Matrix Zeros - Brute Force Solution to Optimal Approach

Jul 17, 2024

Matrix Zeros - Brute Force Solution to Optimal Approach

Introduction

  • World's most in-depth DS algo course with 55 modules and over 400 problems.
  • Completing the course guarantees the ability to clear DS algo rounds in any company globally.
  • Today's topic: Problem Set Matrix Zeros.

Problem Statement

  • Given an NxM binary matrix (only 0s and 1s).
  • Task: For every zero, mark its entire row and column with zeros.
  • Important: Only consider initial zeros, changes made during process should not affect others.

Initial Brute Force Solution

Approach

  1. Iterate through NxM matrix.
  2. When a zero is found, mark entire row and column with zeros.
  3. Repeat until all initial zeros are processed.

Implementation Issues

  • Marks previously changed 0s in subsequent iterations, causing incorrect results.
  • Needs careful handling of marking to avoid above issues.

Improved Brute Force Method

  1. Use -1 instead of 0 for marking changes during iterations.
  2. Convert -1 to 0 in the final iteration.

Algorithm Steps

  1. Traverse NxM matrix and mark zeros with -1 where required.
  2. Traverse matrix again to convert all -1 to 0.

Time Complexity

  • Traversal: O(NxM)
  • Marking columns and rows: O(N+M)
  • Overall: O(NxM * (N+M)) ≈ O(N^3)

Optimized Solution

Observation

  • Each zero in the matrix affects its entire row and column.
  • Instead of updating the matrix during the traversal, use two auxiliary arrays to keep track of rows and columns to be zeroed out.

Approach

  1. Use two arrays of size N and M to keep track of affected rows and columns.
  2. Traverse the matrix to populate these arrays.
  3. Update the matrix in a second pass using these arrays.

Algorithm Steps

  1. Traverse the matrix and populate row and column arrays.
  2. Traverse again and update the matrix.

Time and Space Complexity

  • Time: O(NxM)
  • Space: O(N+M) due to additional tracking arrays.

Optimal Solution

Observation

  • Try to use the matrix itself for marking to avoid extra space usage.
  • Use first row and column for marking zeros.

Approach

  1. Utilize the first row and column as auxiliary arrays.
  2. Handle special case where the first cell is part of both first row and column.

Algorithm Steps

  1. Traverse matrix to mark zeros in the first row and column, except the first cell.
  2. Traverse matrix again to mark the matrix using first row and column's markers.
  3. Update the first row and column separately.
  4. Handle the first cell separately due to dual role.

Pseudocode Outline

  1. Initialize space for column flag.
  2. Mark rows and column using initial pass, handle column zero separately.
  3. Second pass to update matrix except the first row and column.
  4. Final pass to update the first row and column based on initial markers.

Time and Space Complexity

  • Time: O(NxM)
  • Space: O(1) (In-place modification)

Conclusion

  • Three approaches discussed: Brute Force, Optimized with auxiliary space, Optimal In-Place solution.
  • Trade-offs between time complexity and space usage.
  • Optimal in-place solution is efficient for interviews.

References

  • Problem description and code links in video description.
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