### A quiz

Alexey and I have recently discussed the following quiz:

You are given an array of length $n>0$ of pairwise independent random real numbers uniformly distributed on $[0,1]$. Suppose you are searching for the maximal element in the array by traversing it from left to right and updating a temporary variable holding the maximal value found so far (the variable is initialized with the 0th element of the array before the loop). How many updates will you be making on average?

Frankly, my own intuition for these kinds of things is pretty weak, which is why I felt challenged and sat down to solve the quiz. Below is what I ended up with. This is how not to solve it.

First, let us formalize the problem. Let $T(n)$ be the answer for an array of length $n$. Let us figure out what exactly the “on average” part means here. Let $u(x)$ denote the number of updates required by an array $x$ of length $n$. If the set $X$ of all possible arrays were finite, say $X=\{x_1, x_2, \dots, x_N\}$, we would count the number of updates for each array, and then $T(n)$ would be the average of these numbers:

$\displaystyle \displaystyle T(n) = \frac{1}{N}\sum_{p=1}^N u(x_p).$

Because $u(x)$ can have as its values only integers between $0$ and $n-1$, we can rearrange the terms in the above formula and write it as follows:

$\displaystyle \displaystyle T(n) = \sum_{i=1}^{n-1}i \cdot \frac{\#\{x\in X \mid u(x) = i\}}{\# X}.$

Here $\# S$ denotes the cardinality of a set $S$. The quotient

$\displaystyle \displaystyle \frac{\#\{x\in X \mid u(x) = i\}}{\# X}$

can be interpreted as the probability that an array picked at random from the set $X$ requires $i$ updates, provided all arrays in $X$ are equally likely. In other words, $T(n)$ becomes the expectation of $u$. Unfortunately, the set $X$ is infinite; in fact, it can be identified with the $n$-dimensional cube $[0, 1]^n$. And yet the logic remains the same. Because each coordinate is uniformly distributed over the interval $[0, 1]$, the distribution of arrays in $X$ is also uniform. That is, the probability that an array picked at random from the set $X$ is contained in a subset $A\subset X$ is the $n$-dimensional volume $\mathrm{vol}_n(A)$ of $A$. In the fancy language of probability theory, the $n$-dimensional volume $\mathrm{vol}_n$ is the product of $n$ copies of the Lebesgue measure on $[0,1]$ corresponding to the uniform distribution of each of the $n$ coordinates. The measure $\mathrm{vol}_n$ is a probability measure on the set $X$, i.e., $\mathrm{vol}_n(X)=1$, and hence the pair $(X,\mathrm{vol}_n)$ is a probability space. Furthermore, $u$ is a function $X\to\mathbb{N}$, i.e., a discrete random variable, and $T(n)$ is the expectation of $u$:

$\displaystyle \displaystyle T(n) = \sum_{i=1}^{n-1}i\cdot\mathrm{vol}_n(u^{-1}(i)).$

Let us describe the set $u^{-1}(i)=\{x\in X \mid u(x)=i\}$. An array $x$ requires $i$ updates if and only if there exists a sequence of indexes $1 \le k_1 < k_2 < \dots < k_i \le n-1$ where the updates happen such that $x(0) < x(k_1) < x(k_2) < \dots < x(k_i)$ and for each $0 \le j \le i$ and index $k_j < k < k_{j+1}$ holds $x(k_j) \ge x(k)$; here we put $k_0=0$ and $k_{i+1}=n$ for uniformity. If the indexes $k_1 < k_2 < \dots < k_i$ are fixed, the volume of the set of arrays $x$ satisfying these inequalities is given by the integral

$\displaystyle \displaystyle\int_0^1\int_0^{t_i}\dots\int_0^{t_1}t_0^{k_1-1}t_1^{k_2-k_1-1} \dots t_{i-1}^{k_i-k_{i-1}-1}t_i^{n-k_i-1}dt_0dt_1\dots dt_i,$

which is fairly easy to compute and which is equal to

$\displaystyle \displaystyle\frac{1}{k_1\cdot k_2\cdot\ldots\cdot k_i\cdot n}.$

The volume of the preimage $u^{-1}(i)$ is equal to the sum of these integrals over all possible sequences of indexes:

$\displaystyle \displaystyle\mathrm{vol}_n(u^{-1}(i))=\frac{1}{n}\sum_{1\le k_1 < k_2 < \dots < k_i \le n-1}\frac{1}{k_1\cdot k_2\cdot\ldots\cdot k_i}.$

The average number of updates is then given by the formula

$\displaystyle T(n) = \displaystyle \frac{1}{n}\sum_{i=1}^{n-1}\sum_{1\le k_1 < k_2 < \dots < k_i \le n-1}\frac{i}{k_1\cdot k_2\cdot\ldots\cdot k_i}.$

Clearly, this cannot possibly be an answer to a job interview quiz. It is too complicated. I wasn’t pleased with my solution, so I emailed the quiz to Alexey to see if he could come up with anything better. And indeed, his solution turned out to be very slick: The probability that the largest item in the array is the last is $1/n$. Whether it is or no, the $[0..n-1]$ prefix is isomorphic to the original problem. Thus we have to do $T(n-1)$ work, and with probability $1/n$ we have one more update after that:

$\displaystyle T(n) = 1/n + T(n-1).$

An array of length $1$ doesn’t require updates, so that $T(1) = 0$. Together these equations imply that $T(n) = 1/2 + 1/3 + \dots + 1/n$, the $n$-th harmonic number less $1$. In particular, this immediately suggests that the number of updates grows logarithmically in $n$, which is very hard to see from my formula.

By the way, is my answer actually correct? At first, I didn’t think so. How can something as hairy as

$\displaystyle \displaystyle\frac{1}{n}\sum_{i=1}^{n-1}\sum_{1 \le k_1 < k_2 < ... < k_i \le n-1}\frac{i}{k_1\cdot k_2\cdot\ldots\cdot k_i}$

possibly be equal to $1/2 + 1/3 + ... + 1/n?$ I tried to prove that by induction and didn’t succeed. However, this is indeed the case, and I have discovered an elegant algebraic proof, which I am pleased to present.

First, observe that the sum

$\displaystyle \displaystyle \sum_{1 \le k_1 < k_2 < ... < k_i \le n-1}\frac{1}{k_1\cdot k_2\cdot\ldots\cdot k_i}$

occurring in $T(n)$ is the $i$-th elementary symmetric polynomial in $1$, $1/2$, …, $1/(n-1)$. Consider the polynomial

$\displaystyle \displaystyle \begin{array}{rcl}P(u) & = & \displaystyle \Bigl(u + 1\Bigr)\Bigl(\frac{u}{2}+ 1\Bigr)\dots\Bigl(\frac{u}{n-1} + 1\Bigr) \\ \\ & = & \displaystyle u^{n-1}\cdot\frac{1}{1\cdot 2\cdot\ldots\cdot(n-1)}\\ \\ & + & \displaystyle\dots \\ \\ & + & \displaystyle u^2\cdot\Bigl(\frac{1}{1\cdot 2} + \frac{1}{1\cdot 3} + \dots + \frac{1}{(n-1)(n-2)}\Bigr) \\ \\ & + & \displaystyle u\cdot\Bigl(1 + \frac{1}{2} + \dots + \frac{1}{n-1}\Bigr) \\ \\ & + & 1. \end{array}$

The coefficients of $P$ are precisely the elementary symmetric polynomials in $1, 1/2, \dots, 1/(n-1)$. The value of $P$ at the point $1$ is the sum of these polynomials, which is almost $T(n)$, except that in $T(n)$ these polynomials show up with coefficients $1, 2, \dots, n-1$. That’s pretty easy to fix. The trick is to consider the derivative of $P$. On the one hand, it equals

$\displaystyle \displaystyle \begin{array}{rcl} P'(u) & = & \displaystyle u^{n-2}\cdot (n-1)\cdot\frac{1}{1\cdot 2\cdot \ldots\cdot (n-1)} \\ \\ & + & \displaystyle\dots \\ \\ & + & \displaystyle u \cdot 2\cdot\Bigl(\frac{1}{1\cdot 2} + \frac{1}{1\cdot 3} + \dots + \frac{1}{(n-1)(n-2)}\Bigr) \\ \\ & + & \displaystyle\Bigl(1 + \frac{1}{2} + \dots + \frac{1}{n-1}\Bigr) \end{array}$

and therefore $T(n) = P'(1)/n$. On the other hand, using the product rule, we obtain

$\displaystyle \displaystyle P'(u) = P(u)\Bigl(\frac{1}{u+1}+\frac{1}{u+2} + \dots + \frac{1}{u+n-1}\Bigr).$

Furthermore, $P(1) = 2 \cdot 3/2 \cdot 4/3 \cdot\ldots\cdot n/(n-1) = n$, and hence $T(n) = P'(1)/n = 1/2 + 1/3 + \dots + 1/n$. Q. E. D.