Probabilistic Programming and Data Structures

  1. Cardinality Estimation
    1. Problem Formulation
      1. Count-Distinct Problem
        1. Streaming Constraints
          1. Accuracy Requirements
          2. Linear Counting
            1. Bit Vector Approach
              1. Hash Function Requirements
                1. Estimation Formula
                  1. Error Analysis
                    1. Memory Requirements
                    2. Probabilistic Counting
                      1. Flajolet-Martin Algorithm
                        1. Bit Pattern Analysis
                          1. Geometric Distribution
                            1. Variance Reduction
                            2. LogLog Counting
                              1. Bucket-Based Approach
                                1. Leading Zero Counting
                                  1. Harmonic Mean Estimation
                                    1. Bias Correction
                                    2. HyperLogLog
                                      1. Algorithm Description
                                        1. Hash Value Processing
                                          1. Bucket Assignment
                                            1. Maximum Leading Zeros
                                            2. Estimation Process
                                              1. Harmonic Mean Formula
                                                1. Bias Correction Factors
                                                  1. Small Range Corrections
                                                  2. Error Bounds
                                                    1. Standard Error Analysis
                                                      1. Confidence Intervals
                                                      2. Practical Considerations
                                                        1. Parameter Selection
                                                          1. Memory Usage
                                                            1. Merging Operations
                                                            2. Extensions
                                                              1. HyperLogLog++
                                                                1. Sparse Representation
                                                                  1. Compressed Counters

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                                                              7. Membership and Set Operations

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                                                              9. Frequency Estimation and Heavy Hitters