bracket-sequence-restoration

{"output": "", "exit_code": 0, "error": null}

<think>Let me analyze this problem step by step.

We have a string with parentheses and question marks. We need to determine if there's exactly one way to replace each '?' with '(' or ')' to form a valid RBS (Regular Bracket Sequence).

Key observations:
1. A valid RBS must have equal number of '(' and ')'
2. For any prefix, the number of '(' must be >= number of ')'

Let me think about when the reconstruction is unique vs when there are multiple ways.

If we think about this as a range of possible values for each position, we can determine uniqueness.

Let's think of it this way: we want to know if there's exactly one valid assignment.

Let me define:
- `min_open[i]` = minimum number of '(' needed up to position i
- `max_open[i]` = maximum number of '(' possible up to position i

For a prefix ending at position i:
- Each '(' contributes +1, each ')' contributes -1, each '?' can be either (+1 or -1)
- `min_open[i]` = count of '(' + min possible from '?' (all ')')
- `max_open[i]` = count of '(' - min possible from '?' (all '(') + count of ')'

Wait, let me reconsider. Let's define:
- `balance` after processing a prefix: 
  - '(' increases balance by 1
  - ')' decreases balance by 1
  - '?' can be '(' (balance + 1) or ')' (balance - 1)

For the sequence to be valid:
1. Final balance must be 0
2. At any prefix, balance >= 0

Now, for uniqueness: we need to check if at any position, the choice for '?' is forced.

Let me think of it differently. Let's consider the "flexibility" at each position.

If we process left to right and track the range of possible balances:
- Initially: balance = 0
- For '(' : range shifts by [+1, +1] → [min+1, max+1]


- For ')' : range shifts by [-1, -1] → [min-1, max-1]
- For '?' : range shifts by [-1, +1] → [min-1, max+1]

The key insight is that uniqueness requires the range of possible balances to collapse to exactly one value at each position. If at any point the range has width > 1, there are multiple valid reconstructions. Conversely, if the range is always exactly 1 wide, the reconstruction is unique.

However, I need to reconsider this more carefully. The critical constraint is that the final balance must be 0 and all prefixes must have non-negative balance. When a range becomes [k, k], that balance is forced. But I should verify whether a unique assignment always exists when the range width is 1 throughout, or if there are edge cases where multiple assignments could still exist despite appearing unique at intermediate steps.

Let me trace through a concrete example. With `()??`:
- Position 0 '(' : range = [1, 1]
- Position 1 ')' : range = [0, 0]
- Position 2 '?' : range = [-1, 1]
- Position 3 '?' : range = [-2, 2]

The range width grows to 2, which suggests multiple valid completions exist. For the sequence to be valid, the final balance must be 0 and all prefixes must maintain balance >= 0. I need to reconsider the approach.

Let me define `mn[i]` and `mx[i]` as the minimum and maximum possible balance at each position. At each step, '?' expands the range by [-1, +1], '(' shifts it to [+1, +1], and ')' shifts it to [-1, -1]. The key insight is that if `mx[i]` becomes negative at any point, no valid sequence exists. When the range width `mx[i] - mn[i]` exceeds what's needed to reach a final balance of 0, multiple valid assignments become possible.

For the sequence to be reconstructable in exactly one way, I need to check: the range must never go negative, the final balance must be exactly 0, and crucially, at each position the range must be constrained to a single value. If the range ever has width greater than 1, multiple valid assignments exist. The challenge is that even with width 1 at each position, I need to verify whether a complete valid sequence actually exists or if the constraints force us into an impossible state.

Let me reconsider the problem more carefully. The key insight is that a unique reconstruction exists if and only if the range of possible balances at every position has w

→ terminal({"command": "mkdir -p /workdir"})

{"bytes_written": 1639, "dirs_created": true}

<think>Now let me create the solution file with the algorithm.
</think>

→ write_file({"path": "/workdir/solution.py", "content": "import sys\n\ndef is_unique_reconstruction(s):\n    \"\"\"\n    Determine if there's exactly one way to replace '?' with '(' or ')' \n    to form a valid RBS.\n    \n    Key insight: Track the range of possible balances at each position.\n    - If range width > 1 at any point, multiple valid assignments exist\n    - If range width == 1 at all points and…)

{"output": "YES\nNO\nNO", "exit_code": 0, "error": null}

<think>Let me test the solution with the sample input to verify it works correctly.
</think>

→ terminal({"command": "cd /workdir && echo -e \"3\\n()()\\n(??)\\n((?)\" | python3 solution.py"})

{"output": "NO", "exit_code": 0, "error": null}

<think>Hmm, the third test case shows "NO" but the expected output is "YES". Let me re-examine.

For `((?)`:
- Position 0 '(' : mn=1, mx=1
- Position 1 '(' : mn=2, mx=2
- Position 2 '?' : mn=1, mx=3
- Position 3 ')' : mn=0, mx=2

Wait, the problem says the sample output is YES for `((?)`. Let me re-read the problem...

Oh wait, the sample input shows:
```
3
()()
(??)
((?)
```

And sample output:
```
YES
NO
YES
```

So `((?)` should return "YES". But my algorithm returns "NO" because the range width becomes 2 at position 2.

Let me think about this more carefully. At position 2 (the '?'), the range is [1, 3]. But can we actually achieve all values in this range?

After `((`:
- mn = 2, mx = 2

For '?' at position 2:
- If '?' is '(': balance = 3
- If '?' is ')': balance = 1
- So range = [1, 3]

Now at position 3 (')'):
- If balance was 3: 3-1 = 2
- If balance was 1: 1-1 = 0
- So range = [0, 2]

The final range is [0, 2]. Can we achieve exactly balance 0? Yes, if the '?' at position 2 was ')'.

But wait, my algorithm checks if the range width > 1 at any position. At position 2, the range width is 3-1=2, which is > 1. So my algorithm says "NO".

But the expected answer is "YES"! This means my understanding of "uniqueness" is wrong.

Let me reconsider. 

The key distinction is that even though multiple balances are mathematically possible at position 2, not all of them can actually lead to a valid final sequence. If I choose '(' at position 2 giving balance 3, then at position 3 with ')', I'd get balance 2—but this can't reach 0 by the end. The only viable path is choosing ')' at position 2, which leads to balance 0. So the "uniqueness" here refers to the final reconstruction being unique, not the intermediate balance.

This means I need to reconsider my approach. I can't just track the range of possible balances at each step—I need to verify that at least one path through those balances actually reaches a valid final state. Let me think about this differently. I should compute the minimum and maximum possible final balances, then check whether a valid sequence exists and whether that sequence is uniquely determined.

The key insight is that if the range of possible balances at any position exceeds what can be compensated for by the remaining characters, then multiple valid sequences must exist. Specifically, if the remaining positions could only close a limited number of brackets, but the range shows we could have more opens than that allows, then some choices at the current position must lead to invalid sequences.

Let me formalize this: let `remaining_close` be the maximum number of ')' we could still add. If the minimum balance we could have is `mn`, then the minimum final balance is `mn - remaining_close`. For a valid sequence, this needs to be ≤ 0. If it's already < 0, then some earlier choice must have been wrong, which means that choice couldn't have led to a valid sequence—so the range must actually be constrained.

The core constraint is that at every position, the balance can't exceed what's sustainable given the remaining characters. If at position i the balance is b, then `b ≤ remaining_opens[i+1]`. This means the actual balance at any position is bounded by the future opens available.

The key insight is that a reconstruction is unique if and only if the balance is forced at every position—meaning there's only one valid balance value at each step. When multiple balances are possible at some position, multiple valid reconstructions exist. For a valid RBS, the balance must never drop below zero and must reach exactly zero at the end.

The challenge is determining when the balance is uniquely determined versus when multiple paths remain viable. I need to check not just the range of possible balances, but also which values in that range can actually lead to a valid completion. The minimum balance at each position represents the best-case scenario (maximizing '(' choices), and if that minimum is non-negative, the sequence c

→ terminal({"command": "cd /workdir && echo -e \"1\\n((?)\" | python3 solution.py"})