Skip to content

Latest commit

 

History

History
115 lines (82 loc) · 5.5 KB

explanation.md

File metadata and controls

115 lines (82 loc) · 5.5 KB
Raw

Minimum Number of Removals to Make Mountain Array

This README explains the step-by-step approach to solving the "Minimum Number of Removals to Make Mountain Array" problem. We'll walk through each language solution—C++, Java, JavaScript, Python, and Go—and detail the logic and methodology for creating an optimized solution.

Problem Description

Given an integer array, the task is to transform it into a "mountain array" by removing the fewest elements possible. A mountain array has a sequence that strictly increases to a peak and then strictly decreases after the peak.

Approach

Step 1: Understanding Mountain Array Requirements

  • A valid mountain array has:
    1. An increasing sequence before a peak element.
    2. A decreasing sequence after the peak.
  • For any given element in the array to serve as a "peak," it should have:
    • At least one element on its left in an increasing order.
    • At least one element on its right in a decreasing order.

Step 2: Define LIS and LDS Arrays

  • LIS (Longest Increasing Subsequence) array:
    • For each element, calculate the length of the longest increasing subsequence ending at that element.
  • LDS (Longest Decreasing Subsequence) array:
    • For each element, calculate the length of the longest decreasing subsequence starting at that element.

Step 3: Calculate LIS for Each Element

  • For each element in the array:
    • Iterate through all previous elements.
    • If the current element is greater than a previous element, update its LIS value to LIS[i] = max(LIS[i], LIS[j] + 1).

Step 4: Calculate LDS for Each Element

  • For each element, starting from the end of the array:
    • Iterate through all subsequent elements.
    • If the current element is greater than the next element, update its LDS value to LDS[i] = max(LDS[i], LDS[j] + 1).

Step 5: Find the Maximum Length of a Mountain Array

  • For each element:
    • Check if it can serve as a valid peak by ensuring both LIS[i] > 1 and LDS[i] > 1.
    • For valid peaks, calculate the mountain length as LIS[i] + LDS[i] - 1.
    • Track the maximum mountain length found.

Step 6: Calculate Minimum Removals

  • The minimum number of removals required to form a mountain array is n - maxMountainLength, where n is the size of the input array.

Step-by-Step Code Explanation

C++ Code

  1. Initialize LIS and LDS arrays with default values of 1 for each element.
  2. Calculate the LIS values for each element by comparing with all previous elements.
  3. Calculate the LDS values for each element by comparing with all following elements.
  4. Identify valid peaks where both LIS and LDS values are greater than 1.
  5. Calculate the maximum mountain length by adding the LIS and LDS values for each valid peak.
  6. Return the minimum removals needed by subtracting the maximum mountain length from the total array size.

Java Code

  1. Initialize LIS and LDS arrays with default values of 1 for each element.
  2. Calculate the LIS for each element by iterating through all previous elements to find the longest increasing subsequence.
  3. Calculate the LDS for each element by iterating through all elements following it to find the longest decreasing subsequence.
  4. Identify valid peaks based on the criteria that both LIS and LDS are greater than 1.
  5. Find the maximum mountain length for valid peaks.
  6. Return the minimum removals required by calculating the difference between the array size and the maximum mountain length.

JavaScript Code

  1. Initialize LIS and LDS arrays with 1 as the default value for each index.
  2. Compute the LIS array for each index by comparing it with all previous elements.
  3. Compute the LDS array for each index by comparing it with all elements that follow.
  4. Identify valid peaks where LIS > 1 and LDS > 1.
  5. Determine the maximum mountain length by summing LIS and LDS for each valid peak.
  6. Return the minimum number of elements to remove by calculating the difference between the array length and the maximum mountain length.

Python Code

  1. Initialize LIS and LDS arrays with 1 as default for each index.
  2. Calculate the LIS:
    • For each element, iterate over all prior elements.
    • Update the LIS value to maintain the longest increasing subsequence up to that element.
  3. Calculate the LDS:
    • For each element, iterate over all elements after it.
    • Update the LDS value for the longest decreasing subsequence starting from that element.
  4. Identify valid peaks by ensuring both LIS[i] > 1 and LDS[i] > 1.
  5. Calculate maximum mountain length by summing LIS and LDS values for valid peaks and tracking the maximum.
  6. Return the result by subtracting the maximum mountain length from the total array length.

Go Code

  1. Initialize LIS and LDS arrays with values of 1 for each element in the array.
  2. Compute LIS for each element:
    • Iterate through all previous elements.
    • Update LIS values to hold the longest increasing subsequence length up to each element.
  3. Compute LDS for each element:
    • Iterate from the end of the array backward.
    • Update LDS values for the longest decreasing subsequence length from each element.
  4. Identify valid peaks where LIS and LDS are both greater than 1.
  5. Find the maximum mountain length by summing the LIS and LDS values for each valid peak.
  6. Calculate and return the minimum removals as the difference between the array length and the maximum mountain length.