4, x = 3}, {x = 5, y = 2, z = 4}, {y = 4, z = 4, x = 3}, {y = 2, z = 3, x = 3}
<pre>
S := [z - 15 z + 85 z - 225 z + 274 z - 120, 2 z y - 14 z y
4 3 2 2 2 + 28 y z - 16 y - 3 z + 26 z - 77 z + 94 z - 40, z y 2 2 2 4 3 - 5 z y + 4 y - 3 z y + 15 y z - 12 y - 2 z + 20 z 2 - 68 z + 90 z - 40, 3 2 4 3 2 2 y - 14 y + 28 y - z + 10 z - 35 z + 50 z - 40, 9 x 2 2 2 4 3 2 + z y + 5 y - 9 z y + 42 y z - 69 y + z - 4 z - z + 25 ]
Here is couple of amusing manipulations:
> gbasis(convert(S ,set) union {x-1},plex(x,y,z)); # polynomial representation
solve(convert(S ,set) union {x-1},{x,y,z}); # relational representation
2 2 [z - 3 z + 2, y z - z - 2 y + 2, y - 3 y + 2, x - 1]
{x = 1, y = 1, z = 1}, {z = 2, x = 1, y = 1}, {z = 2, x = 1, y = 2}
2. Project the previous result to {x,y}:
i) Eliminate variable x by computing Grobner basis in proper monomial ordering. In our case we can just take a shortcut and notice that we can just trow away "x-1" from the
2 2 [z - 3 z + 2, y z - z - 2 y + 2, y - 3 y + 2, x - 1]
ii) > solve({z^2-3*z+2, y*z-z-2*y+2, y^2-3*y+2},{y,z});
{z = 2, y = 2}, {y = 1, z = 1}, {z = 2, y = 1}
Life is not as simple for bags. If we fing grobner basis -- that would be a radical ideal and duplicate information would be lost.
> More importanly, I am not convinced that ideals is a good tool tool to
> represent bags: killing a fly with a gun, unfamiliar to most
> mumbo-jumbo, etc. Could you please list some advantages as compared
> to the "old way" ?
I don't have satisfactory answer yet. Received on Fri Jun 30 2006 - 19:07:02 CDT