Expected Number of Trials to Find Collisions

Oct 13, 2025 by windailys.com 139 views

Introduction

Hash tables are a fundamental data structure in computer science, used for efficient storage and retrieval of data. A dynamic hash table is a type of hash table that can resize itself as the number of elements inserted into it increases or decreases. However, as the number of elements in a hash table grows, the likelihood of collisions also increases. In this article, we will discuss the expected number of trials to find collisions in dynamic hash tables.

What are Collisions in Hash Tables?

Collisions in hash tables occur when two or more elements are mapped to the same index in the hash table. This can happen when the hash function used to map elements to indices is not perfect, or when the number of elements in the hash table is large compared to its size. Collisions can lead to poor performance and even crashes in certain situations.

Expected Number of Trials to Find Collisions

The expected number of trials to find collisions in a dynamic hash table can be calculated using the concept of birthday attacks. A birthday attack is a type of attack that exploits the fact that the probability of two randomly chosen numbers having the same value is proportional to the square root of the number of possible values.

Theorem 1: Expected Number of Trials to Find Collisions

Let $n$ be the number of elements in the hash table, and $p$ be the size of the hash table. Suppose that the hash function used to map elements to indices is a random function, and that the elements are chosen uniformly at random from the set of possible elements. Then, the expected number of trials to find collisions in the hash table is given by:

E[T] = \frac{n^2}{p}

Proof of Theorem 1

To prove Theorem 1, we can use the concept of birthday attacks. Let $X_i$ be the random variable that represents the number of trials needed to find a collision with the $i$ -th element. Then, we can write:

E[X_i] = \frac{n}{p}

Since the elements are chosen uniformly at random from the set of possible elements, we can assume that the hash function used to map elements to indices is a random function. Therefore, the probability that the $i$ -th element collides with any of the previous elements is given by:

P(X_i = 1) = \frac{n-1}{p}

Using the law of total expectation, we can write:

E[X_i] = \frac{n}{p} + \frac{n-1}{p}E[X_{i-1}]

Solving for $E[X_i]$ , we get:

E[X_i] = \frac{n^2}{p}

Corollary 1: Expected Number of Trials to Find Collisions in a Dynamic Hash Table

Let $n$ be the number of elements in the hash table, and $p$ be the size of the hash table. Suppose that the hash table is dynamic, and that the size of the hash table is chosen uniformly at random from the set of prime numbers. Then, the expected number of trials to find collisions in the hash table is given by:

E[T] = \frac{n^2}{p}

Proof of Corollary 1

To prove Corollary 1, we can use the fact that the size of the hash table is chosen uniformly at random from the set of prime numbers. Therefore, the expected value of the size of the hash table is given by:

E[p] = \sum_{p \in S} p \cdot P(p)

where $S$ is the set of prime numbers, and $P(p)$ is the probability that the size of the hash table is equal to $p$ . Since the size of the hash table is chosen uniformly at random from the set of prime numbers, we can assume that $P(p) = \frac{1}{|S|}$ .

Using the law of total expectation, we can write:

E[T] = E\left[\frac{n^2}{p}\right] = \frac{n^2}{E[p]}

Substituting the expression for $E[p]$ , we get:

E[T] = \frac{n^2}{\sum_{p \in S} p \cdot P(p)}

Since the size of the hash table is chosen uniformly at random from the set of prime numbers, we can assume that $P(p) = \frac{1}{|S|}$ . Therefore, we can write:

E[T] = \frac{n^2}{\sum_{p \in S} p \cdot \frac{1}{|S|}}

Simplifying the expression, we get:

E[T] = \frac{n^2}{p}

Conclusion

In this article, we discussed the expected number of trials to find collisions in dynamic hash tables. We proved that the expected number of trials to find collisions is given by $E[T] = \frac{n^2}{p}$ , where $n$ is the number of elements in the hash table, and $p$ is the size of the hash table. We also proved that the expected number of trials to find collisions in a dynamic hash table is given by $E[T] = \frac{n^2}{p}$ , where $p$ is the size of the hash table chosen uniformly at random from the set of prime numbers.

References

[1] "Hash Tables" by Cormen, Leiserson, Rivest, and Stein
[2] "The Art of Computer Programming" by Donald E. Knuth
[3] "Introduction to Algorithms" by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein

Additional Information

Suppose, a dynamic hash table takes its size only as a prime $p$ from the set of primes $S=\{2,3,\ldots,q\}$ sampled using a seed as $p\xleftarrow{\\\$}\mathsf{RandomSampler}(\mathtt{Integer}\;seed,\...$ . Then, the expected number of trials to find collisions in the hash table is given by:

E[T] = \frac{n^2}{p}

Q: What is the expected number of trials to find collisions in a dynamic hash table?

A: The expected number of trials to find collisions in a dynamic hash table is given by $E[T] = \frac{n^2}{p}$ , where $n$ is the number of elements in the hash table, and $p$ is the size of the hash table.

Q: What is the significance of the size of the hash table in determining the expected number of trials to find collisions?

A: The size of the hash table plays a crucial role in determining the expected number of trials to find collisions. A larger size of the hash table reduces the likelihood of collisions, resulting in a smaller expected number of trials to find collisions.

Q: How does the number of elements in the hash table affect the expected number of trials to find collisions?

A: The number of elements in the hash table also affects the expected number of trials to find collisions. A larger number of elements in the hash table increases the likelihood of collisions, resulting in a larger expected number of trials to find collisions.

Q: What is the relationship between the expected number of trials to find collisions and the size of the hash table chosen uniformly at random from the set of prime numbers?

A: The expected number of trials to find collisions is given by $E[T] = \frac{n^2}{p}$ , where $p$ is the size of the hash table chosen uniformly at random from the set of prime numbers. This means that the expected number of trials to find collisions is inversely proportional to the size of the hash table.

Q: How can we minimize the expected number of trials to find collisions in a dynamic hash table?

A: To minimize the expected number of trials to find collisions in a dynamic hash table, we can increase the size of the hash table or reduce the number of elements in the hash table.

Q: What is the impact of using a random hash function on the expected number of trials to find collisions?

A: Using a random hash function can reduce the likelihood of collisions, resulting in a smaller expected number of trials to find collisions.

Q: Can we use a different distribution for the size of the hash table instead of a uniform distribution?

A: Yes, we can use a different distribution for the size of the hash table instead of a uniform distribution. However, this may affect the expected number of trials to find collisions.

Q: How can we estimate the expected number of trials to find collisions in a dynamic hash table?

A: We can estimate the expected number of trials to find collisions in a dynamic hash table using the formula $E[T] = \frac{n^2}{p}$ , where $n$ is the number of elements in the hash table, and $p$ is the size of the hash table.

Q: What are some real-world applications of dynamic hash tables?

A: Dynamic hash tables have many real-world applications, including:

Database indexing
Caching
Web search engines
Social media platforms

Q: How can we implement dynamic hash tables in practice?

A: We can implement dynamic hash tables in practice using a variety of algorithms and data structures, including:

Hash table resizing
Hash table rehashing
Hash table rebalancing

Q: What are some common pitfalls to avoid when implementing dynamic hash tables?

A: Some common pitfalls to avoid when implementing dynamic hash tables include:

Hash table resizing too frequently
Hash table rehashing too infrequently
Hash table rebalancing too aggressively

Q: How can we optimize the performance of dynamic hash tables?

A: We can optimize the performance of dynamic hash tables by:

Using a good hash function
Resizing the hash table efficiently
Rehashing the hash table frequently
Rebalancing the hash table aggressively

Q: What are some future research directions for dynamic hash tables?

A: Some future research directions for dynamic hash tables include:

Developing more efficient hash functions
Improving the performance of hash table resizing
Developing new algorithms for hash table rehashing and rebalancing
Exploring the use of dynamic hash tables in new applications.