Universal hashing is a technique used in computer science and cryptography to design hash functions that minimize collisions and provide good average-case performance. The idea is to randomly select a hash function from a family of hash functions, ensuring that the probability of collisions is low for any given input set. Here's an overview of how universal hashing works:
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Hash Function Family: Define a family of hash functions, denoted as H, that can be used to hash the keys. Each hash function in this family maps keys to hash values.
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Random Selection: When you want to hash a key, you randomly select a hash function from the family H. This random selection introduces an element of randomness that helps distribute the keys more uniformly.
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Collision Probability: The key to the effectiveness of universal hashing is to ensure that the probability of collisions (i.e., two different keys hashing to the same value) is low for any input set of keys. This is achieved by carefully designing the hash function family.
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Analysis: Universal hashing relies on probabilistic analysis to demonstrate that the chosen hash function family provides good average-case performance in terms of minimizing collisions.
To solve or implement universal hashing, you need to perform the following steps:
Step 1: Define Hash Function Family Choose a prime number p and define a hash function family H. For each a and b where a is in the range [1, p-1] and b is in the range [0, p-1], define a hash function h_{a,b}(k) as:
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h_{a,b}(k) = ((a * k + b) % p) % m
Where:
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k is the input key.
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p is a prime number that determines the range of the hash function.
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m is the number of slots or buckets in the hash table.
Step 2: Random Selection For each key you want to hash, randomly choose a and b from the allowed ranges. This random selection is typically done using a truly random source or a pseudorandom number generator.
Step 3: Hashing Given a key k, and the selected a and b, calculate the hash value using the formula mentioned earlier.
Step 4: Collision Analysis To analyze the collision probability, you can use techniques from probability theory and number theory. The goal is to show that for any distinct keys k1 and k2, the probability that h_{a,b}(k1) = h_{a,b}(k2) is small, ideally close to 1/m (where m is the number of buckets).
Step 5: Usage Use the selected hash function to hash keys in your hash table. Remember to re-randomize the a and b values whenever you rehash or resize the hash table.
It's important to note that implementing universal hashing requires a good understanding of number theory, probability, and hash functions. The effectiveness of the technique relies on the careful choice of parameters and the quality of the hash function family H. Universal hashing is often used in situations where security and performance are both important, such as in cryptographic applications or hash-based data structures.
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