Overview

This is a problem asking for the number of attractions that all group members can ride. The condition for everyone to ride is that even the shortest person in the group must meet the height requirement.

Analysis

Key Insight

For all group members to ride an attraction, everyone must meet the height requirement.

Here is an important observation: - If the shortest person in the group can clear the height requirement, then everyone else can definitely clear it too - Conversely, if the shortest person cannot clear it, then not everyone can ride

In other words, if the group’s minimum height \(\geq\) the attraction’s height requirement, then everyone can ride that attraction.

Verification with a Concrete Example

For example, if the group’s heights are \([150, 160, 140]\) cm: - The minimum height is \(140\) cm - An attraction with a height requirement of \(130\) cm → \(130 \leq 140\), so everyone can ride ✓ - An attraction with a height requirement of \(145\) cm → \(145 > 140\), so not everyone can ride ✗

Comparison with the Naive Approach

A naive approach would be to check all members for each attraction: - For each attraction \(i\), verify that all members \(j\) satisfy \(S_i \leq L_j\) - This takes \(O(N \times M)\), requiring up to \(4 \times 10^{10}\) comparisons in the worst case, risking TLE

However, if we realize that we only need to look at the minimum height, we can solve it in \(O(N + M)\).

Algorithm

1. Find the minimum value \(\min(L)\) from the group members’ height list \(L\)
2. For each attraction’s height requirement \(S_i\), count those that satisfy \(S_i \leq \min(L)\)
3. Output that count

N, M = map(int, input().split())
L = list(map(int, input().split()))
S = list(map(int, input().split()))

min_height = min(L)  # Find the minimum height in the group
count = sum(1 for s in S if s <= min_height)  # Count attractions that meet the condition
print(count)

Complexity

• Time complexity: \(O(N + M)\)
  • \(O(N)\) for computing \(\min(L)\)
  • \(O(M)\) for checking each attraction
• Space complexity: \(O(N + M)\)
  • \(O(N)\) for the height list \(L\)
  • \(O(M)\) for the height requirement list \(S\)

Implementation Notes

• Python’s min() function easily retrieves the minimum value of a list
• sum(1 for s in S if s <= min_height) is a method to count elements satisfying a condition using a generator expression. It gives the same result as len([s for s in S if s <= min_height]) but is more memory-efficient
• Since the constraints are as large as \(N, M \leq 2 \times 10^5\), an \(O(N \times M)\) algorithm should be avoided

Source Code

N, M = map(int, input().split())
L = list(map(int, input().split()))
S = list(map(int, input().split()))

min_height = min(L)
count = sum(1 for s in S if s <= min_height)
print(count)

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