This paper (2024) presents Cedar, AWS's new authorization policy language. By providing a clear declarative way to manage access control policies, Cedar addresses the common limitations of embedding authorization logic directly into application code: problems with correctness, auditing, and maintainence. Cedar introduces a domain-specific language (DSL) to express policies that are separate from application code, and improves readability and manageability. In that sense, this is like aspect-oriented programming but for authorization policy.

The language is designed with four main objectives: expressiveness, performance, safety, and analyzability. Cedar balances these goals by supporting role-based (RBAC), attribute-based (ABAC), and relationship-based (ReBAC) access control models.

Policy Structure and Evaluation

Cedar policies consist of three primary components: effect, scope, and conditions. The effect can either be "permit" or "forbid", defining whether access is granted or denied. The scope specifies the principal (user), action, and resource. Conditions provide additional constraints using entity attributes.

Cedar ensures safety by applying a deny-by-default model. If no "permit" policy is applicable, access is denied. Additionally, "forbid" policies always take precedence over "permit" policies.

A schema defines entity types, attributes, and valid actions, and Cedar's policy validator uses optional typing and static capabilities to catch potential errors, like accessing non-existent attributes. Another key feature is policy templates, which allow developers to define reusable policy patterns with placeholders. These placeholders are instantiated with specific entities or attributes at runtime, reducing redundancy and simplifying policy management.

Symbolic Analysis and Proofs

Cedar supports deeper policy analysis through symbolic compilation. Symbolic compilation reduces policies to SMT (Satisfiability Modulo Theories) formulas, enabling automated reasoning about policy behavior. This allows checking for policy equivalence, identifying inconsistencies, and verifying security invariants.

Checking policy equivalence is particularly useful for ensuring that policy refactoring or updates do not introduce unintended behavior. By comparing two versions of a policy set, Cedar can determine if they produce identical authorization decisions for all possible requests. This is crucial for organizations with frequent policy updates to ensure no permissions are inadvertently granted or revoked.

By compiling to SMT, Cedar leverages the rapid advancements in SMT solvers over the past two decades. Improvements in solver algorithms, better heuristics for decision procedures, and more efficient memory management have significantly enhanced SMT solver performance. Tools like Z3 and CVC5 are now capable of solving complex logical formulas quickly, making real-time policy analysis feasible. These advancements enable Cedar to perform sophisticated policy checks with minimal overhead.

Cedar is formalized in the Lean programming language and uses its proof assistant to establish key properties like correctness of the authorization algorithm, sound slicing, and validation soundness. The compiler's encoding is proved to be decidable, sound, and complete. This result, the first of its kind for a non-trivial policy language, is made possible by Cedar's careful control over expressiveness and by leveraging invariants ensured by Cedar's policy validator.

Implementation and Evaluation

Cedar is implemented in Rust and open-sourced at https://github.com/cedar-policy/. The implementation is extensively tested using differential random testing to ensure correctness.

Cedar was evaluated against two prominent open-source, general-purpose authorization languages, OpenFGA and Rego, using three example sets of policies. The evaluation demonstrated that Cedar's policy slicing significantly reduces evaluation time. For large policy sets, Cedar achieved a 28.7x to 35.2x speedup over OpenFGA and a 42.8x to 80.8x speedup over Rego.

Unlike OpenFGA and Rego, which show scaling challenges with increased input sizes, Cedar maintains consistently low evaluation latency. In a simulated Google Drive authorization scenario, Cedar handled requests in a median time of 4.0 microseconds (𝜇s) with 5 Users/Groups/Documents/Folders, increasing only to 5.0𝜇s with 50 Users/Groups/Documents/Folders. Similarly, in a GitHub-like authorization scenario, Cedar maintained a median performance of around 11.0𝜇s across all input ranges. Even at the 99th percentile (p99), Cedar's response times remained low, with under 10𝜇s for Google Drive and under 20𝜇s for GitHub. Further validating its real-world applicability, Cedar is deployed at scale within Amazon Web Services, providing secure and efficient authorization for large and complex systems.

In a previous blog post, I explained how MongoDB TTL indexes work and their optimization to avoid fragmentation during scans. However, I didn’t cover the details of on-disk storage. A recent Reddit question is the occasion to explore this aspect further.

Reproducible example

Here is a small program that inserts documents in a loop, with a timestamp and some random text:

// random string to be not too friendly with compression
function getRandomString(length) {
  const characters = 'ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789';
  let result = '';
  const charactersLength = characters.length;
  for (let i = 0; i < length; i++) {
    result += characters.charAt(Math.floor(Math.random() * charactersLength));
  }
  return result;
}
// MongoDB loop for inserting documents with a random string
while (true) {
  const doc = { text: getRandomString(1000), ts: new Date() };
  db.ttl.insertOne(doc);
  insertedCount++;
}

Before executing this, I ran a background function to display statistics every minute, including the number of records, memory usage, and disk size.

// Prints stats every minute
 let insertedCount = 0;
 let maxThroughput = 0;
 let maxCollectionCount = 0;
 let maxCollectionSize = 0;
 let maxStorageSize = 0;
 let maxTotalIndexSize = 0;
 setInterval(async () => {
      const stats = await db.ttl.stats();
      const throughput = insertedCount / 10; // assuming measure over 10 seconds
      const collectionCount = stats.count;
      const collectionSizeMB = stats.size / 1024 / 1024;
      const storageSizeMB = stats.storageSize / 1024 / 1024;
      const totalIndexSizeMB = stats.totalIndexSize / 1024 / 1024;
      maxThroughput = Math.max(maxThroughput, throughput);
      maxCollectionCount = Math.max(maxCollectionCount, collectionCount);
      maxCollectionSize = Math.max(maxCollectionSize, collectionSizeMB);
      maxStorageSize = Math.max(maxStorageSize, storageSizeMB);
      maxTotalIndexSize = Math.max(maxTotalIndexSize, totalIndexSizeMB);
      console.log(`Collection Name: ${stats.ns}
   Throughput:        ${throughput.toFixed(0).padStart(10)} docs/min (max: ${maxThroughput.toFixed(0)} docs/min)
   Collection Size:   ${collectionSizeMB.toFixed(0).padStart(10)} MB (max: ${maxCollectionSize.toFixed(0)} MB)
   Number of Records: ${collectionCount.toFixed(0).padStart(10)}     (max: ${maxCollectionCount.toFixed(0)} docs)
   Storage Size:      ${storageSizeMB.toFixed(0).padStart(10)}    MB (max: ${maxStorageSize.toFixed(0)} MB)
   Total Index Size:  ${totalIndexSizeMB.toFixed(0).padStart(10)} MB (max: ${maxTotalIndexSize.toFixed(0)} MB)`);
      insertedCount = 0;
 }, 60000); // every minute

I created the collection with a TTL index, which automatically expires data older than five minutes:

// TTL expire after 5 minutes
db.ttl.drop();
db.ttl.createIndex({ ts: 1 }, { expireAfterSeconds: 300 });

I let this running to see how the storage size evolves. Note that this was run in on MongoDB 7.0.16 (without the auto compact job that appeared in 8.0).

Output after 3 hours

The consistent insert rate, combined with TTL expiration, keeps the number of documents in the collection relatively stable. Deletions occur every minute, ensuring that the overall document count remains consistent.

I observe the same from the statistics I log every minute:

The storage size also remains constant, at 244MB for the table and 7MB for the indexes. The size of files has increased for the first 6 minutes and then remained constant:

This is sufficient to show that the deletion and insertion do not have a fragmentation effect that would require additional consideration. About 25% is marked as available for reuse and is effectively reused.

It is possible to force a compaction, to temporarily reclaim more space, but it is not necessary:

Collection Name: test.ttl
   Throughput:              3286 docs/min (max: 3327 docs/min)
   Collection Size:          170 MB (max: 198 MB)
   Number of Records:     171026     (max: 198699 docs)
   Storage Size:             244    MB (max: 244 MB)
   Total Index Size:           7 MB (max: 7 MB)

Collection Name: test.ttl
   Throughput:              3299 docs/min (max: 3327 docs/min)
   Collection Size:          170 MB (max: 198 MB)
   Number of Records:     170977     (max: 198699 docs)
   Storage Size:             244    MB (max: 244 MB)
   Total Index Size:           6 MB (max: 7 MB)

Collection Name: test.ttl
   Throughput:              3317 docs/min (max: 3327 docs/min)
   Collection Size:          170 MB (max: 198 MB)
   Number of Records:     170985     (max: 198699 docs)
   Storage Size:             244    MB (max: 244 MB)
   Total Index Size:           6 MB (max: 7 MB)

test> db.runCommand({ compact: 'ttl' });
{ bytesFreed: 49553408, ok: 1 }

Collection Name: test.ttl
   Throughput:              1244 docs/min (max: 3327 docs/min)
   Collection Size:          150 MB (max: 198 MB)
   Number of Records:     150165     (max: 198699 docs)
   Storage Size:             197    MB (max: 244 MB)
   Total Index Size:           6 MB (max: 7 MB)
Collection Name: test.ttl
   Throughput:              3272 docs/min (max: 3327 docs/min)
   Collection Size:          149 MB (max: 198 MB)
   Number of Records:     149553     (max: 198699 docs)
   Storage Size:             203    MB (max: 244 MB)
   Total Index Size:           6 MB (max: 7 MB)

While this has reduced storage, it eventually returns to its normal volume. It's typical for a B-tree to maintain some free space, which helps to minimize frequent space allocations and reclaim.

Here is a focus when I've run manual compaction:

Conclusion

TTL deletion makes space available for reuse instead of reclaiming it immediately, but this doesn't increase the fragmentation. This space is reused automatically to maintain a total size proportional to the document count, with a constant free space of 25% in my case, to minimize frequent allocations typical of B-Tree implementations.
The TTL mechanism operates autonomously, requiring no manual compaction. If you have any doubt, monitor it. MongoDB offers statistics to compare logical and physical sizes at both the MongoDB layer and WiredTiger storage.

In a previous blog post, I explained how MongoDB TTL indexes work and their optimization to avoid fragmentation during scans. However, I didn’t cover the details of on-disk storage. A recent Reddit question is the occasion to explore this aspect further.

Reproducible example

Here is a small program that inserts documents in a loop, with a timestamp and some random text:

// random string to be not too friendly with compression
function getRandomString(length) {
  const characters = 'ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789';
  let result = '';
  const charactersLength = characters.length;
  for (let i = 0; i < length; i++) {
    result += characters.charAt(Math.floor(Math.random() * charactersLength));
  }
  return result;
}
// MongoDB loop for inserting documents with a random string
while (true) {
  const doc = { text: getRandomString(1000), ts: new Date() };
  db.ttl.insertOne(doc);
  insertedCount++;
}

Before executing this, I ran a background function to display statistics every minute, including the number of records, memory usage, and disk size.

// Prints stats every minute
 let insertedCount = 0;
 let maxThroughput = 0;
 let maxCollectionCount = 0;
 let maxCollectionSize = 0;
 let maxStorageSize = 0;
 let maxTotalIndexSize = 0;
 setInterval(async () => {
      const stats = await db.ttl.stats();
      const throughput = insertedCount / 10; // assuming measure over 10 seconds
      const collectionCount = stats.count;
      const collectionSizeMB = stats.size / 1024 / 1024;
      const storageSizeMB = stats.storageSize / 1024 / 1024;
      const totalIndexSizeMB = stats.totalIndexSize / 1024 / 1024;
      maxThroughput = Math.max(maxThroughput, throughput);
      maxCollectionCount = Math.max(maxCollectionCount, collectionCount);
      maxCollectionSize = Math.max(maxCollectionSize, collectionSizeMB);
      maxStorageSize = Math.max(maxStorageSize, storageSizeMB);
      maxTotalIndexSize = Math.max(maxTotalIndexSize, totalIndexSizeMB);
      console.log(`Collection Name: ${stats.ns}
   Throughput:        ${throughput.toFixed(0).padStart(10)} docs/min (max: ${maxThroughput.toFixed(0)} docs/min)
   Collection Size:   ${collectionSizeMB.toFixed(0).padStart(10)} MB (max: ${maxCollectionSize.toFixed(0)} MB)
   Number of Records: ${collectionCount.toFixed(0).padStart(10)}     (max: ${maxCollectionCount.toFixed(0)} docs)
   Storage Size:      ${storageSizeMB.toFixed(0).padStart(10)}    MB (max: ${maxStorageSize.toFixed(0)} MB)
   Total Index Size:  ${totalIndexSizeMB.toFixed(0).padStart(10)} MB (max: ${maxTotalIndexSize.toFixed(0)} MB)`);
      insertedCount = 0;
 }, 60000); // every minute

I created the collection with a TTL index, which automatically expires data older than five minutes:

// TTL expire after 5 minutes
db.ttl.drop();
db.ttl.createIndex({ ts: 1 }, { expireAfterSeconds: 300 });

I let this running to see how the storage size evolves. Note that this was run in on MongoDB 7.0.16 (without the auto compact job that appeared in 8.0).

Output after 3 hours

The consistent insert rate, combined with TTL expiration, keeps the number of documents in the collection relatively stable. Deletions occur every minute, ensuring that the overall document count remains consistent.

I observe the same from the statistics I log every minute:

The storage size also remains constant, at 244MB for the table and 7MB for the indexes. The size of files has increased for the first 6 minutes and then remained constant:

This is sufficient to show that the deletion and insertion do not have a fragmentation effect that would require additional consideration. About 25% is marked as available for reuse and is effectively reused.

It is possible to force a compaction, to temporarily reclaim more space, but it is not necessary:

Collection Name: test.ttl
   Throughput:              3286 docs/min (max: 3327 docs/min)
   Collection Size:          170 MB (max: 198 MB)
   Number of Records:     171026     (max: 198699 docs)
   Storage Size:             244    MB (max: 244 MB)
   Total Index Size:           7 MB (max: 7 MB)

Collection Name: test.ttl
   Throughput:              3299 docs/min (max: 3327 docs/min)
   Collection Size:          170 MB (max: 198 MB)
   Number of Records:     170977     (max: 198699 docs)
   Storage Size:             244    MB (max: 244 MB)
   Total Index Size:           6 MB (max: 7 MB)

Collection Name: test.ttl
   Throughput:              3317 docs/min (max: 3327 docs/min)
   Collection Size:          170 MB (max: 198 MB)
   Number of Records:     170985     (max: 198699 docs)
   Storage Size:             244    MB (max: 244 MB)
   Total Index Size:           6 MB (max: 7 MB)

test> db.runCommand({ compact: 'ttl' });
{ bytesFreed: 49553408, ok: 1 }

Collection Name: test.ttl
   Throughput:              1244 docs/min (max: 3327 docs/min)
   Collection Size:          150 MB (max: 198 MB)
   Number of Records:     150165     (max: 198699 docs)
   Storage Size:             197    MB (max: 244 MB)
   Total Index Size:           6 MB (max: 7 MB)
Collection Name: test.ttl
   Throughput:              3272 docs/min (max: 3327 docs/min)
   Collection Size:          149 MB (max: 198 MB)
   Number of Records:     149553     (max: 198699 docs)
   Storage Size:             203    MB (max: 244 MB)
   Total Index Size:           6 MB (max: 7 MB)

While this has reduced storage, it eventually returns to its normal volume. It's typical for a B-tree to maintain some free space, which helps to minimize frequent space allocations and reclaim.

Here is a focus when I've run manual compaction:

Conclusion

TTL deletion makes space available for reuse instead of reclaiming it immediately, but this doesn't increase the fragmentation. This space is reused automatically to maintain a total size proportional to the document count, with a constant free space of 25% in my case, to minimize frequent allocations typical of B-Tree implementations.
The TTL mechanism operates autonomously, requiring no manual compaction. If you have any doubt, monitor it. MongoDB offers statistics to compare logical and physical sizes at both the MongoDB layer and WiredTiger storage.

This paper (2021) dives deeper in the Parallel Raft protocol introduced with PolarFS.

PolarFS (VLDB'18) is a distributed file system with ultra-low latency and high availability, developed by Alibaba. Its variant of the Raft consensus protocol, ParallelRaft,  relaxes Raft's strict serialization constraints. ParallelRaft allows state machines to commit and execute log entries out of order.

This study provides a formal specification of ParallelRaft in TLA+, proves its correctness, and identifies consistency issues caused by ghost log entries. To address these issues, the refined ParallelRaft-CE protocol limits parallelism during leader transitions.

Introduction

Raft is a tight-ass. It enforces sequential commitment and execution of log entries. No funny business. Multi-Paxos is a bit rebellious. It allows out-of-order commitment, but it still insists on ordered execution. In order not to leave any performance on the table, PolarFS's ParallelRaft rebels completely: it not only allows out-of-order commitment, but out-of-order execution as well. 

How does ParallelRaft keep consistency of state machine replication across replicas? The trick is to permit out-of-order execution if commands are conflict-free. It is the oldest trick in the book, the generalized Paxos idea. Each command contains a Logical Block Address (LBA). Commands accessing non-overlapping LBAs are conflict-free and can execute in parallel (i.e., out of order). Overlapping commands must execute in log order.

Ok, still this is a big claim. Concurrency is a wild beast! Did you expect this to be simple? What about leader turnovers? There is a lot to consider there. This paper complains that the original PolarFS paper was too quick to call it a day without specifying the protocol completely or providing an opensource implementation. They suspect that, in the presence of leader turnovers, ParallelRaft protocol is vulnerable to inconsistency induced by ghost log entries.

To investigate, they introduce an intermediate protocol, ParallelRaft-SE (Sequential Execution), which allows out-of-order commitment but mandates sequential execution. The point of ParallelRaft-SE is to establish a refinement mapping to Multi-Paxos. They then relax it to get ParallelRaft-CE (Concurrent Execution), which supports both out-of-order commitment and execution. They show that ParallelRaft-CE is prone to state inconsistency due to ghost log entries, and they propose to mitigate this issue using stricter leader election rules and log recovery mechanisms, forbidding out-of-order execution during leader takeover.

Out-of-Order Executions and Ghost Log Entries

A key challenge of out-of-order execution is determining conflicts when log entries are missing. How do you track conflicts with holes in the log? ParallelRaft introduces a Look Behind Buffer (LBF) that tracks the LBAs of up to K preceding log entries. If no holes exceeding K exist, conflict detection is feasible.

However, the paper argues ghost log entries can cause inconsistency as in Figure 1.

In phase 1, leader s1 creates entries 1-4, with only entries 1-2 being committed before s1 fails. When s3 becomes leader in phase 2, it doesn't receive s1's uncommitted entries 3-4 during recovery. S3 then adds its own entries 3-6, commits entry 6 (which sets y←2), and executes this operation before failing.

In phase 3, new leader s2 unexpectedly receives entry 4 from the now-recovered s1 during recovery. After committing this entry, s2 must execute it (setting y←3). This operation should have executed before entry 6 according to ordering rules, but s3 has already executed y←2.

This "ghost log entries" phenomenon--where s1's uncommitted entries disappear during s3's leadership only to reappear during s2's leadership-- creates inconsistent execution order. The actual execution conflicts with what should have happened based on log position, leading to incorrect conflict determination and system inconsistency.

The proposed ParallelRaft-CE patch

ParallelRaft-CE addresses ghost entries by refining ParallelRaft-SE. The key idea is to track the generation moment of log entries by the sync number, and barrier entries from old terms.

The protocol creates a clear boundary between terms by allowing concurrent operations only for log entries with the same term, while enforcing sequential processing for entries from different terms. It tracks entry generation through the sync number and adds a "date" variable to each log entry (similar to Paxos proposal numbers) to determine recency.

During recovery, the new leader collects log entries from a majority of nodes and selects the latest entries based on their "date" values. This ensures that already committed entries remain in the log, previously uncommitted entries are either properly committed or discarded, and no ghost entries can persist undetected across term changes. By introducing this controlled concurrency model where the sync number is always less than or equal to a node's term, ParallelRaft-CE effectively uses leader election as a sequentialization bottleneck to clean up potential inconsistencies.

ParallelRaft-CE has three key components:

1. Log Synchronization Mechanism: Followers accept only log entries matching their sync number (equivalent to the current term). Entries from different terms are rejected.

2. Leader Election Mechanism: Similar to Raft, but with an additional check to ensure no ghost entries exist. Leaders are responsible for reconciling logs and eliminating inconsistencies.

3. Log Recovery Mechanism: During recovery, the new leader collects logs from a majority of nodes and applies a conflict resolution process. Uncommitted entries from the previous term are either committed or discarded. 

In sequential Raft, leader transition is simple, as the selected leader is already caught up: Logs of nodes in Raft do not have holes, and the log matching property between nodes is guaranteed. But in ParallelRaft, since there are holes, leader election needs to be augmented with recovery as in Paxos leader transitions. This is the price to pay for being relaxed in out-of-order commit and execution.

The ParallelRaft-CE protocol limits out-of-order execution to entries within the same term. Log entries from different terms follow a sequential recovery and commitment process to  eliminate ghost entries and guarantee state consistency.

This article examines PostgreSQL operations that benefit from GIN indexes, as listed in Built-in GIN Operator Classes.

I created a table containing one million rows, plus three additional rows relevant to my queries. The goal is to demonstrate the benefits of using indexes, allowing for efficient data retrieval without the need to read all rows.

Sample table in PostgreSQL

Here is a table that simulates a document collection, with all attributes in a JSONB column:

CREATE TABLE data (
    id SERIAL PRIMARY KEY,
    details JSONB
);

INSERT INTO data (details)
SELECT jsonb_build_object(
    'stuff', md5(random()::text)
)
FROM generate_series(1, 1000000);

ANALYZE data;

INSERT INTO data (details) VALUES
('{"name": "Alice", "age": 30, "tags": ["developer", "engineer"]}'),
('{"name": "Bob", "age": 25, "tags": ["designer"]}'),
('{"name": "Carol", "age": 40, "tags": ["developer", "manager"]}');

Sample collection on MongoDB

I am creating the same data in a MongoDB collection:

const bulk = db.data.initializeUnorderedBulkOp();
for (let i = 0; i < 1000000; i++) {
    bulk.insert({
        stuff: Math.random().toString(36).substring(2, 15)
    });
}
bulk.execute();

db.data.insertMany([
    { "name": "Alice", "age": 30, "tags": ["developer", "engineer"] },
    { "name": "Bob", "age": 25, "tags": ["designer"] },
    { "name": "Carol", "age": 40, "tags": ["developer", "manager"] }
]);

@> (jsonb, jsonb): Contains

The following queries search for data where the tag array contains a value:

PostgreSQL:

SELECT * FROM data WHERE details @> '{"tags": ["developer"]}'
;

MongoDB:

db.data.find({ tags: { $all: ["developer"] } })
;

Without an index, this runs a SeqScan on PostgreSQL:

postgres=# explain (analyze, costs off, buffers, serialize text)
SELECT * FROM data WHERE details @> '{"tags": ["developer"]}'
;
                           QUERY PLAN
----------------------------------------------------------------
 Seq Scan on data (actual time=144.002..144.004 rows=2 loops=1)
   Filter: (details @> '{"tags": ["developer"]}'::jsonb)
   Rows Removed by Filter: 1000001
   Buffers: shared hit=5715 read=4595
 Planning Time: 0.065 ms
 Serialization: time=0.008 ms  output=1kB  format=text
 Execution Time: 144.236 ms

and a COLLSCAN on MongoDB:

test> print( 
db.data.find({ tags: { $all: ["developer"] } })
.explain('executionStats').executionStats 
);
{
  executionSuccess: true,
  nReturned: 2,
  executionTimeMillis: 437,
  totalKeysExamined: 0,
  totalDocsExamined: 1000003,
  executionStages: {
    stage: 'COLLSCAN',
    filter: { tags: { '$eq': 'developer' } },
    nReturned: 2,
    executionTimeMillisEstimate: 391,
    works: 1000004,
    advanced: 2,
    direction: 'forward',
    docsExamined: 1000003
  }
}

I create a jsonb_path_ops GIN index on PostgreSQL:

CREATE INDEX idx_details ON data USING GIN (details jsonb_path_ops)
;

and a multi-key index sparse index on MongoDB for the array of tags:

db.data.createIndex({ tags: 1 } , { sparse: true })
;

PostgreSQL uses the GIN index to find the two index entries, though a Bitmap Index Scan, and then the two rows where the Recheck Cond didn't have to filter more:

postgres=# explain (analyze, costs off, buffers, serialize text)
SELECT * FROM data WHERE details @> '{"tags": ["developer"]}'
;
                                    QUERY PLAN                                    
----------------------------------------------------------------------------------
 Bitmap Heap Scan on data (actual time=0.019..0.021 rows=2 loops=1)
   Recheck Cond: (details @> '{"tags": ["developer"]}'::jsonb)
   Heap Blocks: exact=1
   Buffers: shared hit=5
   ->  Bitmap Index Scan on idx_details (actual time=0.010..0.010 rows=2 loops=1)
         Index Cond: (details @> '{"tags": ["developer"]}'::jsonb)
         Buffers: shared hit=4
 Planning:
   Buffers: shared hit=1
 Planning Time: 0.079 ms
 Serialization: time=0.005 ms  output=1kB  format=text
 Execution Time: 0.041 ms

MongoDB efficiently reads two index entries and retrieves two documents without the need of bitmaps. This method preserves the index order, which is advantageous when handling multiple rows.

test> print( 
db.data.find({ tags: { $all: ["developer"] } })
. explain('executionStats').executionStats 
);
{
  executionSuccess: true,
  nReturned: 2,
  executionTimeMillis: 0,
  totalKeysExamined: 2,
  totalDocsExamined: 2,
  executionStages: {
    stage: 'FETCH',
    nReturned: 2,
    executionTimeMillisEstimate: 0,
    works: 3,
    advanced: 2,
    docsExamined: 2,
    inputStage: {
      stage: 'IXSCAN',
      nReturned: 2,
      executionTimeMillisEstimate: 0,
      works: 3,
      advanced: 2,
      keyPattern: { tags: 1 },
      indexName: 'tags_1',
      isMultiKey: true,
      multiKeyPaths: { tags: [ 'tags' ] },
      isSparse: true,
      isPartial: false,
      direction: 'forward',
      indexBounds: { tags: [ '["developer", "developer"]' ] },
      keysExamined: 2,
      seeks: 1,
      dupsTested: 2
    }
  }
}

@? (jsonb, jsonpath): JSON Path Match

The following query searches for developers, as well as non developers younger than 35:

On PostgreSQL:

SELECT * FROM data 
 WHERE details @? '$?(@.tags[*] == "developer" || @.age < 35)'
;

On MongoDB:

db.data.find({
  $or: [
    { tags: { $elemMatch: { $eq: "developer" } } },
    { age: { $lt: 35 } }
  ]
});

Without an additional index on "age", MongoDB chooses a COLLSCAN:

test> print( 
db.data.find({
  $or: [
    { tags: { $elemMatch: { $eq: "developer" } } },
    { age: { $lt: 35 } }
  ]
}).explain('executionStats').executionStats 
);
{
  executionSuccess: true,
  nReturned: 3,
  executionTimeMillis: 585,
  totalKeysExamined: 0,
  totalDocsExamined: 1000003,
  executionStages: {
    isCached: false,
    stage: 'SUBPLAN',
    nReturned: 3,
    executionTimeMillisEstimate: 547,
    works: 1000004,
    advanced: 3,
    needTime: 1000000,
    inputStage: {
      stage: 'COLLSCAN',
      filter: {
        '$or': [
          { tags: { '$elemMatch': [Object] } },
          { age: { '$lt': 35 } }
        ]
      },
      nReturned: 3,
      executionTimeMillisEstimate: 517,
      works: 1000004,
      advanced: 3,
      needTime: 1000000,
      direction: 'forward',
      docsExamined: 1000003
    }
  }
}

PostgreSQL uses the GIN index but not efficiently as it reads all index entries to remove them later by recheck:

postgres=# explain (analyze, costs off, buffers, serialize text)
SELECT * FROM data 
WHERE details @? '$?(@.tags[*] == "developer" || @.age < 35)'
;
                                         QUERY PLAN                                          
---------------------------------------------------------------------------------------------
 Bitmap Heap Scan on data (actual time=582.323..582.327 rows=3 loops=1)
   Recheck Cond: (details @? '$?(@."tags"[*] == "developer" || @."age" < 35)'::jsonpath)
   Rows Removed by Index Recheck: 1000000
   Heap Blocks: exact=10310
   Buffers: shared hit=14703
   ->  Bitmap Index Scan on idx_details (actual time=123.755..123.755 rows=1000003 loops=1)
         Index Cond: (details @? '$?(@."tags"[*] == "developer" || @."age" < 35)'::jsonpath)
         Buffers: shared hit=4393
 Planning:
   Buffers: shared hit=1
 Planning Time: 0.117 ms
 Serialization: time=0.009 ms  output=1kB  format=text
 Execution Time: 582.575 ms

The solution, in both databases, is to add an index on "age".

An expression-based index in PostgreSQL:

CREATE INDEX idx_age ON data ( ((details->>'age')::int) )
;

A regular index on MongoDB, but sparse as I don't need to index missing values:

db.data.createIndex({ age: 1 }, { sparse: true} );

Here is the new execution plan in MongoDB which can combine the multi-key index on "tags" and the regular index on "age":

print( 
db.data.find({
  $or: [
    { tags: { $elemMatch: { $eq: "developer" } } },
    { age: { $lt: 35 } }
  ]
}).explain('executionStats').executionStats 
);
{
  executionSuccess: true,
  nReturned: 3,
  executionTimeMillis: 0,
  totalKeysExamined: 4,
  totalDocsExamined: 5,
  executionStages: {
    isCached: false,
    stage: 'SUBPLAN',
    nReturned: 3,
    executionTimeMillisEstimate: 0,
    works: 6,
    advanced: 3,
    inputStage: {
      stage: 'FETCH',
      nReturned: 3,
      executionTimeMillisEstimate: 0,
      works: 6,
      advanced: 3,
      docsExamined: 3,
      alreadyHasObj: 2,
      inputStage: {
        stage: 'OR',
        nReturned: 3,
        executionTimeMillisEstimate: 0,
        works: 6,
        advanced: 3,
        dupsTested: 4,
        dupsDropped: 1,
        inputStages: [
          {
            stage: 'FETCH',
            filter: { '$or': [Array] },
            nReturned: 2,
            executionTimeMillisEstimate: 0,
            works: 3,
            advanced: 2,
            docsExamined: 2,
            alreadyHasObj: 0,
            inputStage: {
              stage: 'IXSCAN',
              nReturned: 2,
              executionTimeMillisEstimate: 0,
              works: 3,
              advanced: 2,
              keyPattern: [Object],
              indexName: 'tags_1',
              isMultiKey: true,
              multiKeyPaths: [Object],
              isUnique: false,
              isSparse: true,
              isPartial: false,
              indexVersion: 2,
              direction: 'forward',
              indexBounds: [Object],
              keysExamined: 2,
              seeks: 1,
              dupsTested: 2,
              dupsDropped: 0
            }
          },
          {
            stage: 'IXSCAN',
            nReturned: 2,
            executionTimeMillisEstimate: 0,
            works: 3,
            advanced: 2,
            keyPattern: { age: 1 },
            indexName: 'age_1',
            isMultiKey: false,
            multiKeyPaths: { age: [] },
            isUnique: false,
            isSparse: true,
            isPartial: false,
            indexVersion: 2,
            direction: 'forward',
            indexBounds: { age: [Array] },
            keysExamined: 2,
            seeks: 1,
            dupsTested: 0,
            dupsDropped: 0
          }
        ]
      }
    }
  }
}

I've detailed this OR expansion in a previous blog post:

PostgreSQL cannot do the same and you need to write the query differently:

postgres=# explain (analyze, costs off, buffers, serialize text)
-- Query for using GIN index on "details" column
SELECT * FROM data 
WHERE details @? '$?(@.tags[*] == "developer")'
UNION
-- Query for using B-tree index on "age" column
SELECT * FROM data 
WHERE (details->>'age')::int < 35
;
                                          QUERY PLAN                                          
----------------------------------------------------------------------------------------------
 HashAggregate (actual time=0.134..0.220 rows=3 loops=1)
   Group Key: data.id, data.details
   Batches: 1  Memory Usage: 1561kB
   Buffers: shared hit=9
   ->  Append (actual time=0.023..0.035 rows=4 loops=1)
         Buffers: shared hit=9
         ->  Bitmap Heap Scan on data (actual time=0.022..0.024 rows=2 loops=1)
               Recheck Cond: (details @? '$?(@."tags"[*] == "developer")'::jsonpath)
               Heap Blocks: exact=1
               Buffers: shared hit=5
               ->  Bitmap Index Scan on idx_details (actual time=0.012..0.012 rows=2 loops=1)
                     Index Cond: (details @? '$?(@."tags"[*] == "developer")'::jsonpath)
                     Buffers: shared hit=4
         ->  Bitmap Heap Scan on data data_1 (actual time=0.009..0.010 rows=2 loops=1)
               Recheck Cond: (((details ->> 'age'::text))::integer < 35)
               Heap Blocks: exact=1
               Buffers: shared hit=4
               ->  Bitmap Index Scan on idx_age (actual time=0.008..0.008 rows=2 loops=1)
                     Index Cond: (((details ->> 'age'::text))::integer < 35)
                     Buffers: shared hit=3
 Planning:
   Buffers: shared hit=1
 Planning Time: 0.160 ms
 Serialization: time=0
                                    
                                    
                                    

                                        by Franck Pachot