Question

Consider the following algorithm (which takes no input): \(j \leftarrow 1\) repeat $$\begin{aligned} j & \leftarrow j+1, n \stackrel{\phi}{\leftarrow}\\{0, \ldots, j-1\\} \\ \text { until } n &=0 \end{aligned}$$ Show that the expected running time of this algorithm is infinite (even though it does halt with probability 1 ).

Short answer

Question: Show that the algorithm described has an infinite expected running time, even though it halts with probability 1. Answer: The expected running time of the algorithm is infinite, as shown by the sum of the probabilities of halting at each specific step, multiplied by the step number, which does not converge to any finite value. However, it halts with probability 1 since each step has a non-zero probability of halting, and as the number of steps increases, the likelihood of halting increases.

Step-by-step solution

Understand the algorithm

The algorithm starts with a variable \(j\) initialized to 1. In each iteration, \(j\) is incremented by 1 and a random number \(n\) is chosen uniformly from the set \(\{0,\ldots, j-1\}\). If \(n = 0\), the algorithm halts.

Calculate the probability of halting at each step

At each step, the probability of halting depends on the probability of picking a 0 from the set \(\{0,\ldots, j-1\}\). Since there is only one 0 in the set and the set has \(j\) elements, the probability of halting at each step is: $$P(n=0\ |\ j) = \frac{1}{j}$$

Calculate the expected number of steps

To calculate the expected number of steps for the algorithm to halt, we will analyze the sum of the probabilities of halting at each specific step, multiplied by the step number. Let \(X_i\) be a random variable representing the step at which the algorithm halts, and let \(i \in \{1,2,\ldots\}\): $$\begin{aligned} E[X_i] &= \sum_{i=1}^{\infty} i \cdot P(n=0\ |\ j=i) \\ &= \sum_{i=1}^{\infty} i \cdot \frac{1}{i} \\ &= \sum_{i=1}^{\infty} 1 \\ \end{aligned}$$

Analyze the expected number of steps

The expected number of steps for the algorithm to halt is the sum of an infinite series of 1's. Therefore, the sum does not converge to any finite value: $$E[X_i] = \infty$$

Conclusion

The expected running time of the algorithm is infinite, even though it halts with probability 1. Each step has a non-zero probability of halting, and as the number of steps increases, the likelihood of halting increases. However, because the expected number of steps is infinite, the algorithm's running time is also infinite.

Key concepts

Expected Running Time

Understanding the expected running time of an algorithm allows us to predict how long an algorithm will take to complete. It's an average measure that encompasses all possible paths the execution might take, each weighted by its probability. In probabilistic scenarios, like the given algorithm that halts when a random number equals zero, the expected running time can sometimes be counter-intuitive.

In the provided exercise, to find the expected running time, we take into account the probability of stopping at each iteration, which decreases as the iteration number, or rather the value of the variable j, increases. The math behind this involves creating a series based on the probabilities and summing it up. As the iterations continue indefinitely and the sum of probabilities diverges, our conclusion is that the expected running time is infinite.

Probability Theory

Probability theory plays a vital role in algorithm analysis, especially in algorithms that involve randomness. This branch of mathematics deals with calculating the likelihood of different outcomes. In the context of the given algorithm, the probability of the algorithm halting at each iteration is crucial for determining the expected running time.

For this specific case, since the algorithm halts when it selects a particular value (0 in this instance), and as long as there is more than one option to choose from, the probability will not be 100%. This leads to the realization that while the algorithm will almost certainly halt eventually (with a probability of 1), the expected number of steps to get there is infinite, which is an interesting paradox that often occurs in probability theory.

Number Theory

Number theory, traditionally concerned with properties of integers, comes into play in algorithmic contexts when we deal with discrete structures, such as sets of integers or operations on integers. In our exercise, the choice of the random number n from the set \(\{0,\ldots, j-1\}\) involves elementary concepts from number theory — specifically, understanding that the set contains j distinct elements and deducing the uniform probability of selecting any single element from this set. Number theory provides the groundwork for comprehending why selecting 0 from a progressively increasing set retains a decreasing probability.

Random Variables

Random variables are foundational to understanding algorithms with stochastic processes, such as the one given. In the context of our problem, a random variable represents an outcome of a random phenomenon — here, it's the step number at which the algorithm halts. The concept of a random variable allows for the quantification of uncertainty and the use of probability distributions to express and analyze this uncertainty.

In the illustrated step-by-step solution, we define a random variable \(X_i\) to represent the stopping point of the algorithm. We use this variable to set up an expected value calculation, by summing over the product of the value of i and its associated probability, leading to insights about the behavior of the algorithm.

Infinite Series

Infinite series are sums of infinitely many terms and are frequently found in various areas of mathematics, including algorithm analysis. When we sum over all possible outcomes of a random process, as we do when determining the expected running time of an algorithm, we may end up with an infinite series.

The solution to our exercise hinges on recognizing that the sum \(\sum_{i=1}^{\infty} 1\) is an infinite series that does not converge to a finite number. This concept is pivotal because it reveals the nature of the expected running time of the algorithm — since the series diverges, the expected running time does as well, although every single term in the series represents a finite quantity (in this case, a probability).