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name: core-algorithms-system-design description: Mathematical and architectural analysis of Consistent Hashing, Rate Limiting, and Bloom Filters.

System Design Core Algorithms

1. Consistent Hashing and Virtual Nodes

Standard modulo hashing (hash(key) % N) causes catastrophic remapping when a node is added or removed, invalidating (N-1)/N keys. Consistent Hashing maps keys and nodes onto a conceptual ring (e.g., integer range [0, 2^32-1]). A key belongs to the first node encountered clockwise.

• Rebalancing: Node addition/removal only affects its immediate predecessor's keys, remapping only K/N keys.
• Virtual Nodes (vnodes): To solve data skew, each physical node is assigned multiple hash points (vnodes) on the ring. The variance in load distribution decreases logarithmically with the number of vnodes.

2. Rate Limiting Algorithms

• Token Bucket: Focuses on burst capacity. A bucket holds up to B tokens. Tokens are added at a constant rate R. A request consumes N tokens. If empty, the request is rejected. It allows bursts up to the bucket capacity while sustaining long-term rate R.
• Leaky Bucket: Focuses on smoothing traffic. Requests enter a queue (bucket) of size Q. A background process dequeues and processes them at a constant rate R. Burst traffic fills the queue; excess is dropped (traffic shaping).

3. Bloom Filters: Probabilistic Verification

A Bloom Filter is a space-efficient probabilistic data structure used to test set membership.

• Mechanics: An array of m bits initialized to 0. k independent hash functions map an item to k bit indices, setting them to 1.
• Verification: To check membership, hash the item k times. If all k bits are 1, it is probably in the set. If any bit is 0, it is definitely not.
• Mathematics: The probability of a false positive p depends on bits m, inserted elements n, and hash functions k. Optimal k = (m/n) * ln(2). Highly utilized in LSM-Trees to skip reading SSTables that do not contain a key.

%%{init: {"theme": "default", "flowchart": {"useMaxWidth": true}}}%%
flowchart TD
    A[Core Algorithms]
    
    A --> B[Consistent Hashing]
    B --> B1[Hash Ring Topology]
    B --> B2[Clockwise Key Lookup]
    B --> B3[Virtual Nodes: Load Balancing]
    
    A --> C[Rate Limiting]
    C --> C1[Token Bucket: Burst Allowed]
    C --> C2[Leaky Bucket: Traffic Smoothing]
    
    A --> D[Bloom Filters]
    D --> D1[m bit array & k hash functions]
    D --> D2[Zero False Negatives]
    D --> D3[Optimal k = m/n * ln2]