Probabilistic Analysis & Randomized Algorithms — Indicator Variables, Secretary Problem, and Randomized Sorting | Chapter 5 of Intro to Algorithms

Randomness is a powerful tool in algorithm design. Chapter 5 of Introduction to Algorithms introduces probabilistic analysis and randomized algorithms—two approaches that help us analyze performance or even improve efficiency when input distributions are unknown. This chapter covers key ideas such as indicator variables, linearity of expectation, random permutations, and optimal strategies for classic problems like hiring and the secretary problem.

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The Hiring Problem and Probabilistic Analysis

A classic illustration of probabilistic analysis is the hiring problem: a deterministic HIRE-ASSISTANT algorithm hires a candidate only if they’re better than the current hire. In the worst case (inputs in ascending order), every candidate is hired—costing O(n). But if candidates arrive in random order, probabilistic analysis shows the expected number of hires is the sum of 1/i for i = 1 to n (the harmonic series), yielding an expected cost of O(ln n). This is accomplished using indicator variables to formalize which candidates are hired and leveraging linearity of expectation for a straightforward calculation.

Randomized Algorithms & Random Permutation

Randomized algorithms use internal randomness to avoid worst-case scenarios or to simplify logic. For instance, the RANDOMIZED-HIRE-ASSISTANT algorithm begins by randomly permuting candidates before running HIRE-ASSISTANT. This ensures truly random execution and protects against adversarial input. A correct randomly permute algorithm relies on swapping elements in place (Theta(n) time) and careful proof using loop invariants to guarantee all permutations are equally likely.

Indicator Variables and Linearity of Expectation

Indicator variables are binary variables set to 1 if a certain event occurs and 0 otherwise. Their expected value equals the probability of the event, and linearity of expectation allows us to sum expected values—even when variables are dependent. This technique underlies many probabilistic analyses, enabling us to calculate averages for randomized or random-input algorithms with minimal effort.

Real-World Examples: Birthday Paradox, Balls-and-Bins, Coin Tosses, and the Secretary Problem

• Birthday Paradox: The probability that two people in a group share a birthday rises quickly with group size. The expected number of birthday collisions is k(k–1)/(2n), and the first collision is likely when k ≈ √(2n).
• Balls and Bins (Coupon Collector): Throwing n balls into b bins, the expected tosses to fill all bins is about b ln b. This is known as the coupon collector problem.
• Coin Toss Streaks: The expected longest streak of heads in n coin flips is about log n, a handy result for analyzing random sequences and streaks.
• Secretary Problem: You must hire exactly one candidate with no option to revisit previous ones. The optimal strategy is to skip the first k candidates and select the next one who is better than all seen so far, with k ≈ n/e. This approach guarantees at least a 1/e probability of hiring the best candidate.

Why Use Randomized Algorithms?

Randomized algorithms are often simpler to design and analyze, and can deliver superior performance—especially when worst-case inputs would sabotage deterministic algorithms. Examples include randomized quicksort, randomized primality testing, and efficient hashing. By randomizing either the input or the decision process, algorithms become robust against adversarial data and, in many cases, achieve better expected efficiency.

Key Concepts and Terms

• Probabilistic Analysis: Uses random input models to estimate average performance
• Randomized Algorithm: Relies on internal random choices to drive computation
• Expected Running Time: The average over all possible random outcomes
• Indicator Variable: Binary (0/1) variable for event tracking in analysis
• Linearity of Expectation: Sum of expected values, even for dependent variables
• Harmonic Series: The sum 1 + 1/2 + ... + 1/n, which ≈ ln n
• Uniform Random Permutation: Each arrangement of n elements is equally likely
• Coupon Collector Problem: Classic scenario for expected waiting times
• Secretary Problem: A famous stopping rule problem in hiring
• Loop Invariant: Assertion used to prove correctness of loops
• k-permutation: Ordered selection of k items from n
• Boole’s Inequality: Bound on probabilities for union of events
• Expected Value: Weighted average across all possible outcomes

Conclusion: The Power of Probability in Algorithm Design

Chapter 5 of Introduction to Algorithms demonstrates how randomness can be harnessed both in analyzing and improving algorithms. Concepts like indicator variables, linearity of expectation, and randomized strategies open the door to solving complex problems efficiently and with elegance. Probabilistic and randomized approaches are now fundamental tools for researchers, programmers, and students working with modern algorithms.

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