| Paper Abstract and Keywords |
| Presentation |
2006-03-23 11:05
Improved Lower Bounds for Families of ε -Approximate k-Restricted Min-Wise Independent Permutations
Toshiya Itoh, Tatsuya Nagatani (Tokyo Inst. of Tech.) |
| Abstract |
(in Japanese) |
(See Japanese page) |
| (in English) |
A family ${\cal F}$ of min-wise independent permutations is known to be a useful tool~of~indexing replicated documents on the Web. For any integer $n>0$, let
$S_{n}$ be the family of all~permutations on $[1,n]=\{1,2,\ldots, n\}$. For any integer $k \in [1,n]$ and any real $\varepsilon >0$, we say that a family ${\cal F} \subseteq S_{n}$ of permutations is $\varepsilon$-approximate k$-restricted
min-wise independent if for any (nonempty) $X \subseteq [1,n]$ such that
$\|X\| \leq k$ and any $x \in X$, $|\Pr [\min \{\pi(X)\} = \pi(x)]- 1/\|X\| |\leq
\varepsilon/\|X\|$, when $\pi$ is chosen from ${\cal F}$ uniformly at random (where $\|A\|$ is the cardinality of a finite set $A$). For the size of families ${\cal F} \subseteq S_{n}$ of $\varepsilon$-approximate $k$-restricted min-wise independent permutations, the following results are known: For any integer $k\in [1,n]$ and any real $\varepsilon > 0$, (constructive upper bound)
$\|{\cal F}\|=2^{4k+o(k)}k^{2 \log \log (n/\varepsilon)}$; (nonconstructive upper bound) $\|{\cal F}\|=O(\frac{k^{2}}{\varepsilon^{2}} \log (n/k))$; (lower bound) $\|{\cal F}\|=\Omega(k^{2}(1-\sqrt{8 \varepsilon}))$. In this paper, we first derive an upper bound for the Ramsey number of the edge coloring with $m \geq 2$ colors of a complete graph $K_{\ell}$ of $\ell$ vertices, and by the linear algebra method, we then derive a slightly improved lower bound, i.e., we show that
for any family ${\cal F} \subseteq S_{n}$ of $\varepsilon$-approximate $k$-restricted min-wise independent permutations, $\|{\cal F}\| =
\Omega(k \sqrt{\frac{1}{\varepsilon}\log (n/k)})$. |
| Keyword |
(in Japanese) |
(See Japanese page) |
| (in English) |
Min-Wise Independence / Positive Definite / Ramsey Number / Linear Algebra Method / / / / |
| Reference Info. |
IEICE Tech. Rep., vol. 105, no. 680, COMP2005-66, pp. 23-30, March 2006. |
| Paper # |
COMP2005-66 |
| Date of Issue |
2006-03-16 (COMP) |
| ISSN |
Print edition: ISSN 0913-5685 |
| Download PDF |
|
| Conference Information |
| Committee |
COMP |
| Conference Date |
2006-03-22 - 2006-03-23 |
| Place (in Japanese) |
(See Japanese page) |
| Place (in English) |
The University of Electro-Communications |
| Topics (in Japanese) |
(See Japanese page) |
| Topics (in English) |
|
| Paper Information |
| Registration To |
COMP |
| Conference Code |
2006-03-COMP |
| Language |
English (Japanese title is available) |
| Title (in Japanese) |
(See Japanese page) |
| Sub Title (in Japanese) |
(See Japanese page) |
| Title (in English) |
Improved Lower Bounds for Families of ε -Approximate k-Restricted Min-Wise Independent Permutations |
| Sub Title (in English) |
|
| Keyword(1) |
Min-Wise Independence |
| Keyword(2) |
Positive Definite |
| Keyword(3) |
Ramsey Number |
| Keyword(4) |
Linear Algebra Method |
| Keyword(5) |
|
| Keyword(6) |
|
| Keyword(7) |
|
| Keyword(8) |
|
| 1st Author's Name |
Toshiya Itoh |
| 1st Author's Affiliation |
Tokyo Institute of Technology (Tokyo Inst. of Tech.) |
| 2nd Author's Name |
Tatsuya Nagatani |
| 2nd Author's Affiliation |
Tokyo Institute of Technology (Tokyo Inst. of Tech.) |
| 3rd Author's Name |
|
| 3rd Author's Affiliation |
() |
| 4th Author's Name |
|
| 4th Author's Affiliation |
() |
| 5th Author's Name |
|
| 5th Author's Affiliation |
() |
| 6th Author's Name |
|
| 6th Author's Affiliation |
() |
| 7th Author's Name |
|
| 7th Author's Affiliation |
() |
| 8th Author's Name |
|
| 8th Author's Affiliation |
() |
| 9th Author's Name |
|
| 9th Author's Affiliation |
() |
| 10th Author's Name |
|
| 10th Author's Affiliation |
() |
| 11th Author's Name |
|
| 11th Author's Affiliation |
() |
| 12th Author's Name |
|
| 12th Author's Affiliation |
() |
| 13th Author's Name |
|
| 13th Author's Affiliation |
() |
| 14th Author's Name |
|
| 14th Author's Affiliation |
() |
| 15th Author's Name |
|
| 15th Author's Affiliation |
() |
| 16th Author's Name |
|
| 16th Author's Affiliation |
() |
| 17th Author's Name |
|
| 17th Author's Affiliation |
() |
| 18th Author's Name |
|
| 18th Author's Affiliation |
() |
| 19th Author's Name |
|
| 19th Author's Affiliation |
() |
| 20th Author's Name |
|
| 20th Author's Affiliation |
() |
| Speaker |
Author-1 |
| Date Time |
2006-03-23 11:05:00 |
| Presentation Time |
35 minutes |
| Registration for |
COMP |
| Paper # |
COMP2005-66 |
| Volume (vol) |
vol.105 |
| Number (no) |
no.680 |
| Page |
pp.23-30 |
| #Pages |
8 |
| Date of Issue |
2006-03-16 (COMP) |
|