Randomized Algorithms: Foundations

Agenda

1. A Simple Mathematical Fact — the inner product mod 2.
2. Application: Freivalds’ Algorithm — verifying matrix products in O(n^2) time.
3. Formal Foundations — state machines & random tape models.
4. Two Flavors — Las Vegas vs. Monte Carlo.
5. Boosting — one-sided error amplification.

A Simple Mathematical Fact

Let’s begin with a fundamental algebraic property of random vectors:

• Let d \in \{0, 1\}^n be a fixed, non-zero boolean vector.
• Let r \in \{0, 1\}^n be a random boolean vector, where each r_i \in \{0, 1\} is chosen independently and uniformly at random: \Pr[r_i = 1] = \Pr[r_i = 0] = \frac{1}{2}

Proof of the Mod-2 Lemma

Since d \neq 0, there exists at least one coordinate k where d_k = 1.

• We can partition the sum: d \cdot r = d_k r_k + \sum_{i \neq k} d_i r_i \equiv r_k + \sum_{i \neq k} d_i r_i \pmod 2

• Let S = \sum_{i \neq k} d_i r_i \bmod 2. Its value is determined entirely by the choices of r_i for i \neq k.

• To make d \cdot r \equiv 0 \pmod 2, we must have: r_k \equiv S \pmod 2

• Since r_k is chosen uniformly and independently: \Pr[r_k = S] = \frac{1}{2}

Application: Freivalds’ Algorithm

Let’s apply this mathematical property to verify matrix multiplication: AB = C.

• We want to verify the product without spending O(n^3) or O(n^{2.37}) time to multiply them.
• Freivalds’ Protocol:
  1. Pick a random vector r \in \{0, 1\}^n.
  2. Compute A(Br) and check if it equals Cr.
• Analysis:
  • If AB = C, then (AB)r = Cr is always true.
  • If AB \neq C, let D = AB - C \neq 0. There is a non-zero row in D. Its product with r is 0 with probability \le 1/2 by the Mod-2 Lemma.
  • Runtime: O(n^2) time!

What is a Randomized Algorithm?

The protocols we just designed are examples of randomized algorithms:

Las Vegas Algorithms

We classify randomized algorithms based on where the uncertainty lies.

• Correctness: Always 100% correct.
• Runtime: A random variable T. We analyze the expected running time \mathbb{E}[T] or high-probability bounds.
• Classic Example: Randomized Quicksort
  • Pivot is chosen uniformly at random.
  • Array is always sorted correctly.
  • Expected running time is O(n \log n) on any input array.

Monte Carlo Algorithms

• Correctness: Bounded error probability \delta.
• Runtime: Deterministic / strictly bounded.
• The Core Trade-off:
  • We swap absolute certainty for computational speed and efficiency.
• Classic Example: Freivalds’ Algorithm
  • Strictly O(n^2) running time.
  • Error probability is at most 1/2 when AB \neq C.

Comparing the Two Flavors

One-Sided Error Amplification

We can amplify success by running the algorithm k times independently.

• Suppose the algorithm errs only on “Yes” inputs with probability \le \delta.

• Output “Yes” if any of the k runs output “Yes”.

• A failure requires every single run to err.

• The error probability drops exponentially: \Pr(\text{error}) \le \delta^k

• For \delta = 1/2 and k = 30, error is \le 2^{-30} \approx 10^{-9}.

Next Lecture

Two-Sided Error Amplification & Randomized Complexity Classes (RP, co-RP, BPP, ZPP).