Dimension Reduction Technique

2D Matrix Problem

Let's start with a 2D matrix where we need to find a submatrix with a sum of zero.

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This is a 2×3 matrix. Finding a zero-sum submatrix directly in 2D can be complex.

Fixing Rows for Dimension Reduction

We can reduce this 2D problem to multiple 1D problems by fixing rows. Let's fix the top row (row 0) and the bottom row (row 1).

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Creating a Temporary Array

We compress the columns between our fixed rows into a temporary 1D array. Each value in this array is the sum of the column elements between our rows.

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↓

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For this example, the temporary array becomes [-1, 1, -1].

Finding a Zero-Sum Subarray

Now, we can apply 1D algorithms to find a subarray with a sum of zero in our temporary array.

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In our example, the subarray [1, -1] has a sum of zero.

Converting Back to 2D

We can now map this 1D solution back to our original 2D matrix.

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The zero-sum submatrix is highlighted. This technique of dimension reduction converted a complex 2D problem to a simpler 1D problem!