636 lines
No EOL
23 KiB
Markdown
636 lines
No EOL
23 KiB
Markdown
# HyperLogLog-Based Tombstone Garbage Collection for Distributed Systems
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## Abstract
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When synchronizing records in a distributed network, deletion presents a fundamental challenge. If nodes simply delete their local copies, other nodes may resynchronize the original data, reverting the deletion. This occurs due to non-simultaneous events between nodes or nodes temporarily disconnecting and reconnecting with outdated state. The traditional solution creates "tombstone" records that persist after deletion to prevent resurrection of deleted data.
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While effective, this approach requires every node to indefinitely maintain an ever-growing collection of tombstone records. Typically, after an arbitrarily large time period, tombstones are assumed safe to clear since no rogue nodes should retain the original data.
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This paper presents a methodology using the HyperLogLog algorithm to estimate how many nodes have received a record, comparing this estimate against the count of nodes that have received the corresponding tombstone. This enables pruning tombstones across the network to a minimal set of "keeper" nodes, extending the viable tombstone retention period while significantly reducing storage overhead.
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## 1. Introduction
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Distributed systems face an inherent tension between data consistency and storage efficiency when handling deletions. Traditional tombstone-based approaches guarantee correctness but impose unbounded storage growth. Time-based garbage collection (GC) offers storage efficiency but risks data resurrection if stale nodes reconnect after the GC window.
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This paper introduces a probabilistic approach using HyperLogLog (HLL) cardinality estimation[^1] to achieve both goals: safe garbage collection that provably prevents resurrection while minimizing the number of nodes that must retain tombstones.
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[^1]: [TODO: Cite HyperLogLog paper - Flajolet et al., "HyperLogLog: the analysis of a near-optimal cardinality estimation algorithm"]
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### 1.1 Core Concept
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The algorithm operates in three phases:
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```mermaid
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sequenceDiagram
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participant A as Node A
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participant B as Node B
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participant C as Node C
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Note over A,C: Phase 1: Record Propagation
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A->>B: record + recordHLL
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B->>A: update recordHLL estimate
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B->>C: record + recordHLL
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Note over A,C: Phase 2: Tombstone Propagation
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A->>A: Create tombstone with recordHLL and delete record
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C->>B: update recordHLL estimate
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A->>B: tombstone + tombstoneHLL + recordHLL
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B->>B: tombstone updated with new recordHLL and delete record
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B->>C: tombstone + tombstoneHLL + recordHLL
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Note over A,C: Phase 3: Keeper Election and tombstone garbage collection
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C->>C: tombstoneCount >= recordCount, become keeper and deletes record
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C->>B: updates with node tombstone count estimate
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B->>B: sees higher estimate, step down and garbage collects its own tombstone record
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B->>A: update connected node with tombstoneHLL
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A->>A: garbage collects its own tombstone record
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```
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**Phase 1**: Records propagate through the network via gossip, with each node adding itself to the record's HLL. Nodes then talk between themselves to slowly turn local estimates for the records count into global ones.
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**Phase 2**: When deletion occurs, the deleting node creates a tombstone containing a copy of the record's HLL as the target count. The tombstone propagates similarly, with nodes adding themselves to the tombstone's HLL. During propagation, the target recordHLL is updated to the highest estimate encountered.
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**Phase 3**: When a node detects that `tombstoneCount >= recordCount`, it becomes a "keeper" responsible for continued propagation. As keepers communicate, those with lower estimates step down and garbage collect, converging toward a minimal keeper set.
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## 2. Data Model
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Records and tombstones are maintained as separate entities with distinct tracking mechanisms:
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```ts
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interface DataRecord<Data> {
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readonly id: string;
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readonly data: Data;
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readonly recordHLL: HyperLogLog; // Tracks nodes that have received this record
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}
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interface Tombstone {
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readonly id: string;
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readonly recordHLL: HyperLogLog; // Target count: highest observed record distribution
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readonly tombstoneHLL: HyperLogLog; // Tracks nodes that have received the tombstone
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}
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```
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## 3. Algorithm
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### 3.1 Record Creation and Distribution
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When a node creates or receives a record, it adds itself to the record's HLL:
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```ts
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const createRecord = <Data>(id: string, data: Data, nodeId: string): DataRecord<Data> => ({
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id,
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data,
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recordHLL: hllAdd(createHLL(), nodeId),
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});
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const receiveRecord = <Data>(
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node: NodeState<Data>,
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incoming: DataRecord<Data>
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): NodeState<Data> => {
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// Reject records that have already been deleted
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if (node.tombstones.has(incoming.id)) {
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return node;
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}
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const existing = node.records.get(incoming.id);
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const updatedRecord: DataRecord<Data> = existing
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? { ...existing, recordHLL: hllAdd(hllMerge(existing.recordHLL, incoming.recordHLL), node.id) }
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: { ...incoming, recordHLL: hllAdd(hllClone(incoming.recordHLL), node.id) };
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const newRecords = new Map(node.records);
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newRecords.set(incoming.id, updatedRecord);
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return { ...node, records: newRecords };
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};
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```
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### 3.2 Tombstone Creation
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When deleting a record, a node creates a tombstone containing a copy of the record's HLL as the initial target count:
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```ts
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const createTombstone = <Data>(record: DataRecord<Data>, nodeId: string): Tombstone => ({
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id: record.id,
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recordHLL: hllClone(record.recordHLL),
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tombstoneHLL: hllAdd(createHLL(), nodeId),
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});
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```
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### 3.3 Garbage Collection Status Check
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The core decision logic determines whether a node should become a keeper, step down, or continue as-is:
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```ts
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const checkGCStatus = (
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tombstone: Tombstone,
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incomingTombstoneEstimate: number | null,
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myTombstoneEstimateBeforeMerge: number,
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myNodeId: string,
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senderNodeId: string | null
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): { shouldGC: boolean; stepDownAsKeeper: boolean } => {
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const targetCount = hllEstimate(tombstone.recordHLL);
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const isKeeper = myTombstoneEstimateBeforeMerge >= targetCount;
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if (isKeeper) {
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// Keeper step-down logic:
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// If incoming tombstone has reached the target count, compare estimates.
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// If incoming estimate >= my estimate before merge, step down.
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// Use node ID as tie-breaker: higher node ID steps down when estimates are equal.
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if (incomingTombstoneEstimate !== null && incomingTombstoneEstimate >= targetCount) {
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if (myTombstoneEstimateBeforeMerge < incomingTombstoneEstimate) {
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return { shouldGC: true, stepDownAsKeeper: true };
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}
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// Tie-breaker: if estimates are equal, the lexicographically higher node ID steps down
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if (myTombstoneEstimateBeforeMerge === incomingTombstoneEstimate &&
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senderNodeId !== null && myNodeId > senderNodeId) {
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return { shouldGC: true, stepDownAsKeeper: true };
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}
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}
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return { shouldGC: false, stepDownAsKeeper: false };
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}
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// Not yet a keeper - will become one if tombstone count reaches target after merge
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return { shouldGC: false, stepDownAsKeeper: false };
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};
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```
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### 3.4 Tombstone Reception and Processing
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```mermaid
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graph TD
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A[Receive tombstone deletion message] --> B{Do I have<br/>this record?}
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B -->|No| C[Ignore: record not found]
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B -->|Yes| D[Merge HLLs and select<br/>highest record estimate]
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D --> E{Am I already a keeper?<br/>my tombstone count >= target}
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E -->|Yes| F{Is incoming tombstone<br/>count higher than mine?}
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F -->|Yes| G[Step down as keeper:<br/>delete tombstone]
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F -->|No| H{Same count but<br/>sender has lower node ID?}
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H -->|Yes| G
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H -->|No| I[Remain keeper:<br/>update tombstone]
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E -->|No| J{Does my tombstone<br/>count reach target?}
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J -->|Yes| K[Become keeper:<br/>store tombstone]
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J -->|No| L[Store tombstone<br/>but not keeper yet]
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G --> M[Forward tombstone to peers]
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I --> M
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K --> M
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L --> M
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```
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The complete tombstone reception handler:
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```ts
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const receiveTombstone = <Data>(
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node: NodeState<Data>,
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incoming: Tombstone,
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senderNodeId: string
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): NodeState<Data> => {
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// Don't accept tombstones for unknown records
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const record = node.records.get(incoming.id);
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if (!record) {
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return node;
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}
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const existing = node.tombstones.get(incoming.id);
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// Merge tombstone HLLs and add self
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const mergedTombstoneHLL = existing
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? hllAdd(hllMerge(existing.tombstoneHLL, incoming.tombstoneHLL), node.id)
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: hllAdd(hllClone(incoming.tombstoneHLL), node.id);
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// Select the best (highest estimate) record HLL as target count
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// This ensures we use the most complete view of record distribution
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let bestRecordHLL = incoming.recordHLL;
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if (existing?.recordHLL) {
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bestRecordHLL = hllEstimate(existing.recordHLL) > hllEstimate(bestRecordHLL)
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? existing.recordHLL
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: bestRecordHLL;
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}
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if (hllEstimate(record.recordHLL) > hllEstimate(bestRecordHLL)) {
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bestRecordHLL = hllClone(record.recordHLL);
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}
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const updatedTombstone: Tombstone = {
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id: incoming.id,
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tombstoneHLL: mergedTombstoneHLL,
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recordHLL: bestRecordHLL,
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};
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const myEstimateBeforeMerge = existing ? hllEstimate(existing.tombstoneHLL) : 0;
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const gcStatus = checkGCStatus(
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updatedTombstone,
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hllEstimate(incoming.tombstoneHLL),
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myEstimateBeforeMerge,
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node.id,
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senderNodeId
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);
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// Always delete the record when we have a tombstone
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const newRecords = new Map(node.records);
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newRecords.delete(incoming.id);
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if (gcStatus.stepDownAsKeeper) {
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// Step down: delete both record and tombstone
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const newTombstones = new Map(node.tombstones);
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newTombstones.delete(incoming.id);
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return { ...node, records: newRecords, tombstones: newTombstones };
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}
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const newTombstones = new Map(node.tombstones);
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newTombstones.set(incoming.id, updatedTombstone);
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return { ...node, records: newRecords, tombstones: newTombstones };
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};
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```
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### 3.5 Cascading Step-Down via Forwarding
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When a keeper steps down, it immediately forwards the tombstone to all connected peers, creating a cascade effect that rapidly eliminates redundant keepers:
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```ts
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const forwardTombstoneToAllPeers = <Data>(
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network: NetworkState<Data>,
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forwardingNodeId: string,
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tombstone: Tombstone,
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excludePeerId?: string
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): NetworkState<Data> => {
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const forwardingNode = network.nodes.get(forwardingNodeId);
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if (!forwardingNode) return network;
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let newNodes = new Map(network.nodes);
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for (const peerId of forwardingNode.peerIds) {
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if (peerId === excludePeerId) continue;
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const peer = newNodes.get(peerId);
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if (!peer || !peer.records.has(tombstone.id)) continue;
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const updatedPeer = receiveTombstone(peer, tombstone, forwardingNodeId);
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newNodes.set(peerId, updatedPeer);
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// If this peer also stepped down, recursively forward
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if (!updatedPeer.tombstones.has(tombstone.id) && peer.tombstones.has(tombstone.id)) {
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const result = forwardTombstoneToAllPeers({ nodes: newNodes }, peerId, tombstone, forwardingNodeId);
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newNodes = new Map(result.nodes);
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}
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}
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return { nodes: newNodes };
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};
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```
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## 4. Design Rationale
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### 4.1 Why Propagate the Record HLL with Tombstones?
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Without a shared target count, each node would compare against its own local recordHLL estimate, leading to premature garbage collection:
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```mermaid
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graph LR
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subgraph Problem: Without Shared Target
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A1["Node A: recordHLL=2"] -->|gossip| B1["Node B: recordHLL=2"]
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B1 --> C1["Both estimate 2 nodes have record"]
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C1 --> D1["tombstoneHLL reaches 2"]
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D1 --> E1["GC triggers prematurely!"]
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E1 --> F1["Node C still has record, resurrection"]
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end
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```
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By propagating the recordHLL with the tombstone and always keeping the highest estimate encountered, all nodes converge on a safe target count. During propagation, if a node has a more complete view of record distribution (higher HLL estimate), that becomes the new target for all subsequent nodes.
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### 4.2 Why Dynamic Keeper Election?
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A fixed originator-as-keeper design creates a single point of failure. If the originator goes offline, tombstone propagation halts.
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Dynamic election allows any node to become a keeper when it detects `tombstoneCount >= recordCount`. Multiple keepers provide redundancy during network partitions or node failures.
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### 4.3 Why Keeper Step-Down?
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Without step-down logic, every node eventually becomes a keeper (since they all eventually observe the threshold condition). This defeats the purpose of garbage collection.
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Step-down creates convergence toward a minimal keeper set:
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```mermaid
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graph TD
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subgraph Keeper Convergence Over Time
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T0["t=0: 0 keepers"]
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T1["t=1: 5 keepers<br/>(first nodes to detect threshold)"]
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T2["t=2: 3 keepers<br/>(2 stepped down after seeing higher estimates)"]
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T3["t=3: 1-2 keepers<br/>(most informed nodes remain)"]
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end
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T0 --> T1 --> T2 --> T3
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```
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### 4.4 Why Node ID Tie-Breaker?
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When HLL estimates converge (all nodes have similar tombstoneHLL values due to full propagation), no node can have a strictly higher estimate. Without a tie-breaker, keepers with equal estimates would never step down.
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The lexicographic node ID comparison ensures deterministic convergence: when two keepers with equal estimates communicate, the one with the higher node ID steps down. This guarantees eventual convergence to a single keeper per connected component.
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### 4.5 Why Forward on Step-Down?
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Without forwarding, keepers only step down when randomly selected for gossip - a slow process. With aggressive forwarding, a stepping-down keeper immediately propagates the "winning" tombstone to all neighbors, creating a cascade effect that rapidly eliminates redundant keepers.
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## 5. Evaluation
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### 5.1 Experimental Setup
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We implemented a discrete-event simulation to evaluate the algorithm under various network conditions. The simulation models:
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- **Gossip protocol**: Each round, every node with a record or tombstone randomly selects one peer and exchanges state
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- **HLL precision**: 10 bits (1024 registers, ~1KB per HLL)
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- **Convergence criteria**: Records deleted, followed by 100 additional rounds for keeper convergence
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### 5.2 Test Scenarios
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#### 5.2.1 Single Node Deletion
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**Scenario**: A single node creates a record, propagates it through gossip, then initiates deletion.
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```mermaid
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graph TD
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subgraph Network Topology 15 nodes 40 percent connectivity
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N0((node-0<br/>originator))
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N1((node-1))
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N2((node-2))
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N3((node-3))
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N4((node-4))
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N5((node-5))
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N6((node-6))
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N7((node-7))
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N0 --- N1
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N0 --- N3
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N1 --- N2
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N1 --- N4
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N2 --- N5
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N3 --- N4
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N3 --- N6
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N4 --- N5
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N5 --- N7
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N6 --- N7
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end
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```
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**Protocol**:
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1. Node-0 creates record and propagates for 20 rounds
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2. Node-0 creates tombstone and initiates deletion
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3. Simulation runs until convergence
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**Results** (averaged over 50 trials):
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| Metric | Value |
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|--------|-------|
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| Nodes | 15 per trial (750 total) |
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| Records deleted | 100% success |
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| Rounds to delete records | 11 |
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| Total rounds (including convergence) | 121 |
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| Final tombstones | 116 (~15.5% of nodes) |
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**Analysis**: Record deletion completes rapidly (11 rounds). Tombstone keeper count converges to approximately 2-3 keepers per trial, demonstrating effective garbage collection while maintaining redundancy.
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#### 5.2.2 Early Tombstone Creation
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**Scenario**: Tombstone created before record fully propagates, testing the algorithm's handling of partial record distribution.
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```mermaid
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sequenceDiagram
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participant N0 as Node-0
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participant N1 as Node-1
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participant N2 as Node-2
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participant Nx as Nodes 3-19
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Note over N0,Nx: Record only partially propagated
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N0->>N1: record (round 1)
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N1->>N2: record (round 2)
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N2->>N0: record (round 3)
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Note over N0: Create tombstone after only 3 rounds
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N0->>N1: tombstone
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N1->>N2: tombstone
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Note over Nx: Most nodes never receive record
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```
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**Results**:
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| Metric | Value |
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|--------|-------|
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| Nodes | 20 |
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| Records deleted | Yes |
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| Rounds to delete records | 10 |
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| Total rounds | 120 |
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| Final tombstones | 3 (15% of nodes) |
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**Analysis**: Even with partial record propagation, the algorithm correctly handles deletion. The propagated recordHLL accurately captures the distribution, updating as the tombstone encounters nodes with more complete views. Tombstones converge to nodes that actually received the record.
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#### 5.2.3 Bridged Network (Two Clusters)
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**Scenario**: Two densely-connected clusters joined by a single bridge node, simulating common real-world topologies.
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```mermaid
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graph TD
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subgraph Cluster A 15 nodes
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A0((A-0<br/>bridge))
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A1((A-1))
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A2((A-2))
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A3((A-3))
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A0 --- A1
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A0 --- A2
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A1 --- A2
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A1 --- A3
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A2 --- A3
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end
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subgraph Cluster B 15 nodes
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B0((B-0<br/>bridge))
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B1((B-1))
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B2((B-2))
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B3((B-3))
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B0 --- B1
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B0 --- B2
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B1 --- B2
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B1 --- B3
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B2 --- B3
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end
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A0 ===|single bridge| B0
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```
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**Results**:
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| Metric | Cluster A | Cluster B | Total |
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|--------|-----------|-----------|-------|
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| Nodes | 15 | 15 | 30 |
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| Records deleted | Yes | Yes | Yes |
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| Rounds to delete | - | - | 10 |
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| Final tombstones | 4 | 3 | 7 (23%) |
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**Analysis**: The single-bridge topology creates a natural partition point. Each cluster independently elects keepers, resulting in 2-4 keepers per cluster. This provides fault tolerance - if the bridge fails, each cluster retains tombstones independently.
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#### 5.2.4 Concurrent Tombstones
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**Scenario**: Multiple nodes simultaneously initiate deletion of the same record, simulating concurrent delete operations.
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```mermaid
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sequenceDiagram
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participant N0 as Node-0
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participant N5 as Node-5
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participant N10 as Node-10
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participant Others as Other Nodes
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Note over N0,Others: Record fully propagated (30 rounds)
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par Concurrent deletion
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N0->>N0: Create tombstone
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N5->>N5: Create tombstone
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N10->>N10: Create tombstone
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end
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Note over N0,Others: Three tombstones propagate and merge
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N0->>Others: tombstone (from N0)
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N5->>Others: tombstone (from N5)
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N10->>Others: tombstone (from N10)
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|
|
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Note over N0,Others: HLLs merge, keepers converge
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|
```
|
|
|
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**Results**:
|
|
|
|
| Metric | Value |
|
|
|--------|-------|
|
|
| Nodes | 20 |
|
|
| Concurrent deleters | 3 |
|
|
| Records deleted | Yes |
|
|
| Rounds to delete | 10 |
|
|
| Final tombstones | 2 (10% of nodes) |
|
|
|
|
**Analysis**: The algorithm handles concurrent tombstone creation gracefully. Multiple tombstones merge via HLL union operations, and keeper election converges as normal. The final keeper count (2) is actually lower than single-deleter scenarios, likely due to faster HLL convergence from multiple sources.
|
|
|
|
#### 5.2.5 Network Partition and Heal
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|
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**Scenario**: Network partitions after record propagation, tombstone created in one partition, then network heals.
|
|
|
|
```mermaid
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|
sequenceDiagram
|
|
participant CA as Cluster A
|
|
participant Bridge as Bridge
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|
participant CB as Cluster B
|
|
|
|
Note over CA,CB: Phase 1: Record propagates to all nodes
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|
CA->>Bridge: record
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|
Bridge->>CB: record
|
|
|
|
Note over CA,CB: Phase 2: Network partitions
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|
Bridge--xCB: connection lost
|
|
|
|
Note over CA: Cluster A creates tombstone
|
|
CA->>CA: tombstone propagates within A
|
|
Note over CB: Cluster B still has record
|
|
|
|
Note over CA,CB: Phase 3: Network heals
|
|
Bridge->>CB: tombstone propagates to B
|
|
CB->>CB: record deleted, keepers elected
|
|
```
|
|
|
|
**Results**:
|
|
|
|
| Metric | Cluster A | Cluster B | Total |
|
|
|--------|-----------|-----------|-------|
|
|
| Nodes | 10 | 10 | 20 |
|
|
| Records deleted | Yes | Yes | Yes |
|
|
| Rounds to delete | - | - | 10 |
|
|
| Total rounds (partition + heal) | - | - | 720 |
|
|
| Final tombstones | 3 | 2 | 5 (25%) |
|
|
|
|
**Analysis**: The extended total rounds (720) includes the partition period where only Cluster A processes the tombstone. Upon healing, Cluster B rapidly receives and processes the tombstone. Each cluster maintains independent keepers, providing partition tolerance.
|
|
|
|
#### 5.2.6 Sparse Network
|
|
|
|
**Scenario**: Low connectivity (15%) network, testing algorithm behavior under challenging propagation conditions.
|
|
|
|
```mermaid
|
|
graph TD
|
|
subgraph Sparse Network 25 nodes 15 percent connectivity
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|
N0((0)) --- N3((3))
|
|
N3 --- N7((7))
|
|
N7 --- N12((12))
|
|
N0 --- N5((5))
|
|
N5 --- N9((9))
|
|
N9 --- N15((15))
|
|
N12 --- N18((18))
|
|
N18 --- N22((22))
|
|
N1((1)) --- N4((4))
|
|
N4 --- N8((8))
|
|
N2((2)) --- N6((6))
|
|
N6 --- N11((11))
|
|
N11 --- N16((16))
|
|
N16 --- N20((20))
|
|
N20 --- N24((24))
|
|
end
|
|
|
|
style N0 fill:#f96
|
|
style N24 fill:#9f9
|
|
```
|
|
|
|
**Results** (averaged over 20 trials):
|
|
|
|
| Metric | Value |
|
|
|--------|-------|
|
|
| Nodes | 25 per trial (500 total) |
|
|
| Connectivity | 15% |
|
|
| Records deleted | 100% success |
|
|
| Rounds to delete | 13 |
|
|
| Total rounds | 123 |
|
|
| Final tombstones | 102 (~20.4% of nodes) |
|
|
|
|
**Analysis**: Sparse networks require more rounds for propagation (13 vs. 10-11 for denser networks) and retain more keepers (~20% vs. ~15%). The higher keeper retention provides additional redundancy appropriate for networks where nodes may have limited connectivity.
|
|
|
|
### 5.3 Summary of Results
|
|
|
|
| Scenario | Nodes | Deletion Rounds | Keeper % | Key Insight |
|
|
|----------|-------|-----------------|----------|-------------|
|
|
| Single Node Deletion | 15 | 11 | 15.5% | Baseline performance |
|
|
| Early Tombstone | 20 | 10 | 15% | Handles partial propagation |
|
|
| Bridged Network | 30 | 10 | 23% | Independent keepers per cluster |
|
|
| Concurrent Tombstones | 20 | 10 | 10% | Faster convergence with multiple sources |
|
|
| Partition and Heal | 20 | 10 | 25% | Partition-tolerant |
|
|
| Sparse Network | 25 | 13 | 20.4% | Graceful degradation |
|
|
|
|
### 5.4 Key Findings
|
|
|
|
1. **Consistent deletion**: Records are deleted within 10-13 gossip rounds across all scenarios
|
|
2. **Effective GC**: Tombstones converge to 10-25% of nodes as keepers
|
|
3. **Topology adaptation**: Bridged and partitioned networks maintain ~1-4 keepers per cluster
|
|
4. **Graceful degradation**: Lower connectivity increases keeper retention, providing appropriate redundancy
|
|
5. **Concurrent safety**: Multiple simultaneous deleters do not cause conflicts
|
|
|
|
## 6. Trade-offs
|
|
|
|
| Aspect | Impact |
|
|
|--------|--------|
|
|
| **Memory** | ~1KB per tombstone (HLL at precision 10) |
|
|
| **Bandwidth** | HLLs transmitted with each gossip message (~2KB per tombstone message) |
|
|
| **Latency** | GC delayed until keeper convergence (~100 rounds after deletion) |
|
|
| **Consistency** | Eventual - temporary resurrection attempts are blocked but logged |
|
|
|
|
## 7. Properties
|
|
|
|
The algorithm provides the following guarantees:
|
|
|
|
- **Safety**: Tombstones are never prematurely garbage collected. A tombstone is only deleted when the node has received confirmation (via HLL estimates) that the tombstone has propagated to at least as many nodes as received the original record.
|
|
|
|
- **Liveness**: Keepers eventually step down, enabling garbage collection. The tie-breaker mechanism ensures convergence even when HLL estimates are identical.
|
|
|
|
- **Fault tolerance**: No single point of failure. Multiple keepers provide redundancy, and any keeper can propagate the tombstone.
|
|
|
|
- **Convergence**: Keeper count monotonically decreases over time within each connected component.
|
|
|
|
## 8. Conclusion
|
|
|
|
This paper presented a HyperLogLog-based approach to tombstone garbage collection in distributed systems. By tracking record and tombstone propagation through probabilistic cardinality estimation, the algorithm enables safe garbage collection while reducing storage overhead by 75-90%.
|
|
|
|
The simulation results demonstrate consistent behavior across diverse network topologies and failure scenarios, with records deleted in 10-13 gossip rounds and tombstones converging to 10-25% of nodes as keepers. The algorithm gracefully handles partial propagation, network partitions, and concurrent deletions.
|
|
|
|
Future work may explore adaptive HLL precision based on network size, integration with vector clocks for stronger consistency guarantees, and optimization of the keeper convergence rate.
|
|
|
|
## References
|
|
|
|
A working simulation implementing this algorithm is available at `simulations/hyperloglog-tombstone/simulation.ts`. |