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Given a positive real value $\delta$, a set $P$ of points along a line and adistance function $d$, in the movement to independence problem, we wish to movethe points to new positions on the line such that for every two points$p_{i},p_{j} \in P$, we have $d(p_{i},p_{j}) \geq \delta$ while minimizing thesum of movements of all points. This measure of the cost for moving the pointswas previously unsolved in this setting. However for different cost measuresthere are algorithms of $O(n \log(n))$ or of $O(n)$. We present an $O(n\log(n))$ algorithm for the points on a line and thus conclude the setting inone dimension.

Minimizing the Total Movement for Movement to Independence Problem on a Line