Sketching Algorithms

Core concepts and principles

Sketching algorithms maintain a compact summary (sketch) of data that supports specific query types while adhering to several key properties:

1. Bounded memory usage - The sketch size remains constant regardless of input data volume
2. Single-pass processing - Data is processed sequentially without requiring multiple scans
3. Mergeable summaries - Sketches can be combined to represent aggregated data
4. Probabilistic guarantees - Results have proven error bounds with high probability

The mathematical foundation relies on random projections and hash functions to compress high-dimensional data into low-dimensional sketches while preserving essential properties for specific queries.

Common sketch types and applications

Count-Min Sketch

Used for frequency estimation, the Count-Min Sketch maintains multiple hash tables to track approximate counts:

$\text{estimate}(x) = \min_{i=1}^d h_i(x)$

Where $h_i$ represents independent hash functions mapping items to counters.

HyperLogLog

Estimates cardinality (unique values) using bit patterns:

$E = \alpha_m \cdot m^2 \cdot \left(\sum_{j=1}^m 2^{-M_j}\right)^{-1}$

Where $m$ is the number of registers and $M_j$ is the maximum observed leading zeros in register $j$.

Implementation considerations

Memory-accuracy tradeoffs

The relationship between sketch size and accuracy typically follows:

$\text{Error} \leq \frac{c}{\sqrt{m}}$

Where $m$ is the memory allocation and $c$ is a constant depending on the specific sketch type.

Parallel processing

Modern implementations leverage vectorized operations and parallel processing:

struct Sketch {
    registers: Vec<u32>,
    hash_functions: Vec<HashFn>,
}

impl Sketch {
    fn update(&mut self, value: &[u8]) {
        for hash_fn in &self.hash_functions {
            let idx = hash_fn(value) % self.registers.len();
            self.registers[idx] += 1;
        }
    }
}

Applications in time-series systems

Real-time analytics

Sketches enable efficient computation of:

• Unique user counts
• Heavy hitter detection
• Frequency estimation
• Quantile approximation

Anomaly detection

Anomaly detection systems use sketches to maintain distribution profiles and identify deviations efficiently.

Performance monitoring

Real-time analytics platforms use sketches to track system metrics with minimal overhead:

-- Example using Count-Min Sketch for approximate counts
SELECT approx_count_distinct(user_id)
FROM events
WHERE timestamp > now() - interval '1 hour'
GROUP BY category;

Integration with time-series databases

Query optimization

Query planners use sketches to estimate:

• Cardinality for join optimization
• Value distributions
• Data skew

Storage efficiency

Sketches complement compression algorithms by providing compact summaries for specific query patterns.

Best practices and limitations

When to use sketches

• High-volume streaming data
• Approximate answers are acceptable
• Memory constraints are strict
• Real-time processing requirements

When to avoid sketches

• Exact answers required
• Small datasets
• Complex multi-dimensional queries
• Highly variable data distributions

Future developments

Emerging trends include:

• Learned sketches using machine learning
• Quantum-resistant hash functions
• Adaptive sketch sizing
• Multi-dimensional sketch structures

Modern sketching algorithms continue to evolve, particularly in areas of:

• Distributed systems
• Privacy-preserving analytics
• Edge computing
• Real-time machine learning