Hong Life CalculatorsUnderstanding the hash table load factor calculator — concept and principle
The load factor α of a hash table is the number of stored items divided by the number of buckets. It is the single most important number for predicting how fast lookups will be. This calculator gives α and Knuth's expected probe counts for linear probing, the average chain length for separate chaining, and the expected number of empty buckets.
How it is calculated
- Load factor: α = n ÷ m (items ÷ buckets).
- Linear probing, successful search: ½ · (1 + 1 ÷ (1 − α)) probes.
- Linear probing, unsuccessful search or insert: ½ · (1 + 1 ÷ (1 − α)²) probes.
- Chaining, average chain length: α.
- Expected empty buckets: m · e^(−α).
If α ≥ 1, open addressing has no free slot, so the calculator stops with a warning.
The principle
With a good hash function each item lands in a bucket almost at random. The number of items in one bucket then follows a Poisson distribution with mean α, so the chance that a bucket is empty is e^(−α). In linear probing, a collision sends the item to the next slot, and occupied slots merge into long runs (primary clustering). Knuth showed that the cost grows like 1 ÷ (1 − α) for hits and 1 ÷ (1 − α)² for misses, so performance degrades sharply as the table fills.
Worked example
- Inputs: 7,000 items, 10,000 buckets.
- α = 7,000 ÷ 10,000 = 0.7.
- Successful search: ½ · (1 + 1 ÷ 0.3) = ½ · 4.333 = 2.166666667 probes.
- Unsuccessful search: ½ · (1 + 1 ÷ 0.09) = ½ · 12.111 = 6.055555556 probes.
- Chaining: average chain length 0.7.
- Empty buckets: 10,000 · e^(−0.7) = 10,000 × 0.4966 ≈ 4,965.853038.
| Buckets | Load factor α |
|---|---|
| 2,500 | 2.8 |
| 5,000 | 1.4 |
| 7,500 | 0.9333333333 |
| 10,000 | 0.7 |
| 12,500 | 0.56 |
| 15,000 | 0.4666666667 |
| 20,000 | 0.35 |
How to read the result
For linear probing, α up to about 0.5–0.7 keeps both hits and misses cheap; at α = 0.9 a miss costs about 50 probes. Chaining tolerates higher loads, which is why many libraries resize around 0.75; Java's HashMap uses 0.75 as its default load factor. Nearly half the buckets are still empty at α = 0.7, which is normal and not wasted design.
Common mistakes
- Treating α = 0.7 as "70% of buckets are used"; only about 50% are, because some buckets hold several items.
- Applying these formulas with a weak hash function or patterned keys, which cluster far worse than random.
- Forgetting that deletions in open addressing leave tombstones that keep probe counts high until the table is rebuilt.
Frequently asked questions
What load factor should I choose?
For open addressing, about 0.5–0.75 is common; for chaining, about 0.75–1. Lower values use more memory but give shorter probes.
Why does the table double in size when it grows?
Doubling keeps the average cost of each insert constant, since the expensive rehash happens less and less often.
Does the bucket count need to be prime?
It helps with simple modulo hashing; with a well-mixing hash function, a power of two works well and is faster to index.
Sources: D. E. Knuth, The Art of Computer Programming, Vol. 3: Sorting and Searching, Section 6.4 (Hashing), Addison-Wesley · Oracle, Java SE API documentation, java.util.HashMap (default load factor)