What Is Hashing? Core Intuition

A hash function produces a hash value—but that value alone isn't useful until we translate it into an array index. This translation is the bridge between the abstract mathematical concept of hashing and the concrete implementation of a hash table.

Consider this: a hash function might produce the value 2,147,483,647 for some key, but your array only has 1,000 slots. How do we map that large number to a valid array position? How do we ensure this mapping preserves the uniform distribution we worked so hard to achieve in the hash function?

This page answers these questions. We'll explore the techniques for converting arbitrary hash values into usable array indices, understand the mathematical considerations involved, and learn why certain approaches work better than others.

Let's formalize what we need to accomplish. We have:

1. A key k — Could be any data type: string, integer, object, etc.
2. A hash function h(k) — Produces an integer hash code, potentially very large
3. An array of size m — Valid indices are 0 to m-1

We need to compute index(k) = h(k) → {0, 1, ..., m-1}

This transformation must satisfy several requirements:

Requirement 1: Valid Range The output must always be a valid array index. For an array of size m, this means [0, m-1].

Requirement 2: Preservation of Distribution If h(k) distributes keys uniformly across its output range, index(k) should distribute keys uniformly across [0, m-1]. A poor index computation can destroy a good hash function's distribution.

Requirement 3: Determinism The same key must always map to the same index. index(k) must be a pure function of k and m.

Requirement 4: Efficiency The computation should be fast—ideally O(1) using only basic arithmetic.

Example: The Problem in Action

Suppose we use Java's String.hashCode() method:

"hello".hashCode() = 99162322
"world".hashCode() = 113318802
"apple".hashCode() = 93029210

These values are 32-bit integers. If our hash table has only 16 slots, we need to map these large values to the range [0, 15]. How?

The naive approach—just use the hash value directly—fails immediately: 99162322 is not a valid index for a 16-slot array.

We need a systematic method to compress these values into our target range while preserving as much of the original distribution as possible.

The division method is the most common and intuitive approach to index computation:

index(k) = h(k) mod m

where m is the table size.

This method:

• Maps any integer to the range [0, m-1]
• Is computationally efficient (single modulo operation)
• Is deterministic (same hash always produces same index)

Example:

m = 16 (table size)

index("hello") = 99162322 mod 16 = 2
index("world") = 113318802 mod 16 = 2    // Collision!
index("apple") = 93029210 mod 16 = 10

But wait—"hello" and "world" both map to index 2. Is this the division method's fault, or were these hash values already going to collide?

The Choice of m Matters Enormously

The division method's quality depends critically on choosing m well. Poor choices of m can introduce patterns that destroy uniformity.

Why Powers of 2 Can Be Dangerous:

If m = 2^k, then h mod m is equivalent to taking the last k bits of h. This is fast but ignores all higher-order bits.

m = 16 = 2^4  →  index = last 4 bits of hash

If hash function has patterns in low bits:
hash₁ = ...00010000  →  index = 0
hash₂ = ...01110000  →  index = 0
hash₃ = ...10100000  →  index = 0

All three hash to the same index despite being different!

If the hash function has any weakness in its low-order bits, using a power of 2 amplifies that weakness.

Why Primes Are Preferred:

Prime numbers for m tend to scatter keys more uniformly because they have no common factors with typical integer patterns.

m = 17 (prime)

hash₁ = 272 → 272 mod 17 = 0
hash₂ = 273 → 273 mod 17 = 1
hash₃ = 274 → 274 mod 17 = 2
...

With a prime m, consecutive hash values map to consecutive indices (mod m), spreading keys across the table.

Handling Negative Hash Values:

In many languages, hash values can be negative (e.g., Java's String.hashCode() can return negative integers). The modulo operation with negative numbers can produce negative results:

// Java example
int hash = "some string".hashCode();  // Might be negative!
int index = hash % m;                  // Could be negative!

Solutions:

// Approach 1: Bit masking (most efficient)
int index = (hash & 0x7FFFFFFF) % m;  // Clear sign bit first

// Approach 2: Math.abs (beware of Integer.MIN_VALUE!)
int index = Math.abs(hash) % m;  // Edge case: abs(MIN_VALUE) overflows!

// Approach 3: Mathematical correction
int index = ((hash % m) + m) % m;  // Always positive

The first approach (bit masking) is typically preferred for its simplicity and efficiency.

The multiplication method offers an alternative that's less sensitive to the choice of m:

index(k) = ⌊m × (h(k) × A mod 1)⌋

where:

• A is a constant in the range 0 < A < 1
• "h(k) × A mod 1" extracts the fractional part
• ⌊...⌋ is the floor function

Step-by-step example:

Let m = 16, A = 0.6180339887 (related to golden ratio), h(k) = 99162322

1. h(k) × A = 99162322 × 0.6180339887 = 61,284,489.something
2. Fractional part = 61,284,489.xxx mod 1 = 0.285489...
3. m × 0.285489... = 16 × 0.285489... = 4.567...
4. Floor(4.567...) = 4

So index = 4

Why the Multiplication Method Works:

The multiplication method uses the fractional part of h(k) × A. When A is an irrational number (or a good approximation), this operation "scrambles" the hash value in a way that distributes across [0, 1) with minimal patterns.

The Golden Ratio Connection:

Knuth suggests using A ≈ (√5 - 1) / 2 ≈ 0.6180339887 (the golden ratio's inverse). This value has special properties that minimize clustering:

• It's irrational, so no periodic patterns emerge
• Its continued fraction expansion is [0; 1, 1, 1, ...], making it the "most irrational" number
• Consecutive values spread as evenly as possible

Advantages of the Multiplication Method:

1. m can be any value — Unlike division, powers of 2 work fine
2. Less sensitive to hash function weaknesses — Multiplication scrambles patterns
3. Efficient when m = 2^p — Can use bit manipulation:

// Fast implementation when m = 2^p
// Assuming 32-bit hash values
uint32_t hash = compute_hash(key);
uint32_t product = hash * 2654435769U;  // A ≈ 0.618 as fixed-point
uint32_t index = product >> (32 - p);   // Extract top p bits

The magic constant 2654435769 is ⌊(√5 - 1) / 2 × 2^32⌋.

Universal hashing provides provable guarantees against worst-case behavior by randomly selecting the hash function from a carefully designed family.

The Carter-Wegman Universal Hash Family:

For integer keys in range [0, p-1] (where p is a large prime) and table size m:

h_{a,b}(k) = ((a × k + b) mod p) mod m

where:

• a ∈ {1, 2, ..., p-1} (randomly chosen)
• b ∈ {0, 1, ..., p-1} (randomly chosen)
• p is a prime larger than the largest key

Why is this "universal"?

For any two distinct keys k₁ ≠ k₂:

P[h_{a,b}(k₁) = h_{a,b}(k₂)] ≤ 1/m

This probability is over the random choice of a and b. No adversary can construct a bad input because they don't know which a and b we'll choose.

Practical Implementation:

import random

class UniversalHashTable:
    def __init__(self, size, prime=10000019):
        self.m = size
        self.p = prime  # Prime larger than key range
        self.a = random.randint(1, prime - 1)
        self.b = random.randint(0, prime - 1)
        self.table = [None] * size
    
    def _hash(self, key):
        # Convert key to integer if necessary
        if isinstance(key, str):
            key = sum(ord(c) * (31 ** i) for i, c in enumerate(key))
        return ((self.a * key + self.b) % self.p) % self.m
    
    def insert(self, key, value):
        index = self._hash(key)
        # ... collision handling ...

When to Use Universal Hashing:

1. Security-sensitive applications — When adversaries might craft inputs to cause collisions
2. Real-time systems — When worst-case O(n) is unacceptable
3. Theoretical analysis — When you need provable performance bounds

So far we've assumed keys are integers. In practice, keys are often strings, objects, or composite types. The index computation pipeline becomes:

Key → Integer Hash → Array Index

Let's examine common approaches for the first step: converting arbitrary keys to integers.

Strings:

Strings are the most common non-integer key type. Several methods exist:

# Method 1: Sum of characters (POOR - many collisions)
def naive_hash(s):
    return sum(ord(c) for c in s)

# "abc" and "cba" and "bac" all hash to the same value!

# Method 2: Polynomial rolling hash (GOOD)
def polynomial_hash(s, base=31):
    h = 0
    for c in s:
        h = h * base + ord(c)
    return h

# "abc" = 97*31² + 98*31 + 99 = 93217 + 3038 + 99 = 96354
# "cba" = 99*31² + 98*31 + 97 = 95139 + 3038 + 97 = 98274
# Now they're different!

# Method 3: djb2 (GOOD - widely used)
def djb2_hash(s):
    h = 5381
    for c in s:
        h = ((h << 5) + h) + ord(c)  # h * 33 + c
    return h

Composite Objects:

For objects with multiple fields, combine the hash values of individual fields:

// Java example
public class Person {
    private String name;
    private int age;
    private String city;
    
    @Override
    public int hashCode() {
        int result = 17;  // Non-zero initial value
        result = 31 * result + (name != null ? name.hashCode() : 0);
        result = 31 * result + age;
        result = 31 * result + (city != null ? city.hashCode() : 0);
        return result;
    }
}

Why 31?

• It's odd (avoids loss of information from bit shifts)
• It's prime (distributes hash values well)
• 31 * x = (x << 5) - x (efficient to compute)
• Widely used convention in Java

Floating-Point Numbers:

Floating-point keys require special care:

// Java approach: use bit representation
double d = 3.14159;
long bits = Double.doubleToLongBits(d);
int hash = (int)(bits ^ (bits >>> 32));

Caution: Floating-point comparison is tricky due to precision issues. Consider whether you really want exact floating-point keys, or if you should quantize them first.

Arrays and Collections:

For ordered collections, use a rolling polynomial hash:

def array_hash(arr):
    h = 0
    for elem in arr:
        h = h * 31 + hash(elem)
    return h

For unordered collections (sets), use XOR or sum (order-independent operations):

def set_hash(s):
    return sum(hash(elem) for elem in s)  # Order doesn't matter

Index computation seems simple, but subtle bugs are common. Let's examine the most frequent pitfalls:

Testing Your Hash Function:

def test_hash_distribution(hash_func, keys, table_size):
    """Test if a hash function distributes keys uniformly."""
    buckets = [0] * table_size
    for key in keys:
        index = hash_func(key) % table_size
        buckets[index] += 1
    
    expected = len(keys) / table_size
    variance = sum((b - expected) ** 2 for b in buckets) / table_size
    std_dev = variance ** 0.5
    
    print(f"Expected per bucket: {expected:.2f}")
    print(f"Standard deviation: {std_dev:.2f}")
    print(f"Min: {min(buckets)}, Max: {max(buckets)}")
    
    # For good distribution, std_dev should be close to sqrt(expected)
    return std_dev < 2 * (expected ** 0.5)

Run this test with representative data to verify your hash function behaves well.

We've explored the critical bridge between hash values and array positions. This mapping determines whether your hash table achieves O(1) performance or degrades to O(n). Let's consolidate the key insights:

What's Next:

We've now covered how to compute hash values and map them to array indices. The next page explores the full journey from arbitrary data to array positions—seeing the complete picture of how inputs like strings, objects, and composite types become usable storage locations.