Codeforces Sorting Game Solution Guide | Generated by AI

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Question: How can one solve the Codeforces problem “Sorting Game,” where Alice and Bob take turns rearranging non-increasing subsequences of a binary string into non-decreasing order?

Answer:

Problem Analysis

The game is played on a binary string of length .

  1. Valid Move: A player chooses a non-increasing subsequence (e.g., 1, 1, 0, 0) and sorts it into non-decreasing order (e.g., 0, 0, 1, 1). This move must strictly change the string.
  2. Losing Condition: The player who cannot make a valid move loses (Normal Play Convention).
  3. Terminal State: A string is in a terminal state (no moves possible) if and only if it is already sorted (all 0s come before all 1s). If a string is sorted, any non-increasing subsequence would consist of only all 0s or only all 1s, and sorting them would not change the string.

Winning Strategy

Alice wins if she can transform the string into a sorted state in a single move. Because the game ends as soon as a player is presented with a sorted string, if Alice can sort the string on her first turn, Bob will immediately lose.

Can Alice always sort the string in one move? Yes, if the string is not already sorted. To do this, Alice needs to identify all characters that are “out of place” compared to the final sorted version of the string.

  1. Let be the total number of zeros in the string .
  2. In a sorted string of length with zeros, the first characters must be 0 and the remaining characters must be 1.
  3. Alice can pick the set of indices where the characters are currently incorrect:
    • Any index where .
    • Any index where .
  4. Validity Check:
    • The characters at indices (where ) always appear before the indices (where ).
    • Therefore, the subsequence chosen is always a series of 1s followed by a series of 0s, which is non-increasing.
    • Sorting this subsequence swaps the 1s to the back and the 0s to the front, making the entire string sorted.
    • If the string is unsorted, this set will contain at least one 1 and one 0, making the move valid.

Algorithm

  1. Count the total number of zeros () in the string.
  2. Check if the string is already sorted (i.e., all characters from index to are '0').
  3. If the string is already sorted, output Bob.
  4. Otherwise, output Alice.
  5. To provide the move:
    • Find all indices where .
    • Find all indices where .
    • Combine these indices, sort them, and output them as Alice’s first move.

Complexity

References:

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