CF 1579E2 - Array Optimization by Deque

Rating: 1700
Tags: data structures, greedy
Solve time: 3m 30s
Verified: no

Solution

Correctness

The proposed solution does address the actual exercise and identifies the key idea: each pair $(U_{2n-1},U_{2n})$ is transformed by either keeping it unchanged or swapping coordinates, depending on whether the point lies in $G$. This is indeed the mechanism behind the construction.

However, the proof is not complete, and one of its central arguments is incorrect.

For part (a), the solution correctly observes that different pairs are transformed independently, so blocks corresponding to different values of $n$ remain independent. But whiteness requires that the individual variables $V_i$ be mutually independent. The argument given does not establish this. Independence of the blocks $(V_{2n-1},V_{2n})$ is not enough; one must also show that within each block the two coordinates are independent. The proposed proof never does this.

In fact, the statement

Hence, all $(V_{2n-1},V_{2n})$ pairs are independent, and therefore all $V_n$ are independent.

does not follow logically. Independence of the pairs as random vectors does not imply independence of the coordinates inside each pair.

To prove whiteness one would need to show that $(V_{2n-1},V_{2n})$ is still uniformly distributed on the unit square, which implies that its coordinates are independent uniform random variables. The solution only proves marginal uniformity, not joint uniformity.

For part (b), the solution does not actually compute the probability. It introduces regions $G_1$ and $G^c$, then states that

Symmetry arguments show that the total area of the region where $V_1>V_2$ is exactly one-quarter of the unit square.

No symmetry argument is provided. The subsequent sentence

One can check that ... the set of points mapped to $V_1>V_2$ has measure $1/4$.

is likewise an assertion without proof.

Since the exercise explicitly asks to show that the probability equals $1/4$, the area calculation is the heart of the problem. The solution never carries it out.

Gaps and Errors

Critical error: The proof of whiteness is invalid.

The solution proves only that different transformed pairs are independent. It does not prove that $V_{2n-1}$ and $V_{2n}$ are independent. The conclusion that all $V_n$ are independent does not follow from the stated argument. 2. Justification gap: Equidistribution is proved only via marginal distributions.

The argument shows each coordinate is uniform, but does not establish that $(V_{2n-1},V_{2n})$ is uniformly distributed on $[0,1]^2$. A stronger statement is needed for the whiteness claim. 3. Critical error: Part (b) contains no actual computation of the required probability.

The measure $1/4$ is asserted rather than derived. 4. Justification gap: The phrase

Symmetry arguments show ...

is unsupported. The relevant symmetry is neither identified nor used rigorously. 5. Justification gap: The statement

One can check that ...

replaces the essential geometric calculation with an unproved claim. 6. Claim used without proof: The region corresponding to $V_1>V_2$ has area $1/4$.

This is exactly the result that must be established.

Summary

The solution has the right overall intuition about the coordinate-swapping transformation, but it fails to prove whiteness and never actually derives the probability $1/4$. The most important parts of the exercise are asserted rather than demonstrated.

VERDICT: FAIL - the proof does not establish whiteness and does not actually compute the probability $\Pr{V_n>V_{n+1}}=1/4$.