How to Solve Russian Doll Envelopes

The problem requires finding the maximum number of envelopes that can fit inside each other. By sorting envelopes and applying binary search with dynamic programming, we efficiently compute the solution. The challenge lies in managing the transition between states, especially with envelope heights and widths.

Problem Statement

You are given a 2D array of integers envelopes where each element envelopes[i] = [wi, hi] represents the width and height of an envelope. You need to find the maximum number of envelopes that can be Russian-dolled, meaning that one envelope can fit into another if and only if both its width and height are strictly greater than the other envelope's width and height.

Return the maximum number of envelopes that can fit inside each other, forming the longest chain of Russian-dolled envelopes. The input will have envelopes with width and height constraints, and the challenge lies in optimizing the search for the most fitting arrangement.

Examples

Example 1

Input: envelopes = [[5,4],[6,4],[6,7],[2,3]]

Output: 3

The maximum number of envelopes you can Russian doll is 3 ([2,3] => [5,4] => [6,7]).

Example 2

Input: envelopes = [[1,1],[1,1],[1,1]]

Output: 1

Example details omitted.

Constraints

• 1 <= envelopes.length <= 105
• envelopes[i].length == 2
• 1 <= wi, hi <= 105

Solution Approach

Sort Envelopes

First, sort the envelopes primarily by width in ascending order. If two envelopes have the same width, sort them by height in descending order. This sorting ensures that we only need to focus on finding the longest increasing subsequence based on height.

Binary Search with Dynamic Programming

Once sorted, the problem reduces to finding the longest increasing subsequence of heights. We can achieve this efficiently by using binary search on a dynamic programming array, where each element represents the smallest ending height of an increasing subsequence of a given length.

Transition State Management

The core challenge lies in managing the transitions between the state of increasing subsequences. By using dynamic programming to track the best subsequences and updating it through binary search, the solution efficiently handles larger inputs.

Complexity Analysis

The time complexity of this approach is O(n log n) due to the sorting and binary search, where n is the number of envelopes. The space complexity is O(n) for storing the dynamic programming array. Sorting the envelopes is the most computationally expensive part of the algorithm.

What Interviewers Usually Probe

• Look for clear understanding of dynamic programming and binary search usage together.
• Assess the candidate's ability to explain how sorting impacts the overall approach.
• Check for familiarity with state transitions in dynamic programming problems.

Common Pitfalls or Variants

Common pitfalls

• Failing to sort the envelopes correctly, especially handling ties in width by sorting in descending order of height.
• Overcomplicating the problem by trying to use brute force to compare all pairs of envelopes.
• Misunderstanding the role of binary search in the dynamic programming step, potentially leading to inefficient solutions.

Follow-up variants

• What if envelopes have more than two dimensions?
• What if the envelopes are in a randomized order with no restrictions?
• Can the approach be optimized further for lower space complexity?

How GhostInterview Helps

• GhostInterview guides you through state transition management, which is crucial for optimizing dynamic programming solutions.
• It helps you navigate through binary search application in dynamic programming, focusing on how the data structure evolves.
• With a focus on the exact problem pattern, GhostInterview provides useful insights on managing time complexity and space optimization.

Topic Pages

• Array
• Binary Search
• Dynamic Programming
• Sorting

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FAQ

What is the time complexity of the Russian Doll Envelopes problem?

The time complexity is O(n log n), where n is the number of envelopes, due to sorting and the binary search on the dynamic programming array.

How do we deal with envelopes having the same width in the Russian Doll Envelopes problem?

Envelopes with the same width should be sorted in descending order of height to ensure we don’t mistakenly include non-nested envelopes.

What dynamic programming approach is used in Russian Doll Envelopes?

The problem uses a dynamic programming approach where we maintain an array of minimum possible heights for increasing subsequences, and binary search helps efficiently update this array.

What is the binary search used for in the Russian Doll Envelopes problem?

Binary search is used to find the position in the dynamic programming array where the current envelope’s height can extend or replace an existing subsequence.

Can the solution to the Russian Doll Envelopes problem be optimized for space complexity?

Yes, the space complexity can be reduced by using a more compact data structure to track the subsequences, though the core time complexity will remain O(n log n).