When both chunk sampling and column sampling are subject to the possibility of sampling redundancy (i.e., multiple samples targeting the same column), the sampling probabilities and detection analysis need to incorporate this overlap explicitly. Here is a revised analysis considering redundancy for both scenarios:

Key Setup

  1. Data Dimensions:
  2. Adversarial Assumptions:
  3. Sampling Scenarios:

Probability of Redundancy

General Sampling Overlap

For both chunk and column sampling, overlap arises because of random sampling with replacement. The probability of sampling $r$ unique columns out of $s$ samples can be modeled using the following approach:

  1. Expected Number of Unique Columns Sampled:

  2. Distribution of Unique Samples: The number of unique columns sampled follows a distribution when sampling with replacement:

    $P(r \text{ unique columns}) \approx \binom{2n}{r} \cdot \frac{\binom{s}{r} \cdot r!}{(2n)^s}$

Detection Probability Analysis

Chunk-by-Chunk Sampling

  1. Probability of Sampling an Available Chunk: The probability that a sampled chunk belongs to an available column is:

    $P(\text{chunk available}) = \frac{2n - m}{2n}.$

  2. Probability of Sampling $s$ Available Chunks: The probability that all $s$ samples are from available columns:

    $P(\text{all chunks available}) = \left( \frac{2n - m}{2n} \right)^s$.

  3. Probability of Detecting Unavailability: At least one chunk must be from an unavailable column for detection:

    $P_{\text{detect, chunk}} = 1 - P(\text{all chunks available}) = 1 - \left( \frac{2n - m}{2n} \right)^s$.

Column-by-Column Sampling