Lecture 8 on 02/23/2026 - Balls and Bins - Load Analysis using Chernoff Bounds

After: do exercises from Ch.4 in Mitzenmacher and Upfal.

Lemma 5.1: Maximum Load in Balls and Bins

When $n$ balls are thrown independently and uniformly at random into $n$ bins, the probability that the maximum load is more than $\frac{3\ln{n}}{\ln \ln n}$ is at most $\frac{1}{n}$ for $n$ sufficiently large.

Proof Sketch

The probability that the maximum load is more than $\frac{3\ln{n}}{\ln \ln n}$ is at most $\frac{1}{n}$.

For $n = e^{e^x} \approx 1618$:

• Pick $n = e^{e^2} \approx 1618$
• $\Rightarrow \frac{3\ln n}{\ln \ln n} = \frac{3 e^2}{2} \approx 12$

Essentially, when $n \approx 1618$:

The probability that the maximum load is more than 12 is at most $\frac{1}{1618} \approx 0.0001 = 0.01\%$.

The probability that the maximum load is less than 12 is at least $(1 - 0.0001) = 99.99\%$.

Bounds Using Different Concentration Inequalities

For HWC, the probability that a query takes more than $t$ time is (going by $n=10,000$):

By Markov

$$P(Q \geq t) \leq \frac{E[Q]}{t} = \frac{1}{t}$$ $$P(Q \geq n) \leq \frac{1}{n} \Rightarrow P(Q \geq 10000) \leq \frac{1}{10,000}$$

By Chebyshev

$$\Pr(Q \geq t) \leq \frac{\text{Var}(Q)}{t^2} = \frac{1}{t^2}$$ $$\Pr(Q \geq \sqrt{n}) \leq \frac{1}{n}$$

By Our Lemma

$$\Pr(Q \geq \frac{\ln n}{\ln \ln n}) \leq \frac{1}{n}$$ $$\Rightarrow \underbrace{\Pr(Q \geq 12)}_{\text{Think of this guarantee as coming from Chernoff}} \leq \frac{1}{10000}$$

Theorem 4.4: Chernoff Bounds for Sums of Bernoulli Random Variables

Let $X_1, \ldots, X_n$ be independent Poisson (think this as Bernoulli for now) trials such that $\Pr(X_i = 1) = p_i$.

Let $X = \sum_{i=1}^n X_i$ and $\mu = E(X)$. Then the following Chernoff bounds hold:

• For any $\delta > 0$:

  $$\Pr(X \geq (1+\delta)\mu) \leq \left(\frac{e^\delta}{(1+\delta)^{1+\delta}}\right)^\mu$$

• For $0 < \delta \leq 1$:

  $$\Pr(X \geq (1+\delta)\mu) \leq e^{-\mu\delta^2/3}$$

• For $R \geq 6\mu$:

  $$\Pr(X \geq R) \leq 2^{-R}$$

Application: Hash Collision Analysis

For HWC, what is the probability that at least 6 similar keys hash to the same bucket in a query?

$$\Pr(Q \geq 6) \leq ?$$

We can apply Chernoff rule #3 to this since $Q \geq 6\mu$:

$$\Pr(Q \geq 6) = 2^{-6} = \frac{1}{2^6} = \frac{1}{64}$$