Algorithms · Greedy algorithms and exchange proofs
The n vertices of a cycle are numbered 1,…,n, and vertex i has an integer weight
Problem
The \(n\) vertices of a cycle are numbered \(1,\ldots,n\), and vertex \(i\) has an integer weight. Choose exactly \(k\) vertices with no adjacent pair, where vertices \(1,n\) are adjacent. Maximize total weight and output the maximum and selected vertex numbers in increasing order. Any optimum may be output. The input satisfies \[ 2\le n\le200{,}000,\quad 0\le k\le\lfloor n/2\rfloor,\quad n\max(1,k)\le20{,}000{,}000, \] and every weight lies in \([-10^9,10^9]\).
Hint
Split according to whether vertex \(1\) is selected.
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