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I am given two randomized algorithms $f$ and $g$, that take as input a positive integer $m$ and produce a random non-negative integer. The algorithms are the same except that $g$ receives an small "advantage" with a very low probability.

I need to prove or disprove that as $m$ grows, the two algorithms tend to behave the same. More concretely, if $h(m):=g(m)-f(m)$, to show or refute that

$$\lim_{m\to\infty} E[h(m)] = 0$$

But I'm stuck because the algorithm is a complex stochastic process.

What methodologies can be used to formally prove that the hypothesis holds?

Attachments:

This is the algorithm. Arrays are 0-indexed, randint is inclusive and [0]*m means an array of m zeroes.

def algorithm(m, advantage):
    h = [0] * m
    i = 1
    while i < m:
        if advantage and randint(0, m-1) == 0:
            prev = h[i-1]
        else:
            prev = h[i-1] - 1
        j = randint(0, i-1)
        h[i] = max(prev, h[j]+1)
        i += 1
    return h[m-1]

def f(m): return algorithm(m, advantage=False)
def g(m): return algorithm(m, advantage=True)
def h(m): return g(m) - f(m)

And the plots of some Monte Carlo simulation plots, which show that the hypothesis might hold:

plot of f and g plot of h

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  • $\begingroup$ When $m$ tends to $\infty$, the probability that randint(0,m-1) equals 0 tends to 0, if I understand well what randint() means $\endgroup$
    – Damien
    Jul 4, 2019 at 18:47
  • $\begingroup$ Yes, that's exactly the intuition of why they should be similar. But that is not enough for the proof because that probability is compensated with the fact that $m$ samples are drawn. $\endgroup$ Jul 5, 2019 at 2:10

1 Answer 1

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Each time the advantage makes a difference, it increases the value of some element by $1$. Let us denote by $e_i$ the expected total number of such increases when calculating $h[i]$. Clearly $e_0 = 0$, and we can upper bound $$ e_i \leq \frac{1}{m} + \frac{e_0 + \cdots + e_{i-1}}{i}. $$ The solution to this recurrence is $e_i = O\bigl(\frac{\log i}{m}\bigr)$ (see below). In particular, $e_{m-1} = O\bigl(\frac{\log m}{m}\bigr)$ is an upper bound on the expected advantage in value.

Let us now solve the recurrence $$ x_i = 1 + \frac{x_0 + \cdots + x_{i-1}}{i}, $$ with initial value $x_0 = 0$.

Roughly speaking, $x_i$ counts the expected number of steps it takes to get back to $0$; since each step roughly halves the value, the number of steps behaves like $\log i$.

We can compute $x_i$ exactly in various ways. Let $y_i = ix_i$. Then $$ y_{i+1} - y_i = (i + 1) + (x_0 + \cdots + x_i) - i - (x_0 + \cdots + x_{i-1}) = 1 + x_i = 1 + \frac{y_i}{i}, $$ hence $$ y_{i+1} = 1 + \frac{i+1}{i} y_i. $$ Unrolling this recurrence gives $$ y_i = 1 + \frac{i}{i-1} y_{i-1} = 1 + \frac{i}{i-1} + \frac{i}{i-2} y_{i-2} = \cdots = \frac{i}{i} + \frac{i}{i-1} + \cdots + \frac{i}{1} y_1. $$ Since $y_1 = x_1 = 1$, we deduce that $$ x_i = \frac{y_i}{i} = \frac{1}{i} + \cdots + \frac{1}{1} = H_i \approx \log i. $$

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