What is the time complexity of Number of People Aware of a Secret?

The optimal solution has a time complexity of O(n).

What data structures are used to solve Number of People Aware of a Secret?

This problem uses Dynamic Programming, Queue, Simulation. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Number of People Aware of a Secret?

Insight 1: Dynamic Programming: The number of people who know the secret on a given day depends on the number of people who shared the secret in the past and haven't yet forgotten it. This overlapping subproblems property strongly suggests DP. Insight 2: Sharing Count: It's crucial to keep track of the number of people actively sharing the secret each day, as only these people contribute to the increase in the number of people who know the secret.

Which companies ask LeetCode 2327. Number of People Aware of a Secret in interviews?

This problem is frequently asked at Arcesium, NCR.

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Number of People Aware of a Secret - LeetCode 2327 Solution

Number of People Aware of a Secret - Complete Solution Guide

Number of People Aware of a Secret is LeetCode problem 2327, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

On day 1 , one person discovers a secret. You are given an integer delay , which means that each person will share the secret with a new person every day , starting from delay days after discovering the secret. You are also given an integer forget , which means that each person will forget the secret forget days after discovering it. A person cannot share the secret on the same day they forgot it, or on any day afterwards. Given an integer n , return the number of people who know the secret at t

Detailed Explanation

The problem describes a scenario where one person initially knows a secret. Each person who knows the secret shares it with a new person every day starting after a 'delay' period. However, each person forgets the secret after a 'forget' period. The goal is to determine the total number of people who know the secret on day 'n', modulo 10^9 + 7.

Solution Approach

The solution uses dynamic programming to simulate the spread of the secret day by day. A DP array `dp` is maintained, where `dp[i]` represents the number of people who *learn* the secret on day `i`. We also track `sharing_count`, which is the number of people who are currently sharing the secret on a given day. This variable is updated each day by adding the number of people who started sharing the secret and subtracting the number of people who forgot the secret. Finally, the solution sums up the number of people who learned the secret in the last 'forget' days to find the total number of people who know the secret on day 'n'.

Step-by-Step Algorithm

Step 1: Initialize a DP array `dp` of size `n+1` with all values set to 0. Set `dp[1] = 1` because one person knows the secret initially.
Step 2: Initialize `sharing_count` to 0. This variable tracks the number of people currently sharing the secret.
Step 3: Iterate from day 2 to day `n` (inclusive):
Step 4: Update `sharing_count` by adding `dp[i - delay]` (number of people who started sharing on day `i - delay`) and subtracting `dp[i - forget]` (number of people who forgot the secret on day `i - forget`). Remember to use modulo arithmetic after each operation.
Step 5: Set `dp[i] = sharing_count`. This stores the number of people who *learn* the secret on day `i`.
Step 6: After the loop finishes, iterate from day `n - forget + 1` to day `n` (inclusive). Sum up the values of `dp[i]` for these days. These are the people who learned the secret within the last `forget` days and haven't forgotten it yet.
Step 7: Return the total sum modulo 10^9 + 7.

Key Insights

Insight 1: Dynamic Programming: The number of people who know the secret on a given day depends on the number of people who shared the secret in the past and haven't yet forgotten it. This overlapping subproblems property strongly suggests DP.
Insight 2: Sharing Count: It's crucial to keep track of the number of people actively sharing the secret each day, as only these people contribute to the increase in the number of people who know the secret.
Insight 3: Modulo Arithmetic: Because the number of people can become very large, performing all calculations modulo 10^9 + 7 is necessary to prevent integer overflow and ensure correct results.
Insight 4: Focusing on new sharers: The core idea is to compute the number of *new* people learning the secret each day, rather than tracking *everyone* who knows the secret at any given day, which would be more complex.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(n)

Topics

This problem involves: Dynamic Programming, Queue, Simulation.

Companies

Asked at: Arcesium, NCR.

Number of People Aware of a Secret | LeetCode 2327

Medium

Dynamic Programming Queue Simulation

Time: O(n)Space: O(n)

Solution Code

Available in:

class Solution {
    public int peopleAwareOfSecret(int n, int delay, int forget) {
        int MOD = 1000000007;
        
        long[] dp = new long[n + 1];
        dp[1] = 1;
        
        long sharing_count = 0;

        for (int i = 2; i <= n; i++) {
            if (i - delay >= 1) {
                sharing_count = (sharing_count + dp[i - delay]) % MOD;
            }
            
            if (i - forget >= 1) {
                sharing_count = (sharing_count - dp[i - forget] + MOD) % MOD;
            }
            
            dp[i] = sharing_count;

        }

        long total_aware = 0;
        for (int i = n - forget + 1; i <= n; i++) {
            total_aware = (total_aware + dp[i]) % MOD;
            
        }
        
        return (int) total_aware;
    }
}

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Problem Info

Description

Problem 2327: Number of People Aware of a Secret - Detailed explanation and solution approaches.

Topics

Dynamic Programming Queue Simulation

Companies

ArcesiumNCR

Stats

Difficulty:Medium

Languages:4

Companies:2

Problem #:2327

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

7 steps

1
Step 1: Initialize a DP array `dp` of size `n+1` with all values set to 0. Set `dp[1] = 1` because one person knows the secret initially.
2
Step 2: Initialize `sharing_count` to 0. This variable tracks the number of people currently sharing the secret.
3
Step 3: Iterate from day 2 to day `n` (inclusive):
4
Step 4: Update `sharing_count` by adding `dp[i - delay]` (number of people who started sharing on day `i - delay`) and subtracting `dp[i - forget]` (number of people who forgot the secret on day `i - forget`). Remember to use modulo arithmetic after each operation.
5
Step 5: Set `dp[i] = sharing_count`. This stores the number of people who *learn* the secret on day `i`.
6
Step 6: After the loop finishes, iterate from day `n - forget + 1` to day `n` (inclusive). Sum up the values of `dp[i]` for these days. These are the people who learned the secret within the last `forget` days and haven't forgotten it yet.
7
Step 7: Return the total sum modulo 10^9 + 7.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n)

👍 Linear - Good! Visits each element once.

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The algorithm iterates through the days from 2 to n, which takes O(n) time. The space complexity is also O(n) because we use a DP array of size n+1.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Queue-based approach

A queue can be used to store the days when people started sharing the secret. When a person forgets the secret, their corresponding entry is removed from the queue. While generally offering no time complexity advantages, visualizing the problem like this may help some users understand the secret sharing/forgetting events as they happen.

Time: O(n)Space: O(n)

Trade-offs: This might be slightly more intuitive for some as it models the process more explicitly, but it doesn't change the time/space complexity.

✨

Key Insights

1Insight 1: Dynamic Programming: The number of people who know the secret on a given day depends on the number of people who shared the secret in the past and haven't yet forgotten it. This overlapping subproblems property strongly suggests DP.
2Insight 2: Sharing Count: It's crucial to keep track of the number of people actively sharing the secret each day, as only these people contribute to the increase in the number of people who know the secret.
3Insight 3: Modulo Arithmetic: Because the number of people can become very large, performing all calculations modulo 10^9 + 7 is necessary to prevent integer overflow and ensure correct results.
4Insight 4: Focusing on new sharers: The core idea is to compute the number of *new* people learning the secret each day, rather than tracking *everyone* who knows the secret at any given day, which would be more complex.

📚

Key Terms Explained

For Students

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Queue

FIFO structure - First In, First Out. Like a line at a store!

Edge Cases

Edge case 1: delay = 1, forget = n. In this case, almost everyone will know the secret at the end. The solution correctly calculates the sum of all people who learned the secret during the last forget (n) days.
Edge case 2: delay is close to forget. The solution correctly handles this case by adding people to the `sharing_count` at the correct time (delay) and subtracting them at the correct time (forget).

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to use modulo arithmetic. Since the answer can be very large, it's crucial to perform all calculations modulo 10^9 + 7 to avoid integer overflow.

✗Common mistake 2: Incorrectly calculating the range of days to sum up at the end. The correct range is from `n - forget + 1` to `n`, representing people who learned the secret within the last `forget` days.

✗Common mistake 3: Not accounting for the initial person who knows the secret, leading to an off-by-one error. The DP array should be initialized with `dp[1] = 1`.

✗Common mistake 4: Confusing who knows the secret on any given day with who starts sharing the secret on that day. The algorithm builds upon who starts sharing the secret. `dp[i]` represents the *new* people sharing the secret at day `i`, not all people who know the secret.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Modeling the spread of information in a social network. The 'delay' represents the time it takes for a person to share the information, and 'forget' represents how long they remember it.

🏦Real-world scenario 2: Simulating the spread of a disease. 'delay' could represent the incubation period, and 'forget' could represent the recovery time. The problem can be adapted to estimate the number of infected individuals at a given time.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

Start with recursion, then memoize. The hardest part is identifying the state. Ask yourself: "What information do I need to make the current decision?"

🎯 Interview Tip

Consider multiple approaches before coding. Mentioning alternatives shows breadth of knowledge.

⚠️ Watch Out

Forgetting base cases or initializing the DP array incorrectly. Start by solving the smallest subproblems.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: November 15, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 14

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJul 7

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJun 30

❓

Frequently Asked Questions

Should I use top-down (memoization) or bottom-up (tabulation)?

Start with top-down (recursion + memoization) as it directly maps to the recursive thinking and handles only needed subproblems. Convert to bottom-up for better space efficiency and to avoid stack overflow. Bottom-up is often faster due to no recursion overhead.

How much time should I spend on a medium problem in interviews?

In a 45-minute interview, aim for: 5-10 minutes understanding and clarifying, 5-10 minutes discussing approach, 20-25 minutes coding, and 5-10 minutes testing. If stuck after 15 minutes of thinking, verbalize your thought process and ask for hints.

What is the importance of time and space complexity analysis?

Complexity analysis is crucial because: 1) Interviewers always ask about it, 2) It helps you choose between approaches, 3) It demonstrates CS fundamentals. Always state both time and space complexity, and be prepared to explain how you derived them.

What is the optimal approach for "Number of People Aware of a Secret"?

For this medium-level Dynamic Programming problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether queue techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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