What is the time complexity of Count the Hidden Sequences?

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

What data structures are used to solve Count the Hidden Sequences?

This problem uses Array, Prefix Sum. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Count the Hidden Sequences?

Insight 1: The problem can be solved by calculating the prefix sum of the `differences` array. Each prefix sum represents the difference between an element in the hidden array and the first element. Insight 2: To find the number of possible hidden sequences, we need to determine the minimum and maximum possible values any element in the sequence can take, relative to the first element.

Which companies ask LeetCode 2145. Count the Hidden Sequences in interviews?

This problem is frequently asked at Zomato.

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Count the Hidden Sequences - LeetCode 2145 Solution

Count the Hidden Sequences - Complete Solution Guide

Count the Hidden Sequences is LeetCode problem 2145, 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

You are given a 0-indexed array of n integers differences , which describes the differences between each pair of consecutive integers of a hidden sequence of length (n + 1) . More formally, call the hidden sequence hidden , then we have that differences[i] = hidden[i + 1] - hidden[i] . You are further given two integers lower and upper that describe the inclusive range of values [lower, upper] that the hidden sequence can contain. For example, given differences = [1, -3, 4] , lower = 1 , upper =

Detailed Explanation

The problem asks us to find the number of possible hidden sequences that satisfy a given set of difference constraints and a range constraint. We are given an array `differences` where `differences[i] = hidden[i+1] - hidden[i]`. We are also given a lower and upper bound, `lower` and `upper`, respectively, that all numbers in the hidden sequence must fall within (inclusive). Our goal is to count how many different possible starting values of the hidden sequence exist that lead to a valid sequence adhering to both the difference constraints and the range constraints.

Solution Approach

The solution calculates the prefix sums of the differences array to track how the elements of the hidden array change relative to the first element (the initial value). It then finds the minimum and maximum prefix sums. This gives the range that all elements of the hidden array cover relative to the first element. Then, we determine how many starting values will make the entire sequence fall between `lower` and `upper`. We calculate how much we can shift the minimum and maximum value by, and if there are any valid shifts, we find the number of possible starting values.

Step-by-Step Algorithm

Step 1: Initialize `current_sum`, `min_sum`, and `max_sum` to 0. These will track the current prefix sum, the minimum prefix sum, and the maximum prefix sum, respectively.
Step 2: Iterate through the `differences` array. In each iteration, update `current_sum` by adding the current difference to it.
Step 3: Update `min_sum` and `max_sum` by taking the minimum and maximum of their current values and `current_sum`, respectively. This keeps track of the lowest and highest values in the hidden sequence relative to the starting value.
Step 4: Calculate the `sequence_span` which is `max_sum - min_sum`. This gives the total range of numbers in the hidden sequence relative to the starting value.
Step 5: Calculate `allowed_span` which is `upper - lower`. This gives the allowed range that each value in the hidden sequence can take.
Step 6: If `sequence_span` is greater than `allowed_span`, it is impossible for the hidden sequence to fit within the specified bounds. Return 0.
Step 7: Return `allowed_span - sequence_span + 1`. This represents the number of possible starting values that satisfy the constraints. We add 1 since the range is inclusive.

Key Insights

Insight 1: The problem can be solved by calculating the prefix sum of the `differences` array. Each prefix sum represents the difference between an element in the hidden array and the first element.
Insight 2: To find the number of possible hidden sequences, we need to determine the minimum and maximum possible values any element in the sequence can take, relative to the first element.
Insight 3: The possible starting values for the hidden sequence depend on ensuring that all elements in the generated hidden sequence fall within the specified [lower, upper] range. We need to find the valid range for the first element that satisfies this.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(1)

Topics

This problem involves: Array, Prefix Sum.

Companies

Asked at: Zomato.

Count the Hidden Sequences | LeetCode 2145

Medium

Array Prefix Sum

Time: O(n)Space: O(1)

Solution Code

Available in:

class Solution {
    public int numberOfArrays(int[] differences, int lower, int upper) {
        long current_sum = 0;
        long min_sum = 0;
        long max_sum = 0;
        
        for (int diff : differences) {
            current_sum += diff;
            min_sum = Math.min(min_sum, current_sum);
            max_sum = Math.max(max_sum, current_sum);
        }
        
        long sequence_span = max_sum - min_sum;
        long allowed_span = (long)upper - (long)lower;
        
        if (sequence_span > allowed_span) {
            return 0;
        }
        
        return (int)(allowed_span - sequence_span + 1);
    }
}

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Problem Info

Description

Problem 2145: Count the Hidden Sequences - Detailed explanation and solution approaches.

Topics

Array Prefix Sum

Companies

Zomato

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:2145

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 `current_sum`, `min_sum`, and `max_sum` to 0. These will track the current prefix sum, the minimum prefix sum, and the maximum prefix sum, respectively.
2
Step 2: Iterate through the `differences` array. In each iteration, update `current_sum` by adding the current difference to it.
3
Step 3: Update `min_sum` and `max_sum` by taking the minimum and maximum of their current values and `current_sum`, respectively. This keeps track of the lowest and highest values in the hidden sequence relative to the starting value.
4
Step 4: Calculate the `sequence_span` which is `max_sum - min_sum`. This gives the total range of numbers in the hidden sequence relative to the starting value.
5
Step 5: Calculate `allowed_span` which is `upper - lower`. This gives the allowed range that each value in the hidden sequence can take.
6
Step 6: If `sequence_span` is greater than `allowed_span`, it is impossible for the hidden sequence to fit within the specified bounds. Return 0.
7
Step 7: Return `allowed_span - sequence_span + 1`. This represents the number of possible starting values that satisfy the constraints. We add 1 since the range is inclusive.

💡

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(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The dominant operation is the single loop through the `differences` array, making the time complexity linear. The space complexity is constant since the amount of memory used does not depend on the input size.

📖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

Brute Force

Iterate through all possible starting values within the range [lower, upper]. For each starting value, generate the hidden sequence based on the differences and check if all the elements are within the range [lower, upper].

Time: O(n * (upper - lower)), where n is the length of the `differences` array. The outer loop iterates through the possible starting values, and the inner loop generates the hidden sequence.Space: O(n), to store the hidden sequence for each starting value.

Trade-offs: The brute force approach is simpler to understand but much slower, especially when the range `upper - lower` is large. The optimized approach is much faster but might be slightly harder to grasp initially. Brute force is only viable for small constraints.

✨

Key Insights

1Insight 1: The problem can be solved by calculating the prefix sum of the `differences` array. Each prefix sum represents the difference between an element in the hidden array and the first element.
2Insight 2: To find the number of possible hidden sequences, we need to determine the minimum and maximum possible values any element in the sequence can take, relative to the first element.
3Insight 3: The possible starting values for the hidden sequence depend on ensuring that all elements in the generated hidden sequence fall within the specified [lower, upper] range. We need to find the valid range for the first element that satisfies this.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Edge Cases

Edge case 1: `differences` array is empty. The provided code will still execute correctly, but if you were initializing the first element of the array to some initial value you would need to handle this case.
Edge case 2: `upper` is less than `lower`. The `allowed_span` would be negative. The existing solution handles it correctly in such case it would return 0 because no valid sequences exist.
Edge case 3: The `sequence_span` is larger than `allowed_span`. No solution can exist and algorithm will return 0 which is correct.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Integer Overflow: Not using `long` to store prefix sums. The problem constraints allow for large differences which could cause overflow if using `int`. The provided solutions correctly use `long` in Java and C++.

✗Common mistake 2: Incorrect calculation of valid starting values: Not adding 1 when calculating the number of valid starting values: `allowed_span - sequence_span` instead of `allowed_span - sequence_span + 1`. Remember to include the boundaries since the interval is inclusive.

✅

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: Stock Price Analysis: Analyzing stock price fluctuations over a period of time. The `differences` array could represent the daily changes in stock price, and the goal is to find the number of initial prices that would keep the stock price within a certain range.

🏦Real-world scenario 2: Sensor Data Calibration: Calibrating sensor readings where the `differences` array represents the offsets between consecutive sensor readings. The goal is to find the number of possible baseline readings that would keep all sensor readings within a valid range.

💡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

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

Medium problems are the bread and butter of technical interviews. Practice explaining your approach before coding.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

👨‍💻 From Experience

This problem pattern appears frequently in real codebases. I've used similar logic when building production systems.

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 13, 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

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 13

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 6

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

@InterviewProJun 29

❓

Frequently Asked Questions

When should I use in-place modification vs creating a new array?

Use in-place modification when space complexity matters (O(1) space requirement) and the original array can be modified. Create a new array when you need to preserve the original data or when the problem involves significant restructuring that would complicate in-place logic.

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 "Count the Hidden Sequences"?

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

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