Finding minimum sum of absolute differences between heights of $n$ boys and $m$ girls Given two sets $A$ and $B$, $A$ has heights of $n$ boys and $B$ has heights of $m$ girls, $m \ge n$.
We have to find one solution of pairing up $n$ boys with $n$ (out of $m$) girls so that the sum of the absolute difference between the heights of boy and girl in each pair is minimized.
For example:
$$A = \{10,10,12,13,16\}$$
$$B = \{6,7,9,10,11,12,17\}$$
so ans should be $4$
one possible soln. of pairing is $\{(10,10),(10,9),(12,11),(13,12),(16,17)\}$
Each tuple is $(A,B)$
 A: If two boys have heights $b_1 \leq b_2$ and two girls have heights $g_1 \leq g_2$ and if you have a matching where $b_1$ is paired with $g_2$ and $b_2$ is paired with $g_1$, then you can find at least as good of a matching by instead matching $b_1$ with $g_1$ and $b_2$ with $g_2$.  Thus you can safely assume the best matching must have matched girls in sorted order of height when their corresponding boys are put in sorted order of height.  If you want a computer programming approach to solve the problem, this dramatically reduces the complexity of your search, e.g. you can use dynamic programming to find the optimal matching by building up optimal matchings that have the first $j$ boys in sorted height order all paired up somehow among the first $k$ girls in sorted height order (where $j \leq k$), and using the optimal matchings for the values $(j,k-1)$ and $(j-1,k-1)$ to decide the optimal matching for the values $(j,k)$, based on whether you pair up boy $j$ with girl $k$ or not.  This dynamic programming solution has $O(nm)$ time and space complexity if you have $n$ boys and $m$ girls.
