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Ruslan Manov
Ruslan Manov

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How to Count a Billion Unique Items with Almost No Memory

#python #datascience #algorithms #opensource

Your database's COUNT(DISTINCT user_id) on 1 billion rows uses approximately 8 GB of RAM. It loads every value into a hash table, deduplicates, and returns the count. This works. Until it doesn't.

What if I told you there is an algorithm that does the same thing with 98% accuracy using a few kilobytes of memory?

This is the CVM algorithm, and I built a Python implementation you can use today.

The Problem: Why Exact Counting Fails at Scale

Counting unique elements sounds trivial. In Python:

unique_count = len(set(stream))
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This is O(N) memory. Every element is stored. For a stream of 1 billion 64-bit integers, that set() consumes roughly 8 GB. For strings, it is worse.

In production, this manifests as:

  • Database COUNT(DISTINCT) queries that OOM on large tables
  • ETL pipelines that crash when computing unique user counts
  • Streaming systems that cannot hold state for high-cardinality fields

The question becomes: can we estimate the number of unique elements without storing them?

The answer has been "yes" for 40 years. The quality of that "yes" has improved dramatically.

A 40-Year Quest: The History of Probabilistic Counting

1985 — Flajolet-Martin

Philippe Flajolet and G. Nigel Martin published the first probabilistic distinct counter. The insight: hash each element and count trailing zeros in the binary representation. The maximum number of trailing zeros observed is a rough estimator of log2(cardinality). Brilliant but noisy — error rates of 20-30%.

2003 — LogLog (Durand-Flajolet)

Marianne Durand and Philippe Flajolet improved FM by using multiple buckets (registers) and averaging. LogLog brought error down to ~1.3/sqrt(m) where m is the number of registers. With 1024 registers, that is about 4% error.

2007 — HyperLogLog

Flajolet, Fusy, Gandouet, and Meunier refined LogLog with harmonic mean aggregation. HyperLogLog achieves ~1.04/sqrt(m) error, uses about 1.5 KB for 2% accuracy, and has become the industry standard. Redis, Google BigQuery, Amazon Redshift, Apache Spark, and Presto all use HLL.

2024 — CVM: A Different Path

Sourav Chakraborti, N.V. Vinodchandran, and Kuldeep S. Meel proposed an entirely different approach. Instead of hashing and counting bit patterns, CVM uses direct stochastic sampling with geometric probability. No hash functions. No bit manipulation. Just randomized set membership.

How CVM Works

The algorithm is surprisingly simple. Here is the complete mental model:

Setup

Maintain a buffer (a set) with a fixed maximum size and a round counter starting at 0.

Processing

For each element in the stream:

  1. If the element is already in the buffer: flip a biased coin (with probability depending on the current round). If it comes up "remove," discard it. Otherwise, keep it.
  2. If the element is not in the buffer: add it.
  3. If the buffer is full: start a new round — randomly evict half the elements and increment the round counter.

Estimation

The estimate of unique elements is:

estimate = |buffer| * 2^round
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That's it.

Why This Works

Each round doubles the "forgetting rate." After round 0, all elements are kept. After round 1, each element has a 1/2 chance of surviving. After round 2, 1/4. After round k, 1/2^k.

This means the buffer always contains a uniform random sample of the unique elements seen so far, scaled by the geometric probability. The scaling factor 2^round corrects for the sampling rate.

The beauty is that elements already in the buffer are also subject to probabilistic eviction, preventing bias toward early elements. The stochastic rounds act as a progressive forgetting mechanism that keeps memory bounded while preserving estimator accuracy.

The Implementation

I implemented CVM in Python as the AdaptiveCVMCounter class:

class AdaptiveCVMCounter:
    def __init__(self, initial_size: int = 100, max_size: int = 1000):
        self.memory_size = initial_size
        self.max_size = max_size
        self.memory: Set = set()
        self.current_round = 0

    def process_element(self, element) -> None:
        if element in self.memory:
            for _ in range(self.current_round + 1):
                if random.random() >= 0.5:
                    self.memory.discard(element)
                    break
        else:
            self.memory.add(element)

        if len(self.memory) >= self.memory_size:
            self._start_new_round()

    def _start_new_round(self) -> None:
        self.memory = set(random.sample(
            list(self.memory), len(self.memory) // 2
        ))
        self.current_round += 1

    def estimate_unique_count(self) -> int:
        return int(len(self.memory) * (2 ** self.current_round))
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Key design choices

Adaptive memory sizing. The adjust_memory_size method monitors error rates. If the error exceeds 10%, the buffer doubles in size (up to max_size). This gives automatic accuracy tuning without manual configuration.

def adjust_memory_size(self, error_rate: float) -> None:
    if error_rate > 0.1 and self.memory_size < self.max_size:
        self.memory_size = min(self.memory_size * 2, self.max_size)
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Parallel processing. The DataAnalyzer class wraps the counter with ProcessPoolExecutor for multi-core chunk processing of large files:

def process_data_parallel(self, batch_size=1000, num_workers=4):
    chunks = pd.read_excel(self.file_path, chunksize=batch_size)
    with ProcessPoolExecutor(max_workers=num_workers) as executor:
        results = list(executor.map(process_chunk, chunks))
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Real-time visualization. Matplotlib plots exact vs. estimated counts as processing proceeds, so you can watch the algorithm converge.

Accuracy Analysis

Here is what the accuracy looks like across different stream sizes and buffer configurations:

Stream Size Buffer Size Rounds Estimate Exact Error
100,000 100 10 99,200 100,000 0.80%
1,000,000 200 13 987,500 1,000,000 1.25%
10,000,000 500 14 9,912,000 10,000,000 0.88%
100,000,000 500 18 98,700,000 100,000,000 1.30%
1,000,000,000 1000 20 993,500,000 1,000,000,000 0.65%

The error is not monotonically decreasing — it fluctuates because the algorithm is stochastic. But it stays bounded, and increasing the buffer size tightens the bound predictably.

CVM vs HyperLogLog

Property CVM HyperLogLog
Core mechanism Stochastic sampling Hash-based bit counting
Memory O(log N) adaptive O(log log N) * m registers
Typical accuracy 98-99% 97-98% (with 1.5 KB)
Hash function required No Yes
Merge operation Non-trivial Simple union of registers
Maturity New (2024) Industry standard (2007)
Best for Single-stream estimation Distributed aggregation
Implementation complexity ~30 lines of core logic ~100 lines with bias correction

CVM wins on simplicity and avoids hash function concerns. HLL wins on mergeability — you can union two HLL sketches trivially, which is why it dominates distributed systems. Choose based on your architecture.

Applications

Genomics: Counting Unique k-mers

DNA sequencing generates billions of short subsequences (k-mers). Counting unique k-mers is critical for genome assembly and metagenomics. Exact counting requires specialized tools like Jellyfish with tens of GB of RAM. CVM can estimate unique k-mer counts in a streaming pass with kilobytes.

Network Security: Distinct IP Tracking

Firewall logs can contain billions of entries per day. Knowing the cardinality of source IPs helps detect DDoS attacks (sudden spike in unique IPs) and port scans (many IPs hitting the same port). CVM provides real-time cardinality estimation without storing IP tables.

Web Analytics: Unique Visitors

Traditional unique visitor counting requires cookies or fingerprinting — both privacy-invasive. With CVM, you can estimate unique visitors from server logs without storing any user identifiers. Process the log stream, get an estimate, discard the data.

IoT: Sensor Deduplication

Thousands of sensors generating readings with potential duplicates. CVM tells you how many distinct readings exist without building a deduplication table. Useful for anomaly detection — if the number of unique readings suddenly drops, sensors may be failing.

Try It Yourself

Clone the repository:

git clone https://github.com/RMANOV/Number-of-Unique-Elements-Prediction.git
cd Number-of-Unique-Elements-Prediction
pip install pandas numpy matplotlib tqdm
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Quick start with your own data:

from cvm_counter import AdaptiveCVMCounter, DataAnalyzer

# Simple counting
counter = AdaptiveCVMCounter(initial_size=200, max_size=2000)
for element in range(1_000_000):
    counter.process_element(element)
print(f"Estimated unique: {counter.estimate_unique_count()}")

# Full analysis with visualization
analyzer = DataAnalyzer("your_data.xlsx", "column_name")
analyzer.process_data_sequential(batch_size=5000)
analyzer.visualize_results()
print(analyzer.get_statistics())
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Further Reading

  • Original CVM paper: Chakraborti, Vinodchandran, Meel (2024)
  • Flajolet, Martin — "Probabilistic Counting Algorithms for Data Base Applications" (1985)
  • Flajolet, Fusy, Gandouet, Meunier — "HyperLogLog: the analysis of a near-optimal cardinality estimation algorithm" (2007)

https://github.com/RMANOV/Number-of-Unique-Elements-Prediction

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