Minimum Sum of Maximums

Bessie has $N$ ($2\le N\le 300$) tiles in a line with ugliness values $a_1,a_2,\dots ,a_N$ in that order ($1\le a_i\le 10^6$). $K$ ($0\le K\le \min (N,6)$) of the tiles are stuck in place; specifically, those at indices $x_1,\dots ,x_K$ ($1\le x_1<x_2<\cdots <x_K\le N$).

Bessie wants to minimize the total ugliness of the tiles, which is defined as the sum of the maximum ugliness over every consecutive pair of tiles; that is, ${\sum }_{i=1}^{N-1}\max (a_i,a_{i+1})$. She is allowed to perform the following operation any number of times: choose two tiles, neither of which are stuck in place, and swap them.

Determine the minimum possible total ugliness Bessie can achieve if she performs operations optimally.

Bessie can swap the second and third tiles so that $a=[1,10,100]$, achieving total ugliness $\max (1,10)+\max (10,100)=110$. Alternatively, she could swap the first and second tiles so that $a=[100,1,10]$, also achieving total ugliness $\max (100,1)+\max (1,10)=110$.

Bessie could swap the first and second tiles so that $a=[100,1,10]$, achieving total ugliness $\max (100,1)+\max (1,10)=110$.

The initial total ugliness of the tiles is $\max (1,100)+\max (100,10)=200$. Bessie is only allowed to swap the first and third tiles, which does not allow her to reduce the total ugliness.

Problem credits: Benjamin Qi

Source: USA Computing Olympiad.

Input Format

Output Format

Sample Input 1

Sample Output 1

Sample Input 2

Sample Output 2

Sample Input 3

Sample Output 3

Sample Input 4

Sample Output 4

Submit