CF1654E Arithmetic Operations

You are given an array of integers $ a_1, a_2, \ldots, a_n $ . You can do the following operation any number of times (possibly zero): - Choose any index $ i $ and set $ a_i $ to any integer (positive, negative or $ 0 $ ). What is the minimum number of operations needed to turn $ a $ into an arithmetic progression? The array $ a $ is an arithmetic progression if $ a_{i+1}-a_i=a_i-a_{i-1} $ for any $ 2 \leq i \leq n-1 $ .

N/A

N/A

In the first test, you can get the array $ a = [11, 10, 9, 8, 7, 6, 5, 4, 3] $ by performing $ 6 $ operations: - Set $ a_3 $ to $ 9 $ : the array becomes $ [3, 2, 9, 8, 6, 9, 5, 4, 1] $ ; - Set $ a_2 $ to $ 10 $ : the array becomes $ [3, 10, 9, 8, 6, 9, 5, 4, 1] $ ; - Set $ a_6 $ to $ 6 $ : the array becomes $ [3, 10, 9, 8, 6, 6, 5, 4, 1] $ ; - Set $ a_9 $ to $ 3 $ : the array becomes $ [3, 10, 9, 8, 6, 6, 5, 4, 3] $ ; - Set $ a_5 $ to $ 7 $ : the array becomes $ [3, 10, 9, 8, 7, 6, 5, 4, 3] $ ; - Set $ a_1 $ to $ 11 $ : the array becomes $ [11, 10, 9, 8, 7, 6, 5, 4, 3] $ . $ a $ is an arithmetic progression: in fact, $ a_{i+1}-a_i=a_i-a_{i-1}=-1 $ for any $ 2 \leq i \leq n-1 $ . There is no sequence of less than $ 6 $ operations that makes $ a $ an arithmetic progression. In the second test, you can get the array $ a = [-1, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38] $ by performing $ 10 $ operations. In the third test, you can get the array $ a = [100000, 80000, 60000, 40000, 20000, 0, -20000, -40000, -60000, -80000] $ by performing $ 7 $ operations.