Question

If a hash file is partitioned into 10 buckets, what is the probability of at least two of three arbitrary records hashing to the same bucket? (Assume the hash function gives no bucket priority over the others.) How many records must be stored in the file until it is more likely for collisions to occur than not?

Short answer

The probability of at least two collisions is 0.28. More than 10 records are needed for likely collisions.

Step-by-step solution

Understanding Probability Basics

In this problem, each record has an equal probability of being placed into any of the 10 buckets. This is akin to placing three balls into 10 boxes, where each ball has a 1/10 chance of landing in any specific box in a trial.

Calculate Probability of No Collisions

For no two records to hash into the same bucket, the first record can go into any of the 10 buckets, the second into any of the remaining 9, and the third into any of the remaining 8. The probability is calculated as follows: \( P(\text{no collision}) = \frac{10}{10} \times \frac{9}{10} \times \frac{8}{10} = \frac{9 \times 8}{100} = \frac{72}{100} = 0.72 \).

Compute Probability of At Least One Collision

To find the probability of at least one collision, recognize that it is complementary to the probability of no collision. Thus, \( P(\text{at least one collision}) = 1 - P(\text{no collision}) = 1 - 0.72 = 0.28 \).

Determine Number of Records for Higher Collision Probability

The problem now is to determine the number of records required for it being more likely than not to have at least one collision. Applying the pigeonhole principle, since there are 10 buckets, a 11th record guarantees a collision. \( P(\text{collision}) > \frac{1}{2} \) when the curve bends upwards for increasing number of records. Therefore, with 11 or more records, collisions are more probable than not.

Key concepts

Probability

Probability is the likelihood or chance of an event occurring, expressed in values ranging from 0 to 1. A probability of 0 means the event will not happen, whereas a probability of 1 means it will definitely occur. In this context, the probability is used to calculate the odds of placing records into the same bucket when a hash function is employed.

When records are hashed into buckets, each has an equal probability of settling into any one of the buckets. For instance, in a scenario with 10 buckets, the probability of a single record hashing into any bucket is 1/10. This straightforward probability concept forms the foundation of predicting hash collisions.

• Probability, in this exercise, helps in determining the likelihood of collisions.
• Using the rule of complementary probabilities aids in finding both collision and no-collision outcomes.
• Calculated probabilities can provide insights into structured data organization.

Understanding these probabilities allow for efficient hash function design and collision management in storage systems.

Hash Collisions

A hash collision occurs when two different pieces of data, called keys, hash to the same bucket. Hash functions map data of arbitrary size to fixed-size values, but since there are finite bucket destinations, collisions are inevitable when more data is hashed than there are buckets. This is akin to two different keys unlocking the same door.

In our exercise, with three records being hashed into 10 buckets, we are prompted to compute the chance of at least two records ending up in the same bucket. To determine collision probability, we start by calculating the probability of no collisions, then deduce its complement:

• The probability of colliding (at least one collision occurs) is intentionally calculated using complementary probability theory.
• This collision probability reflects system reliability and performance in real-world scenarios.
• As the number of records increases, the potential for collisions also rises, which carries implications for data retrieval and processing speeds.

Recognizing hash collisions' characteristics helps in designing data structures that minimize the negative impact on performance.

Pigeonhole Principle

The pigeonhole principle is a fundamental concept of combinatorics and logic that states if you put more items into containers than the number of available containers, then at least one container must contain more than one item. This principle is key to understanding and predicting hash collisions.

When more records are hashed into fewer buckets, as demonstrated in the exercise, eventually some records will share buckets due to the pigeonhole principle. This concept reveals that having only 10 buckets cannot accommodate more than 10 records without duplication. Hence:

• This principle explains why the 11th record in a 10-bucket system guarantees a collision.
• Past this point, the probability of collisions ❯ 0.5, meaning it's more likely to have a collision.
• The pigeonhole principle informs how many records can be safely hashed before collisions become a certainty.

Thus, utilizing the pigeonhole principle allows us to evaluate system capacity and plan for efficient data storage management.