CF 1879A - Rigged!

CF 1879A - Rigged!

Rating: 800
Tags: greedy
Solve time: 5m 30s
Verified: yes

Solution

Problem Understanding

We have a weightlifting competition with n athletes. Athlete i has strength s_i and endurance e_i.

A barbell weight w is chosen. Any athlete whose strength is less than w cannot lift the barbell at all. Any athlete whose strength is at least w lifts it exactly e_i times. The winner is the unique athlete with the highest number of lifts. If several athletes tie for the highest number, nobody wins.

Polycarp is athlete 1. We must determine whether there exists some weight w that makes Polycarp the unique winner. If such a weight exists, we may output any valid value of w. Otherwise we output -1.

The constraints are very small. There are at most 100 athletes per test case and at most 100 test cases. Even an O(n²) solution is trivial here. The large strength values do not matter because we never need to iterate through all possible weights. We only need to reason about which athletes remain eligible for a given choice of w.

A common mistake is to focus on finding an explicit winning weight immediately. Consider:

Polycarp: (7, 4)
Other:    (9, 3)

Choosing w = 5 allows both athletes to compete, but Polycarp wins because 4 > 3.

Another easy mistake is forgetting ties. Consider:

Polycarp: (1337, 3)
Other:    (1337, 3)

Any weight that allows Polycarp also allows the other athlete. Their lift counts are equal, so there is never a unique winner. The answer is -1.

A third subtle case occurs when another athlete is stronger and has at least as much endurance:

Polycarp: (4, 6)
Other:    (100, 100)

Any weight that lets Polycarp participate also lets the other athlete participate, because the other athlete is stronger. Since the other athlete lifts more times, Polycarp can never win.

Approaches

A brute force viewpoint is to try every relevant barbell weight and simulate the competition. Since strengths can be as large as 10^9, iterating over all possible weights is impossible. We could reduce the search to the distinct strength values that appear in the input, because the set of eligible athletes only changes when we cross a strength threshold. That would already be fast enough for these constraints.

There is an even simpler observation.

Suppose another athlete has both strength at least Polycarp's strength and endurance at least Polycarp's endurance. Then whenever Polycarp can lift the barbell, that athlete can also lift it. Since the athlete's endurance is not smaller, Polycarp can never be the unique winner.

Conversely, if no athlete satisfies both conditions simultaneously, choosing

w = s₁

makes Polycarp eligible. Any athlete with strength below s₁ is eliminated immediately. Any athlete with strength at least s₁ must have endurance strictly less than e₁, otherwise they would satisfy both conditions. Hence Polycarp has strictly larger endurance than every remaining competitor and becomes the unique winner.

This turns the entire problem into checking whether there exists an athlete i > 1 such that

s_i ≥ s₁ and e_i ≥ e₁.

If such an athlete exists, the answer is -1. Otherwise w = s₁ works.

Approach	Time Complexity	Space Complexity	Verdict
Brute Force over relevant weights	O(n²)	O(1)	Accepted
Dominance observation	O(n)	O(1)	Accepted

Algorithm Walkthrough

1. Read Polycarp's strength s₁ and endurance e₁.
2. For every other athlete, check whether both conditions hold:

s_i ≥ s₁
e_i ≥ e₁

3. If such an athlete exists, print -1.

This athlete is at least as strong and at least as enduring as Polycarp. Whenever Polycarp can participate, this athlete can participate too, preventing a unique Polycarp victory. 4. If no such athlete exists, print s₁.

Choosing w = s₁ guarantees that Polycarp participates. Every other participating athlete must have endurance strictly less than e₁, so Polycarp wins uniquely.

Why it works

An athlete can only threaten Polycarp if they are still eligible whenever Polycarp is eligible. That requires strength at least s₁.

Among those athletes, the only ones capable of preventing Polycarp from being the unique winner are those whose endurance is at least e₁.

Thus the existence of an athlete with

s_i ≥ s₁ and e_i ≥ e₁

is exactly the condition that makes victory impossible.

If no such athlete exists, selecting w = s₁ eliminates all weaker athletes and leaves only athletes with strictly smaller endurance than Polycarp. Therefore Polycarp becomes the unique winner.

Python Solution

import sys
input = sys.stdin.readline

def solve():
    t = int(input())

    for _ in range(t):
        n = int(input())

        s1, e1 = map(int, input().split())
        possible = True

        for _ in range(n - 1):
            s, e = map(int, input().split())

            if s >= s1 and e >= e1:
                possible = False

        print(s1 if possible else -1)

solve()

The implementation follows the proof directly.

We store Polycarp's parameters and scan through the remaining athletes once. If we ever find an athlete whose strength and endurance are both at least Polycarp's values, we mark the answer as impossible.

There is no need to store all athletes because each competitor is examined independently. The solution uses constant extra memory.

The comparison must use >= rather than >. An athlete with exactly the same endurance still causes a tie and prevents a unique winner.

Worked Examples

Example 1

Input:

4
7 4
9 3
4 6
2 2

Athlete	Strength ≥ 7?	Endurance ≥ 4?	Blocks Polycarp?
(9, 3)	Yes	No	No
(4, 6)	No	Yes	No
(2, 2)	No	No	No

No athlete satisfies both conditions.

Output:

7

A valid sample answer is 5, but any valid weight is accepted. Choosing w = 7 also makes Polycarp the unique winner.

Example 2

Input:

2
4 6
100 100

Athlete	Strength ≥ 4?	Endurance ≥ 6?	Blocks Polycarp?
(100, 100)	Yes	Yes	Yes

A blocking athlete exists.

Output:

-1

This example demonstrates the core impossibility condition.

Complexity Analysis

Measure	Complexity	Explanation
Time	O(n)	One pass through the athletes
Space	O(1)	Only a few variables are stored

Since n ≤ 100, the solution is vastly below the time limit. Even with the maximum number of test cases, the total amount of work is tiny.

Test Cases

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

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

    out = []
    t = int(input())

    for _ in range(t):
        n = int(input())

        s1, e1 = map(int, input().split())
        ok = True

        for _ in range(n - 1):
            s, e = map(int, input().split())
            if s >= s1 and e >= e1:
                ok = False

        out.append(str(s1 if ok else -1))

    return "\n".join(out)

# provided sample
assert run(
"""3
4
7 4
9 3
4 6
2 2
2
4 6
100 100
2
1337 3
1337 3
"""
) == "7\n-1\n-1"

# equal athlete causes tie
assert run(
"""1
2
10 5
10 5
"""
) == "-1"

# stronger but lower endurance does not block
assert run(
"""1
2
10 5
100 4
"""
) == "10"

# weaker but higher endurance does not block
assert run(
"""1
2
10 5
9 100
"""
) == "10"

# minimum n
assert run(
"""1
2
1 1
2 2
"""
) == "-1"

Test input	Expected output	What it validates
Equal athlete	-1	Tie prevents a unique winner
Stronger, lower endurance	Polycarp strength	Both conditions are required
Weaker, higher endurance	Polycarp strength	Strength condition matters
Minimum size case	-1	Smallest valid input

Edge Cases

Consider:

1
2
10 5
10 5

The second athlete has identical parameters. The algorithm detects

10 ≥ 10
5 ≥ 5

and outputs -1. Any weight that allows Polycarp also allows the other athlete, producing a tie.

Consider:

1
2
10 5
100 4

The second athlete is stronger but has lower endurance. The algorithm sees that the endurance condition fails and outputs 10. Choosing w = 10 allows both athletes to compete, but Polycarp lifts the barbell five times while the other athlete lifts it four times.

Consider:

1
2
10 5
9 100

The second athlete has enormous endurance but insufficient strength. Choosing w = 10 eliminates that athlete completely. The algorithm correctly outputs 10.

Consider:

1
2
4 6
100 100

The second athlete dominates Polycarp in both strength and endurance. No matter which valid weight is chosen, if Polycarp can lift the barbell then the second athlete can too, and the second athlete lifts more times. The algorithm detects this domination and outputs -1.