6.3 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Discrepancy of hypergraphs | 2/2 | https://en.wikipedia.org/wiki/Discrepancy_of_hypergraphs | reference | science, encyclopedia | 2026-05-05T11:02:54.676918+00:00 | kb-cron |
Since a random coloring with positive probability has discrepancy at most
λ
{\displaystyle \lambda }
, in particular, there are colorings that have discrepancy at most
λ
{\displaystyle \lambda }
. Hence
disc
(
H
)
≤
λ
.
◻
{\displaystyle \operatorname {disc} ({\mathcal {H}})\leq \lambda .\ \Box }
-
For any hypergraph
H{\displaystyle {\mathcal {H}}}
with n vertices and m edges such that
m
≥
n
{\displaystyle m\geq n}
:
disc
(
H
)
∈
O
(
n
)
.
{\displaystyle \operatorname {disc} ({\mathcal {H}})\in O({\sqrt {n}}).}
To prove this, a much more sophisticated approach using the entropy function was necessary. Of course this is particularly interesting for
m
=
O
(
n
)
{\displaystyle m=O(n)}
. In the case
m
=
n
{\displaystyle m=n}
,
disc
(
H
)
≤
6
n
{\displaystyle \operatorname {disc} ({\mathcal {H}})\leq 6{\sqrt {n}}}
can be shown for n large enough. Therefore, this result is usually known to as 'Six Standard Deviations Suffice'. It is considered to be one of the milestones of discrepancy theory. The entropy method has seen numerous other applications, e.g. in the proof of the tight upper bound for the arithmetic progressions of Matoušek and Spencer or the upper bound in terms of the primal shatter function due to Matoušek.
== Hypergraphs of bounded degree == Better discrepancy bounds can be attained when the hypergraph has a bounded degree, that is, each vertex of
H
{\displaystyle {\mathcal {H}}}
is contained in at most t edges, for some small t. In particular:
Beck and Fiala proved that
disc
(
H
)
<
2
t
{\displaystyle \operatorname {disc} ({\mathcal {H}})<2t}
; this is known as the Beck–Fiala theorem. They conjectured that
disc
(
H
)
=
O
(
t
)
{\displaystyle \operatorname {disc} ({\mathcal {H}})=O({\sqrt {t}})}
. Bednarchak and Helm and Helm improved the Beck-Fiala bound in tiny steps to
disc
(
H
)
≤
2
t
−
3
{\displaystyle \operatorname {disc} ({\mathcal {H}})\leq 2t-3}
(for a slightly restricted situation, i.e.
t
≥
3
{\displaystyle t\geq 3}
). Bukh improved this in 2016 to
2
t
−
log
∗
t
{\displaystyle 2t-\log ^{*}t}
, where
log
∗
t
{\displaystyle \log ^{*}t}
denotes the iterated logarithm. A corollary of Beck's paper – the first time the notion of discrepancy explicitly appeared – shows
disc
(
H
)
≤
C
t
log
m
log
n
{\displaystyle \operatorname {disc} ({\mathcal {H}})\leq C{\sqrt {t\log m}}\log n}
for some constant C. The latest improvement in this direction is due to Banaszczyk:
disc
(
H
)
=
O
(
t
log
n
)
{\displaystyle \operatorname {disc} ({\mathcal {H}})=O({\sqrt {t\log n}})}
.
== Special hypergraphs == Better bounds on the discrepancy are possible for hypergraphs with a special structure, such as:
Discrepancy of permutations - when the vertices are the integers 1,...,n, and the hyperedges are all the intervals of some m given permutations on the integers. Geometric discrepancy - when the vertices are points in a Euclidean space, and the hyperedges are geometric objects, such as rectangles or half-spaces. Arithmetic progressions (Roth, Sárközy, Beck, Matoušek & Spencer) Six Standard Deviations Suffice (Spencer)
== Major open problems == Komlós Conjecture
== Applications == Numerical Integration: Monte Carlo methods in high dimensions. Computational Geometry: Divide and conquer algorithms. Image Processing: Halftoning
== Notes ==
== References == Beck, József; Chen, William W. L. (2009). Irregularities of Distribution. Cambridge University Press. ISBN 978-0-521-09300-2. Chazelle, Bernard (2000). The Discrepancy Method: Randomness and Complexity. Cambridge University Press. ISBN 0-521-77093-9. Doerr, Benjamin (2005). Integral Approximation (PDF) (Habilitation thesis). University of Kiel. OCLC 255383176. Retrieved 20 October 2019. Matoušek, Jiří (1999). Geometric Discrepancy: An Illustrated Guide. Springer. ISBN 3-540-65528-X.