Direction Assignments with Exactly K Visible People - Video Solutions
Leetcode 3881 | Direction Assignments with Exactly K Visible People | Leetcode biweekly contest 179
Direction Assignments with Exactly K Visible People - Video Solution
Watch 2 video solutions for Direction Assignments with Exactly K Visible People, a medium level problem involving Math, Combinatorics. This walkthrough by CodeWithMeGuys has 528 views views. Want to try solving it yourself? Practice on FleetCode or read the detailed text solution.
Problem Statement
You are given three integers n, pos, and k.
There are n people standing in a line indexed from 0 to n - 1. Each person independently chooses a direction:
'L': visible only to people on their right'R': visible only to people on their left
pos sees others as follows:
- A person
i < posis visible if and only if they choose'L'. - A person
i > posis visible if and only if they choose'R'.
Return the number of possible direction assignments such that the person at index pos sees exactly k people.
Since the answer may be large, return it modulo 109 + 7.
Example 1:
Input: n = 3, pos = 1, k = 0
Output: 2
Explanation:
- Index 0 is to the left of
pos = 1, and index 2 is to the right ofpos = 1. - To see
k = 0people, index 0 must choose'R'and index 2 must choose'L', keeping both invisible. - The person at index 1 can choose
'L'or'R'since it does not affect the count. Thus, the answer is 2.
Example 2:
Input: n = 3, pos = 2, k = 1
Output: 4
Explanation:
- Index 0 and index 1 are left of
pos = 2, and there is no index to the right. - To see
k = 1person, exactly one of index 0 or index 1 must choose'L', and the other must choose'R'. - There are 2 ways to choose which index is visible from the left.
- The person at index 2 can choose
'L'or'R'since it does not affect the count. Thus, the answer is2 + 2 = 4.
Example 3:
Input: n = 1, pos = 0, k = 0
Output: 2
Explanation:
- There are no indices to the left or right of
pos = 0. - To see
k = 0people, no additional condition is required. - The person at index 0 can choose
'L'or'R'. Thus, the answer is 2.
Constraints:
1 <= n <= 1050 <= pos, k <= n - 1
Approach Overview
Problem Overview: You are given a line of people and must assign each person a viewing direction (left or right). A person is considered visible if nobody taller blocks their view in the direction they face. The task is to count how many direction assignments produce exactly K visible people.
Approach 1: Brute Force Direction Enumeration (O(2^n) time, O(1) space)
The most direct idea is to try every possible assignment of directions. For each of the 2^n configurations, scan the line and check whether each person can see in the direction they face. Visibility can be determined by walking toward that direction until a taller person blocks the view. After computing the number of visible people for that configuration, increment the count if it equals K. This approach works only for very small n because the number of assignments doubles with each additional person.
Approach 2: Combinatorics + Enumeration (O(nk) time, O(n) space)
A more practical solution observes that visibility depends on relative height order rather than the exact configuration of directions. The tallest people determine how far visibility can extend because they block everyone behind them. Instead of enumerating all direction assignments, enumerate how many people contribute to visibility from the left and from the right.
Precompute combinations such as C(n, r) to efficiently count how many ways people can be arranged so that exactly i are visible from the left and j from the right where i + j = K. For each split of K, count the number of valid placements and multiply by the number of direction assignments that keep those people visible. This converts an exponential search into a combinatorial counting problem.
The algorithm iterates over all valid splits of visible people and sums their contributions using precomputed factorials or dynamic programming for combinations. This reduces the search to polynomial time and avoids checking every direction configuration explicitly. Concepts from combinatorics, enumeration, and math-based counting drive the optimization.
Recommended for interviews: Interviewers expect the combinatorics + enumeration approach. Starting with the brute force shows you understand the visibility rule and the search space. Transitioning to counting arrangements using combinations demonstrates algorithmic insight and reduces the complexity from O(2^n) to roughly O(nk), which is practical for typical constraints.
Complexity Analysis
| Approach | Time | Space | When to Use |
|---|---|---|---|
| Brute Force Direction Enumeration | O(2^n) | O(1) | Useful for understanding the visibility rule or validating small test cases |
| Combinatorics + Enumeration | O(nk) | O(n) | General solution that counts valid visibility configurations without enumerating all assignments |
| Combinatorics with Precomputed Factorials | O(nk) | O(n) | Preferred when many combination queries are needed or constraints are larger |