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POMinimize maximum absolute difference in pairs of numbers
 

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    <p><strong>The problem statement</strong>:<br> Give <code>n</code> variables and <code>k</code> pairs. The variables can be distinct by assigning a value from 1 to <code>n</code> to each variable. Each pair <code>p</code> contain 2 variables and let the absolute difference between 2 variables in <code>p</code> is <code>abs(p)</code>. Define the upper bound of difference is <code>U=max(Abs(p)|every p)</code>. </p> <p>Find an assignment that minimize <code>U</code>. </p> <pre><code>Limit: n&lt;=100 k&lt;=1000 </code></pre> <p>Each variable appear at least 2 times in list of pairs.</p> <pre><code>A problem instance: Input n=9, k=12 1 2 (meaning pair x1 x2) 1 3 1 4 1 5 2 3 2 6 3 5 3 7 3 8 3 9 6 9 8 9 Output: 1 2 5 4 3 6 7 8 9 (meaning x1=1,x2=2,x3=5,...) </code></pre> <p><strong>Explaination:</strong> An assignment of <code>x1=1,x2=2,x3=3,...</code> will result in <code>U=6</code> (3 9 has greastest abs value). The output assignment will get <code>U=4</code>, the minimum value (changed pair: <code>3 7 =&gt; 5 7, 3 8 =&gt; 5 8</code>, etc. and 3 5 isn't changed. In this case, <code>abs(p)&lt;=4</code> for every pair). </p> <p><strong>There is an important point</strong>: To achieve the best assignments, the variables in the pairs that have greatest abs must be change.<br> Base on this, I have thought of a greedy algorithm: </p> <pre><code>1)Assign every x to default assignment (x(i)=i) 2)Locate pairs that have largest abs and x(i)'s contained in them. 3)For every i,j: Calculate U. Swap value of x(i),x(j). Calculate U'. If U'&lt;U, stop and repeat step 3. If U'&gt;=U for every i,j, end and output the assignment. </code></pre> <p>However, this method has a major pitfall, if we need an assignment like this: </p> <pre><code>x(a)&lt;&lt;x(b), x(b)&lt;&lt;x(c), x(c)&lt;&lt;x(a) </code></pre> <p>, we have to swap in 2 steps, like: <code>x(a)&lt;=&gt;x(b)</code>, then <code>x(b)&lt;=&gt;x(c)</code>, then there is a possibility that <code>x(b)&lt;&lt;x(a)</code> in first step has its abs become larger than U and the swap failed.<br> Is there any efficient algorithm to solve this problem?</p>
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    1. COI don't see a question here, nor code written, or an explanation of a problem. Instead I see what looks like a homework assignment.
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    2. COSo you want to minimize the maximum difference within each pair?
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