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    <p>The idea of using a number to order the modifications is taken from Dukeling's post.</p> <p>We will need 2 maps and 4 binary indexed tree (BIT, a.k.a. Fenwick Tree): 1 map and 2 BITs for rows, and 1 map and 2 BITs for columns. Let us call them <code>m_row</code>, <code>f_row[0]</code>, and <code>f_row[1]</code>; <code>m_col</code>, <code>f_col[0]</code> and <code>f_col[1]</code> respectively.</p> <p>Map may be implemented with array, or tree like structure, or hashing. The 2 maps are used to store the last modification to a row/column. Since there can be at most 10<sup>5</sup> modification, you may use that fact to save space from simple array implementation.</p> <p>BIT has 2 operations:</p> <ul> <li><code>adjust(value, delta_freq)</code>, which adjusts the frequency of the <code>value</code> by <code>delta_freq</code> amount.</li> <li><code>rsq(from_value, to_value)</code>, (rsq stands for range sum query) which finds the sum of the all the frequencies from <code>from_value</code> to <code>to_value</code> inclusive.</li> </ul> <p>Let us declare global variable: <code>version</code></p> <p>Let us define <code>numRow</code> to be the number of rows in the 2D boolean matrix, and <code>numCol</code> to be the number of columns in the 2D boolean matrix.</p> <p>The BITs should have size of at least MAX_QUERY + 1, since it is used to count the number of changes to the rows and columns, which can be as many as the number of queries.</p> <p>Initialization:</p> <pre><code>version = 1 # Map should return &lt;0, 0&gt; for rows or cols not yet # directly updated by query m_row = m_col = empty map f_row[0] = f_row[1] = f_col[0] = f_col[1] = empty BIT </code></pre> <p>Update algorithm:</p> <pre><code>update(isRow, value, idx): if (isRow): # Since setting a row/column to a new value will reset # everything done to it, we need to erase earlier # modification to it. # For example, turn on/off on a row a few times, then # query some column &lt;prevValue, prevVersion&gt; = m_row.get(idx) if ( prevVersion &gt; 0 ): f_row[prevValue].adjust( prevVersion, -1 ) m_row.map( idx, &lt;value, version&gt; ) f_row[value].adjust( version, 1 ) else: &lt;prevValue, prevVersion&gt; = m_col.get(idx) if ( prevVersion &gt; 0 ): f_col[prevValue].adjust( prevVersion, -1 ) m_col.map( idx, &lt;value, version&gt; ) f_col[value].adjust( version, 1 ) version = version + 1 </code></pre> <p>Count algorithm:</p> <pre><code>count(isRow, idx): if (isRow): # If this is row, we want to find number of reverse modifications # done by updating the columns &lt;value, row_version&gt; = m_row.get(idx) count = f_col[1 - value].rsq(row_version + 1, version) else: # If this is column, we want to find number of reverse modifications # done by updating the rows &lt;value, col_version&gt; = m_col.get(idx) count = f_row[1 - value].rsq(col_version + 1, version) if (isRow): if (value == 1): return numRow - count else: return count else: if (value == 1): return numCol - count else: return count </code></pre> <p>The complexity is logarithmic in worst case for both update and count.</p>
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    1. PORange update and querying in a 2D matrix
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    1. COWhat are the methods rsq, add and adjust ? Could you add a brief detail about them
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      1. USWayne Rooney
    2. CO@WayneRooney: Thanks for the comment. There shouldn't be `add`.
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    3. COAlso what is numRow and numCol?? Is rsq the cumulative sum method ?
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