CF 105A - Transmigration

The problem simulates a single transmigration process in a role-playing game. We start with a character who already possesses several skills. Each skill has a name and an experience level. When transmigration occurs, every existing skill has its level reduced by a coefficient k.

codeforcescompetitive-programmingimplementation

CF 105A - Transmigration

Rating: 1700
Tags: implementation
Solve time: 3m 6s
Verified: yes

Solution

Problem Understanding

The problem simulates a single transmigration process in a role-playing game.

We start with a character who already possesses several skills. Each skill has a name and an experience level. When transmigration occurs, every existing skill has its level reduced by a coefficient k. The new level becomes the integer part of k × level.

After this reduction step, any skill whose resulting level is strictly less than 100 is forgotten and disappears completely.

Next, the character receives the skills that belong to the target class. If the character already has one of those skills after the reduction phase, nothing changes. If the skill is not currently present, it is added with level 0.

The final task is to output all remaining skills after these transformations, sorted lexicographically by skill name.

The constraints are very small. Both the number of current skills and the number of target-class skills are at most 20. Even a straightforward implementation that repeatedly scans collections would be easily fast enough. The challenge is not algorithmic complexity, but correctly following the sequence of transformations described in the statement.

Several subtle cases can produce incorrect answers if the rules are applied in the wrong order.

Consider:

1 1 0.50
fire 300
fire

The skill becomes 150 after reduction and survives. Since the class skill already exists, it must remain 150.

Correct output:

1
fire 150

A careless implementation might overwrite it with 0.

Another important case is when a class skill existed originally but was forgotten after reduction:

1 1 0.50
fire 180
fire

The reduced value is 90, so the skill is forgotten. The class then grants fire again at level 0.

Correct output:

1
fire 0

An implementation that checks class skills before removing forgotten skills would produce the wrong result.

The boundary value 100 is also important:

1 0 0.50
fire 200

The reduced value is exactly 100, which survives because only values strictly less than 100 are removed.

Correct output:

1
fire 100

Using <= 100 instead of < 100 would incorrectly delete the skill.

Approaches

A direct simulation immediately suggests itself. We can process every existing skill, compute its reduced level, and keep only those whose new level is at least 100. After that, we process the target class skills and add any missing ones with level 0.

Because there are at most twenty skills in either list, even a brute-force approach using nested scans would be completely acceptable. The worst case would involve only a few hundred operations.

A slightly cleaner approach uses a dictionary keyed by skill name. The dictionary naturally represents the set of skills currently possessed by the character. After applying the reduction and forgetting rules, adding class skills becomes a simple membership check.

The key observation is that the problem is purely a state transformation. There is no optimization, graph search, dynamic programming, or complex data structure involved. We only need to carefully follow the rules in the exact order given.

Approach Time Complexity Space Complexity Verdict
Brute Force O((n + m)^2) O(n + m) Accepted
Optimal O(n + m + z log z) O(n + m) Accepted

Here z is the number of skills in the final answer, and the sorting step dominates the running time.

Algorithm Walkthrough

  1. Read n, m, and the coefficient k.
  2. Create an empty dictionary that will store surviving skills after transmigration.
  3. For each of the n current skills, compute:
new_level = floor(k × old_level)

The statement explicitly requires taking the integer part. 4. If new_level is at least 100, insert the skill into the dictionary with that value.

Skills below 100 are forgotten and should not appear in the dictionary. 5. Read all m target-class skills one by one. 6. For each class skill, check whether it already exists in the dictionary.

If it does not exist, insert it with level 0.

If it already exists, leave its current value unchanged. 7. Extract all dictionary entries and sort them lexicographically by skill name. 8. Print the number of skills and then print each (name, level) pair in sorted order.

Why it works

After step 4, the dictionary contains exactly the skills that survive the reduction phase, because every original skill is transformed according to the coefficient and every skill below 100 is removed.

After step 6, every target-class skill is guaranteed to exist in the dictionary. Existing surviving skills retain their reduced level, while missing class skills are added at level 0, exactly matching the rules of transmigration.

Since the final output is simply the lexicographically ordered version of this final skill set, the algorithm produces the unique correct result.

Python Solution

import sys
input = sys.stdin.readline

def solve():
    n, m, k_str = input().split()

    n = int(n)
    m = int(m)

    num, den = map(int, k_str.split('.'))
    k_num = num * 100 + den

    skills = {}

    for _ in range(n):
        name, exp = input().split()
        exp = int(exp)

        new_exp = (exp * k_num) // 100

        if new_exp >= 100:
            skills[name] = new_exp

    for _ in range(m):
        name = input().strip()

        if name not in skills:
            skills[name] = 0

    items = sorted(skills.items())

    out = [str(len(items))]
    for name, level in items:
        out.append(f"{name} {level}")

    sys.stdout.write("\n".join(out))

if __name__ == "__main__":
    solve()

The implementation follows the simulation directly.

A small detail is how the coefficient k is handled. Using floating-point arithmetic would usually work for these constraints, but integer arithmetic is cleaner and avoids any possibility of precision issues. Since k always contains exactly two decimal places, we convert it into an integer percentage. For example, 0.75 becomes 75, and the reduced level is computed as:

floor(exp × 75 / 100)

using integer division.

The dictionary stores the current set of skills after the forgetting phase. When class skills are processed, a simple membership check determines whether a skill should be added at level 0.

Finally, sorting the dictionary items produces the required lexicographical order.

Worked Examples

Sample 1

Input:

5 4 0.75
axe 350
impaler 300
ionize 80
megafire 120
magicboost 220
heal
megafire
shield
magicboost

After processing existing skills:

Skill Original Reduced Kept?
axe 350 262 Yes
impaler 300 225 Yes
ionize 80 60 No
megafire 120 90 No
magicboost 220 165 Yes

Current dictionary:

Skill Level
axe 262
impaler 225
magicboost 165

Now process class skills:

Class Skill Already Present? Action
heal No Add 0
megafire No Add 0
shield No Add 0
magicboost Yes Keep 165

Final dictionary:

Skill Level
axe 262
heal 0
impaler 225
magicboost 165
megafire 0
shield 0

Sorted output:

6
axe 262
heal 0
impaler 225
magicboost 165
megafire 0
shield 0

This example demonstrates that a skill can disappear during reduction and then be reintroduced as a class skill with level 0.

Additional Example

Input:

2 2 0.50
fire 300
ice 180
fire
wind

Reduction phase:

Skill Original Reduced Kept?
fire 300 150 Yes
ice 180 90 No

Dictionary after reduction:

Skill Level
fire 150

Class skill phase:

Class Skill Already Present? Result
fire Yes Keep 150
wind No Add 0

Final state:

Skill Level
fire 150
wind 0

Output:

2
fire 150
wind 0

This example shows that existing surviving skills are never overwritten by a class skill.

Complexity Analysis

Measure Complexity Explanation
Time O(n + m + z log z) Processing skills is linear, sorting the final list costs O(z log z)
Space O(n + m) Dictionary stores all surviving and added skills

Since n and m are both at most 20, the running time is effectively instantaneous. The solution is far below the limits in both time and memory usage.

Test Cases

# helper: run solution on input string, return output string
import io
import sys

def run(inp: str) -> str:
    sys.stdin = io.StringIO(inp)

    output = io.StringIO()
    old_stdout = sys.stdout
    sys.stdout = output

    def solve():
        input = sys.stdin.readline

        n, m, k_str = input().split()
        n = int(n)
        m = int(m)

        num, den = map(int, k_str.split('.'))
        k_num = num * 100 + den

        skills = {}

        for _ in range(n):
            name, exp = input().split()
            exp = int(exp)

            val = (exp * k_num) // 100
            if val >= 100:
                skills[name] = val

        for _ in range(m):
            name = input().strip()
            if name not in skills:
                skills[name] = 0

        items = sorted(skills.items())

        print(len(items))
        for name, level in items:
            print(name, level)

    solve()

    sys.stdout = old_stdout
    return output.getvalue()

# sample 1
assert run(
"""5 4 0.75
axe 350
impaler 300
ionize 80
megafire 120
magicboost 220
heal
megafire
shield
magicboost
"""
) == """6
axe 262
heal 0
impaler 225
magicboost 165
megafire 0
shield 0
"""

# minimum input
assert run(
"""1 1 0.50
fire 200
fire
"""
) == """1
fire 100
"""

# forgotten then re-added
assert run(
"""1 1 0.50
fire 180
fire
"""
) == """1
fire 0
"""

# surviving class skill keeps level
assert run(
"""1 1 0.50
fire 300
fire
"""
) == """1
fire 150
"""

# all skills forgotten, class introduces new ones
assert run(
"""2 2 0.01
a 9999
b 5000
c
d
"""
) == """2
c 0
d 0
"""
Test input Expected output What it validates
Sample 1 Official answer Full simulation behavior
Single skill reduced to 100 Skill survives Correct < 100 check
Forgotten then re-added Level 0 skill Correct operation order
Surviving class skill Original reduced level No accidental overwrite
Tiny coefficient Only class skills remain Complete forgetting behavior

Edge Cases

A skill whose reduced value becomes exactly 100 must survive. Consider:

1 0 0.50
fire 200

The reduced value is 100. Since only values strictly below 100 are forgotten, the final output is:

1
fire 100

The implementation uses >= 100, which handles this boundary correctly.

A class skill may already exist after the reduction phase. Consider:

1 1 0.50
fire 300
fire

The reduced value is 150, so the skill survives. When processing class skills, the dictionary already contains "fire", so nothing is changed. The final result remains:

1
fire 150

This prevents accidental replacement with level 0.

A skill can disappear and then be granted again by the new class:

1 1 0.50
fire 180
fire

The reduced value becomes 90, so the skill is forgotten. The class skill processing then sees that "fire" is absent and inserts it with level 0. The result is:

1
fire 0

This verifies that forgetting happens before adding class-specific skills, exactly as required by the statement.