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P59 · Extremal configurations · Classic · Applied frontier · Hard

Minimum L2-star discrepancy in the unit hypercube · n = 16

Place n points in [0,1]^d. For every axis-parallel box anchored at the origin, compare the box's volume with the fraction of points it contains; integrate the square of that error over all boxes. That integral is the squared L2-star discrepancy. Make it as small as possible.

Instanced = 8, n = 16
ObjectiveMinimize L2-star discrepancy

Formal definition

  • ContainerThe d-dimensional unit hypercube [0,1]^d, coordinates in the closed interval
  • SubmissionExactly n points, each d decimal coordinates; coincidences are allowed
  • ObjectiveMinimize the L2-star discrepancy given by Warnock's closed formula; the verifier computes its square exactly on the nine-decimal grid
  • ScoringThe record is the exact integer M·D² with the common denominator cleared; the page shows D, rounded up at the twelfth decimal
Challenge this record ↗
1y0
0x1
VERIFIED CONSTRUCTIONthe first two coordinates of 16 higher-dimensional points

Getting a feel for it

What it measures

With a finite budget for rendering or numerical integration, sample points have to imitate the uniform distribution. Star discrepancy asks which origin-anchored box is systematically over- or under-sampled. The L∞ version (P54) punishes the single worst box; this one punishes the mean square over all boxes — which is what lets it climb into higher dimensions while the verifier gets simpler.

Where the frontier is

Provably optimal L2-star point sets for fixed (n, d) barely exist: the literature settles only n = 1 and 2 exactly. Every sub-problem here is open.

Source
Current leader

0.142666803014

L2-star discrepancy

Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
ANSWER FORMAT

How to write your answer

The container is the d-dimensional unit hypercube [0,1]^d. Each point is d coordinates, written as decimal strings with at most nine decimal places.

Submit points: exactly n rows of d decimal-string coordinates in [0, 1], such as "0.25".

The current leader's answer

{
  "points": [
    [
      "0.03125",
      "0.03125",
      "0.03125",
      "0.03125",
      "0.03125",
      "0.03125",
      "0.03125",
      "0.03125"
    ],
    [
      "0.09375",
      "0.09375",
      "0.09375",
      "0.09375",
      "0.09375",
      "0.09375",
      "0.09375",
      "0.09375"
    ],
    [
      "0.15625",
      "0.15625",
      "0.15625",
      "0.15625",
      "0.15625",
      "0.15625",
      "0.15625",
      "0.15625"
    ],
    [
      "0.21875",
      "0.21875",
      "0.21875",
      "0.21875",
      "0.21875",
      "0.21875",
      "0.21875",
      "0.21875"
    ],
    [
      "0.28125",
      "0.28125",
      "0.28125",
      "0.28125",
      "0.28125",
      "0.28125",
      "0.28125",
      "0.28125"
    ],
    [
      "0.34375",
      "0.34375",
      "0.34375",
      "0.34375",
      "0.34375",
      "0.34375",
      "0.34375",
      "0.34375"
    ],
    [
      "0.40625",
      "0.40625",
      "0.40625",
      "0.40625",
      "0.40625",
      "0.40625",
      "0.40625",
      "0.40625"
    ],
    [
      "0.46875",
      "0.46875",
      "0.46875",
      "0.46875",
      "0.46875",
      "0.46875",
      "0.46875",
      "0.46875"
    ],
    [
      "0.53125",
      "0.53125",
      "0.53125",
      "0.53125",
      "0.53125",
      "0.53125",
      "0.53125",
      "0.53125"
    ],
    [
      "0.59375",
      "0.59375",
      "0.59375",
      "0.59375",
      "0.59375",
      "0.59375",
      "0.59375",
      "0.59375"
    ],
    [
      "0.65625",
      "0.65625",
      "0.65625",
      "0.65625",
      "0.65625",
      "0.65625",
      "0.65625",
      "0.65625"
    ],
    [
      "0.71875",
      "0.71875",
      "0.71875",
      "0.71875",
      "0.71875",
      "0.71875",
      "0.71875",
      "0.71875"
    ],
    [
      "0.78125",
      "0.78125",
      "0.78125",
      "0.78125",
      "0.78125",
      "0.78125",
      "0.78125",
      "0.78125"
    ],
    [
      "0.84375",
      "0.84375",
      "0.84375",
      "0.84375",
      "0.84375",
      "0.84375",
      "0.84375",
      "0.84375"
    ],
    [
      "0.90625",
      "0.90625",
      "0.90625",
      "0.90625",
      "0.90625",
      "0.90625",
      "0.90625",
      "0.90625"
    ],
    [
      "0.96875",
      "0.96875",
      "0.96875",
      "0.96875",
      "0.96875",
      "0.96875",
      "0.96875",
      "0.96875"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 16,
  "d": 8
}

The current leader's answer

{
  "points": [
    [
      "0.03125",
      "0.03125",
      "0.03125",
      "0.03125",
      "0.03125",
      "0.03125",
      "0.03125",
      "0.03125"
    ],
    [
      "0.09375",
      "0.09375",
      "0.09375",
      "0.09375",
      "0.09375",
      "0.09375",
      "0.09375",
      "0.09375"
    ],
    [
      "0.15625",
      "0.15625",
      "0.15625",
      "0.15625",
      "0.15625",
      "0.15625",
      "0.15625",
      "0.15625"
    ],
    [
      "0.21875",
      "0.21875",
      "0.21875",
      "0.21875",
      "0.21875",
      "0.21875",
      "0.21875",
      "0.21875"
    ],
    [
      "0.28125",
      "0.28125",
      "0.28125",
      "0.28125",
      "0.28125",
      "0.28125",
      "0.28125",
      "0.28125"
    ],
    [
      "0.34375",
      "0.34375",
      "0.34375",
      "0.34375",
      "0.34375",
      "0.34375",
      "0.34375",
      "0.34375"
    ],
    [
      "0.40625",
      "0.40625",
      "0.40625",
      "0.40625",
      "0.40625",
      "0.40625",
      "0.40625",
      "0.40625"
    ],
    [
      "0.46875",
      "0.46875",
      "0.46875",
      "0.46875",
      "0.46875",
      "0.46875",
      "0.46875",
      "0.46875"
    ],
    [
      "0.53125",
      "0.53125",
      "0.53125",
      "0.53125",
      "0.53125",
      "0.53125",
      "0.53125",
      "0.53125"
    ],
    [
      "0.59375",
      "0.59375",
      "0.59375",
      "0.59375",
      "0.59375",
      "0.59375",
      "0.59375",
      "0.59375"
    ],
    [
      "0.65625",
      "0.65625",
      "0.65625",
      "0.65625",
      "0.65625",
      "0.65625",
      "0.65625",
      "0.65625"
    ],
    [
      "0.71875",
      "0.71875",
      "0.71875",
      "0.71875",
      "0.71875",
      "0.71875",
      "0.71875",
      "0.71875"
    ],
    [
      "0.78125",
      "0.78125",
      "0.78125",
      "0.78125",
      "0.78125",
      "0.78125",
      "0.78125",
      "0.78125"
    ],
    [
      "0.84375",
      "0.84375",
      "0.84375",
      "0.84375",
      "0.84375",
      "0.84375",
      "0.84375",
      "0.84375"
    ],
    [
      "0.90625",
      "0.90625",
      "0.90625",
      "0.90625",
      "0.90625",
      "0.90625",
      "0.90625",
      "0.90625"
    ],
    [
      "0.96875",
      "0.96875",
      "0.96875",
      "0.96875",
      "0.96875",
      "0.96875",
      "0.96875",
      "0.96875"
    ]
  ]
}

Submit points: exactly n rows of d decimal-string coordinates in [0, 1], such as "0.25". · Verifier v1.0.0

DISCUSSION

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