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We've learned the mechanics of chaining and discussed its tradeoffs qualitatively. Now it's time for the mathematics—the rigorous analysis that allows you to predict and optimize hash table performance.
Understanding performance implications isn't just academic. When you're designing a system, you need to answer questions like:
This page gives you the mathematical tools to answer these questions with confidence.
By the end of this page, you will master time complexity analysis for all chaining operations, understand the load factor's mathematical impact on performance, learn expected chain length distributions, and know optimal sizing and resizing strategies.
Let's establish precise time complexity for each operation. We'll use these variables:
The key insight is that α (load factor) determines average chain length, which in turn determines average operation time.
| Operation | Best Case | Average Case | Worst Case | Space |
|---|---|---|---|---|
| Insert | O(1) | O(1) * | O(n) | O(1) |
| Lookup (hit) | O(1) | O(1 + α) | O(n) | O(1) |
| Lookup (miss) | O(1) | O(1 + α) | O(n) | O(1) |
| Delete | O(1) | O(1 + α) | O(n) | O(1) |
| Resize | — | O(n) | O(n) | O(n) |
Insert is O(1) if we insert at the head of the chain without checking for duplicates. If we must check for existing keys first (standard behavior for maps), insert becomes O(1 + α) like lookup, since we traverse the chain to check.
Breaking Down the Complexity:
In practice, with a good hash function and reasonable load factor, α is small (< 1), making average-case operations effectively O(1).
Critical Assumption: These bounds assume a uniform hash function that distributes keys evenly across buckets. Poor hash functions can cause clustering that defeats this analysis.
The load factor (α = n/m) is the single most important variable for hash table performance. It represents the average number of entries per bucket—equivalently, the average chain length.
Interpreting Load Factor:
Unlike open addressing (which cannot exceed α = 1.0), chaining can operate at any load factor. But performance degrades as α increases.
| Load Factor (α) | Avg Chain Length | Expected Probes (Successful) | Expected Probes (Unsuccessful) | Relative Performance |
|---|---|---|---|---|
| 0.25 | 0.25 | 1.125 | 1.25 | Excellent |
| 0.50 | 0.50 | 1.25 | 1.50 | Very Good |
| 0.75 | 0.75 | 1.375 | 1.75 | Good |
| 1.00 | 1.00 | 1.50 | 2.00 | Acceptable |
| 1.50 | 1.50 | 1.75 | 2.50 | Degraded |
| 2.00 | 2.00 | 2.00 | 3.00 | Poor |
| 5.00 | 5.00 | 3.50 | 6.00 | Severely degraded |
The Formulas:
For a hash table with chaining and load factor α:
Unsuccessful Search (key not found):
Expected probes = 1 + α
We traverse the entire chain, on average α elements.
Successful Search (key found):
Expected probes = 1 + α/2
On average, we find the key halfway through the chain.
Most hash table implementations use 0.75 as the default load factor threshold for resizing. This balances memory efficiency (not too many empty buckets) with performance (chains stay short). Java's HashMap, Python's dict, and many others use this threshold.
Why 0.75?
At α = 0.75:
Increasing to 0.9 saves ~17% memory but increases probes by ~20%. Decreasing to 0.5 adds ~50% memory overhead for only ~15% speed improvement. The returns diminish rapidly in both directions.
With a uniform hash function, chain lengths follow a Poisson distribution with parameter λ = α (the load factor). This mathematical model lets us predict the probability of any chain length.
The Poisson Probability Formula:
P(chain length = k) = (α^k × e^(-α)) / k!
Where:
| Chain Length | Probability | Interpretation |
|---|---|---|
| 0 (empty) | 36.79% | About 1/3 of buckets are empty |
| 1 | 36.79% | About 1/3 have exactly one entry |
| 2 | 18.39% | About 1/5 have two entries |
| 3 | 6.13% | Small fraction have three |
| 4 | 1.53% | Rare |
| 5 | 0.31% | Very rare |
| ≥6 | 0.06% | Extremely rare |
Even with α = 1.0 (as many entries as buckets), over 90% of buckets have 0, 1, or 2 entries. Long chains are genuinely rare with a uniform hash function. This is why chaining performs well in practice.
Probability of Long Chains:
The probability that any specific bucket has a chain length ≥ k is approximately:
P(length ≥ k) ≈ (eα/k)^k for k > α
For a table with m buckets, the expected number of chains of length ≥ k is:
E[chains ≥ k] = m × P(length ≥ k)
| Chain Length ≥ | Probability | Expected Count in 1M Buckets |
|---|---|---|
| 5 | ~0.095% | ~950 buckets |
| 8 | ~0.0012% | ~12 buckets |
| 10 | ~0.000019% | ~0.19 buckets (unlikely any) |
| 15 | ~10^-10 | ~0 (effectively impossible) |
Practical Implication:
With a good hash function and reasonable load factor, chains exceeding 10 elements are essentially impossible in tables of any practical size. This bounds the worst-case behavior in practice (though theory must still consider adversarial inputs that could defeat the hash function).
Most hash tables grow dynamically—when the load factor exceeds a threshold, they allocate a larger array and rehash all entries. This is expensive (O(n)), but it happens rarely enough that the cost amortizes to O(1) per operation.
The Amortization Argument:
Each entry 'pays' a constant amount toward the eventual resize.
When resizing, doubling the capacity (m → 2m) is optimal. It ensures the next resize is at least n more insertions away, maintaining O(1) amortized insert. Growing by smaller factors (e.g., 1.5x) uses less memory but resizes more often.
Cost Analysis of Doubling:
Consider inserting 2^k elements into an initially empty table:
| Insertions | Table Size | Resize Cost | Cumulative Resize Cost |
|---|---|---|---|
| 1 | 1 | 0 | 0 |
| 2 | 2 | 1 | 1 |
| 4 | 4 | 2 | 3 |
| 8 | 8 | 4 | 7 |
| 16 | 16 | 8 | 15 |
| 2^k | 2^k | 2^(k-1) | 2^k - 1 |
After n insertions, total resize cost ≈ n. Average cost per insertion = O(1).
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class ChainingHashTable<K, V> { // ... existing code ... /** * Resizes the hash table by creating a new larger array * and rehashing all existing entries. */ private resize(newCapacity: number): void { console.log(`Resizing: ${this.buckets.length} → ${newCapacity}`); // Save reference to old buckets const oldBuckets = this.buckets; // Create new, larger bucket array this.buckets = new Array(newCapacity).fill(null); this.size = 0; // Will be repopulated // Rehash all existing entries for (const head of oldBuckets) { let current = head; while (current !== null) { // Note: this.put() uses the new bucket array this.put(current.key, current.value); current = current.next; } } } /** * Insertion with automatic resizing when load factor exceeded. */ put(key: K, value: V): V | undefined { // ... insertion logic ... // Check if resize needed AFTER insertion if (this.loadFactor > this.loadFactorThreshold) { this.resize(this.buckets.length * 2); } return oldValue; }}While amortized cost is O(1), the individual resize operation is O(n). For latency-sensitive applications (real-time systems, trading), this spike may be unacceptable. Solutions include incremental resizing (rehash a few entries per operation) or pre-allocating based on expected size.
Understanding space complexity helps you predict memory usage and make sizing decisions.
Total Space:
Space = Bucket Array + All Chain Nodes
= m × pointer_size + n × node_size
= O(m + n)
Since m = n/α (buckets = entries / load_factor), and typical α ≈ 0.75:
Space = O(n/α + n) = O(n × (1/α + 1)) = O(n)
Space grows linearly with entries, multiplied by constants depending on load factor and node overhead.
| Component | Per Entry Bytes | Total (1M entries) | Notes |
|---|---|---|---|
| Bucket array (α=0.75) | 8 bytes (pointer) | ~10.7 MB | 1.33M buckets × 8 bytes |
| Node: key (int64) | 8 bytes | 8 MB | The actual key |
| Node: value (int64) | 8 bytes | 8 MB | The actual value |
| Node: next pointer | 8 bytes | 8 MB | Chain link |
| Node overhead (allocator) | 8-16 bytes | 8-16 MB | Memory allocator bookkeeping |
| TOTAL | 40-48 bytes/entry | 40-48 MB | For int64 → int64 map |
Comparing with Open Addressing:
Open addressing stores entries directly in the bucket array—no separate nodes:
Open Addressing Space = m × entry_size
= (n/α) × (key_size + value_size)
For 1M entries at α = 0.5 (open addressing needs lower load factor):
Chaining uses more memory for small entries due to pointer overhead, but the gap narrows for larger entries.
Lower load factors use more memory but improve speed. A table at α=0.5 uses twice the bucket memory as α=1.0, but has half the average chain length. Profile your specific workload to find the right balance.
Knowing how to size your hash table optimally can prevent performance issues and memory waste:
Given Expected Entries (n), Find Optimal Capacity (m):
m = ceiling(n / target_load_factor)
Example:
- Expected 10,000 entries
- Target α = 0.75
- m = ceiling(10,000 / 0.75) = ceiling(13,333) = 13,334 buckets
Practically, round up to a power of 2 for efficient modulo (bit masking):
m = next_power_of_2(n / α) = 16,384 buckets
| Expected Entries | Load Factor | Minimum Capacity | Power-of-2 Capacity |
|---|---|---|---|
| 100 | 0.75 | 134 | 256 |
| 1,000 | 0.75 | 1,334 | 2,048 |
| 10,000 | 0.75 | 13,334 | 16,384 |
| 100,000 | 0.75 | 133,334 | 262,144 (256K) |
| 1,000,000 | 0.75 | 1,333,334 | 2,097,152 (2M) |
If you know the approximate size in advance, pre-allocate! Starting with capacity 16 and growing to 1M entries requires ~17 resizes. Pre-allocating with 2M capacity requires zero resizes, saving significant CPU time during population.
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// BAD: Default capacity, many resizesconst badMap = new ChainingHashTable<string, number>();for (let i = 0; i < 1000000; i++) { badMap.put(`key_${i}`, i);}// Triggers ~17 resize operations! // GOOD: Pre-sized capacity, zero resizesconst expectedSize = 1000000;const loadFactor = 0.75;const optimalCapacity = Math.pow(2, Math.ceil(Math.log2(expectedSize / loadFactor)));// optimalCapacity = 2097152 const goodMap = new ChainingHashTable<string, number>(optimalCapacity);for (let i = 0; i < 1000000; i++) { goodMap.put(`key_${i}`, i);}// No resize operations needed!The O(n) worst case occurs when all keys hash to the same bucket, creating a single chain of length n. This can happen due to:
In the worst case, the hash table degenerates to a linked list, and all operations become O(n).
In 2011, researchers demonstrated 'hash collision DoS attacks' against web frameworks (PHP, Python, Java, Node.js). Attackers sent carefully crafted POST data where all keys hashed to the same bucket, turning O(1) parsing into O(n²) and crashing servers. This led to randomized hash functions and per-process hash seeds becoming standard security measures.
Protection Strategies:
Java's Hybrid Defense:
Java's HashMap provides a reference implementation of tree conversion:
Chain length 0-8: Linked list (simple, fast for short chains)
Chain length 8+: Convert to red-black tree (O(log n) guarantee)
Chain length <6: Convert back to linked list (hysteresis)
This bounds worst-case lookup to O(log n) while maintaining O(1) average case. The tree threshold of 8 is chosen because:
Theory is essential, but let's ground our understanding in real-world performance numbers. These benchmarks give you practical expectations:
Typical Operation Times (Modern Hardware, 2020s):
| Operation | Time (nanoseconds) | Context |
|---|---|---|
| Hash computation (string, 20 chars) | 15-30 ns | Depends on hash algorithm |
| Bucket access (cache hit) | 1-4 ns | L1/L2 cache |
| Bucket access (cache miss) | 50-100 ns | Main memory |
| Chain traversal (per node) | 3-15 ns | Pointer chasing |
| Key comparison (integer) | 1-2 ns | Simple equality |
| Key comparison (string, 20 chars) | 5-20 ns | Depends on match point |
| Full lookup (successful, α=0.75) | 30-100 ns | Typical case |
| Full lookup (cache-cold) | 200-400 ns | First access to table |
Throughput Expectations:
Scaling Behavior:
Hash tables maintain near-constant operation time regardless of size—that's their key advantage. A table with 1 billion entries performs comparably to one with 1,000 entries (assuming both fit in memory).
These numbers are guidelines, not guarantees. Your performance depends on key types, value sizes, access patterns, cache behavior, and hash function choice. Always benchmark with representative data before making architectural decisions.
We've thoroughly analyzed the performance characteristics of hash tables with chaining. Let's consolidate the key insights:
Module Complete:
You've now completed a comprehensive study of collision handling with chaining. You understand:
Next, you'll explore open addressing—the alternative collision resolution strategy that stores all entries directly in the bucket array without separate chains.
Congratulations! You've mastered collision handling with chaining—from concept through implementation to rigorous performance analysis. You can now make informed decisions about hash table design and predict performance characteristics with confidence.