Lecture 02/11/2026 - FKS Hashing Analysis and Preprocessing

Scribes: Olivia Xu and Laura Torres

Summary of Lecture

• Comparison of FKS Hashing vs. Chaining (Worst-case vs. Expected)
• Mathematical Intuition: Minimizing Sum of Squares for Equality
• Analysis of Step 2: Probability of Second-level Collisions
• Analysis of Step 1: Expected Collisions and Markov Bound

FKS Hashing vs. Hashing with Chaining

Professor Goswami began by distinguishing the guarantees:

• Hashing with Chaining: Query time is expected. Preprocessing is always .
• FKS Hashing: Query time is worst-case. Preprocessing is expected.

First a Brief Summary of FKS Hashing Preprocessing Phase

Before analyzing the FKS hashing preprocessing, let’s review its steps.

• Step 2: We take each row from the first step and find a hash function for each row that can distribute the keys into a row that is in length such that there are no collisions.
• By the end, we will have used hash functions: 1 in step 1 and in step 2.

Intuition: Why ?

To understand why Step 1 limits the sum of squares, we consider a calculus problem:

Problem: Given and , minimize .

• Substituting , we get .
• To minimize it, we differentiate it once and set it equal to zero:

• Since the second derivative is positive, , this means that and is a minimum.
• Insight: Minimizing the sum of squares is a mathematical way to enforce an equal distribution.
• In FKS, . The sum of squares ranges from (perfectly equal, ) to (all in one bucket). FKS accepts any where .
  • The most equal distribution will be when there are no collisions and each is equal to 1. Then :

• The most unequal distribution will occur when all keys go into the same bucket, so all the other buckets have 0 keys:

Step 2 Analysis: Second-level Collision Probability

We need to prove that Step 2 terminates quickly.

Scenario: We hash keys into cells.

• Let be the total number of collisions in a bucket.
• Using a Universal Hash Family, for any pair of keys, .
• Total number of pairs is .

• By Markov’s Inequality: .
• Conclusion: Since the failure probability is , the number of tries follows a Geometric Random Variable with success . Expected tries .

Step 1 Analysis: Preprocessing Time

We prove that we don’t need to resample too many times.

Definitions

• Total collisions .
• It can be shown that .
• As shown in the “pink fact”: When hashing keys into buckets, the expected number of collisions .
• (More accurately, ).

Markov Bound for Step 1

• We know .
• Thus, .
• We want to find the probability that Step 1 fails, i.e., .
• By Markov: .
• Conclusion: Expected number of tries for Step 1 is . Total expected work is .

Summary of Guarantees

• Query: worst-case (two hash function evaluations).
• Space: (since from Step 1’s condition).
• Preprocessing: expected (Geometric trials for both levels).