CF 106130A - \%\%\%自动机

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Solution

Problem Understanding

The task reduces to constructing a single string based on an input integer. We are given a number $n$, and we must output a line consisting of exactly $n$ identical characters, where each character is the percent symbol %.

There is no hidden structure, transformation, or dependency between positions in the output. Each position is independent and identical, so the output is fully determined by repetition count alone.

The constraint $1 \le n \le 100$ implies that even the most direct construction method is trivially fast. Any algorithm that runs in linear time in $n$ is more than sufficient, and even repeated string concatenation without optimization would not risk performance issues at this scale. The memory footprint is also negligible since the output size is at most 100 characters.

Edge cases are minimal due to the small constraint range. The only meaningful boundary case is $n = 1$, where the output must be a single % without extra whitespace or newline anomalies. For example, input 1 must produce %. Any implementation that mistakenly appends separators or assumes multiple tokens would fail this case.

Approaches

A brute-force interpretation would treat the problem as constructing the output character by character in a loop, appending % repeatedly to an initially empty string. This is already optimal in both conceptual simplicity and performance for the given constraints.

The only potential inefficiency arises in languages or implementations where repeated string concatenation creates new strings each time. However, even in that case, with $n \le 100$, the overhead is negligible and still well within limits.

The key observation is that the output does not depend on any state or computation beyond the count $n$. Once this is recognized, the problem reduces to a single primitive operation: repetition of a constant character.

There is no need for alternative algorithms or optimizations beyond using built-in string repetition.

Approach	Time Complexity	Space Complexity	Verdict
Repeated concatenation	O(n²) worst-case (naive string builds)	O(n)	Accepted due to tiny constraints
Direct string multiplication	O(n)	O(n)	Accepted

Algorithm Walkthrough

Optimal Construction

Read the integer $n$ from input. This value directly determines the output length, so no further parsing or transformation is required.
Construct a string consisting of the character % repeated exactly $n$ times. This uses the fact that string repetition is a constant-expression operation in Python-like environments.
Output the constructed string as a single line without additional spacing or formatting.

Why it works

The output definition is purely positional and uniform: every position in the result must contain the same character %. Since there are exactly $n$ positions and no conditional variation between them, the string is uniquely determined as a repetition of a single symbol. There is no alternative valid construction.

Python Solution

import sys
input = sys.stdin.readline

n = int(input().strip())
print('%' * n)

The implementation directly maps the problem statement to Python’s string repetition operator. The strip() ensures robustness against trailing newline characters in the input. The multiplication operator constructs the final string efficiently in linear time.

There are no loops or conditional branches, since the output is fully determined by a single expression.

Worked Examples

Example 1

Input:

Output construction:

Step	n	Current string
Read input	3	""
Repeat `%`	3	"%%%"
Output	3	"%%%"

This confirms that three identical characters are produced in sequence, matching the required length.

Example 2

Input:

Output construction:

Step	n	Current string
Read input	1	""
Repeat `%`	1	"%"
Output	1	"%"

This demonstrates correctness at the lower boundary, where only a single character must be produced without duplication or truncation errors.

Complexity Analysis

Measure	Complexity	Explanation
Time	O(n)	Each character is written once in the output string
Space	O(n)	The resulting string stores exactly $n$ characters

Given $n \le 100$, both time and memory usage are effectively constant in practice. The solution is well within all typical competitive programming limits.

Test Cases

import sys, io

def run(inp: str) -> str:
    sys.stdin = io.StringIO(inp)
    n = int(sys.stdin.readline().strip())
    return "%" * n

# provided sample (from statement)
assert run("3\n") == "%%%", "sample 1"

# minimum size
assert run("1\n") == "%", "minimum case"

# small random case
assert run("5\n") == "%%%%%", "basic repetition"

# maximum size
assert run("100\n") == "%" * 100, "upper bound case"

# additional boundary pattern
assert run("2\n") == "%%", "small even case"

Test input	Expected output	What it validates
1	%	Minimum boundary correctness
100	100 `%` chars	Maximum constraint handling
2	%%	Basic repetition correctness

Edge Cases

For $n = 1$, the algorithm reads the input and performs a single repetition of %, producing "%". There is no loop iteration that could introduce extra characters, so the output remains minimal and correct.

For $n = 100$, the repetition expands to exactly 100 characters. Since the construction is linear and deterministic, there is no risk of overflow, truncation, or formatting errors. The output remains a contiguous sequence of % characters with no separators or trailing whitespace.