CF978D Almost Arithmetic Progression

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    • 58提交
  • 题目来源 CodeForces 978D
  • 评测方式 RemoteJudge
  • 标签 剪枝 搜索 枚举,暴力
  • 难度 普及/提高-
  • 时空限制 1000ms / 256MB

题解

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    题意翻译

    定义对于一个长度为n的序列,如果每两个相邻的数后数减去前数的差都相等,那么这就是一个等差数列。只有一个数或两个数的数列也是等差数列。

    现给出一个数列,你可以把每一个数加1,减1或不变,使其变成一个等差数列,求最小的做出的变化数量(一个数保持不变不算变化,加一或减一算一次变化)。

    输入格式:第一行输入一个整数n,数列的长度,第二行输入n个整数,表示序列。

    输出格式:一个数,最小的变化数量。

    数据范围:1<=n<=100000,序列中所有的数<=1e9。

    感谢@nstk0513 提供的翻译

    题目描述

    Polycarp likes arithmetic progressions. A sequence $ [a_1, a_2, \dots, a_n] $ is called an arithmetic progression if for each $ i $ ( $ 1 \le i < n $ ) the value $ a_{i+1} - a_i $ is the same. For example, the sequences $ [42] $ , $ [5, 5, 5] $ , $ [2, 11, 20, 29] $ and $ [3, 2, 1, 0] $ are arithmetic progressions, but $ [1, 0, 1] $ , $ [1, 3, 9] $ and $ [2, 3, 1] $ are not.

    It follows from the definition that any sequence of length one or two is an arithmetic progression.

    Polycarp found some sequence of positive integers $ [b_1, b_2, \dots, b_n] $ . He agrees to change each element by at most one. In the other words, for each element there are exactly three options: an element can be decreased by $ 1 $ , an element can be increased by $ 1 $ , an element can be left unchanged.

    Determine a minimum possible number of elements in $ b $ which can be changed (by exactly one), so that the sequence $ b $ becomes an arithmetic progression, or report that it is impossible.

    It is possible that the resulting sequence contains element equals $ 0 $ .

    输入输出格式

    输入格式:

    The first line contains a single integer $ n $ $ (1 \le n \le 100\,000) $ — the number of elements in $ b $ .

    The second line contains a sequence $ b_1, b_2, \dots, b_n $ $ (1 \le b_i \le 10^{9}) $ .

    输出格式:

    If it is impossible to make an arithmetic progression with described operations, print -1. In the other case, print non-negative integer — the minimum number of elements to change to make the given sequence becomes an arithmetic progression. The only allowed operation is to add/to subtract one from an element (can't use operation twice to the same position).

    输入输出样例

    输入样例#1: 复制
    4
    24 21 14 10
    
    输出样例#1: 复制
    3
    
    输入样例#2: 复制
    2
    500 500
    
    输出样例#2: 复制
    0
    
    输入样例#3: 复制
    3
    14 5 1
    
    输出样例#3: 复制
    -1
    
    输入样例#4: 复制
    5
    1 3 6 9 12
    
    输出样例#4: 复制
    1
    

    说明

    In the first example Polycarp should increase the first number on $ 1 $ , decrease the second number on $ 1 $ , increase the third number on $ 1 $ , and the fourth number should left unchanged. So, after Polycarp changed three elements by one, his sequence became equals to $ [25, 20, 15, 10] $ , which is an arithmetic progression.

    In the second example Polycarp should not change anything, because his sequence is an arithmetic progression.

    In the third example it is impossible to make an arithmetic progression.

    In the fourth example Polycarp should change only the first element, he should decrease it on one. After that his sequence will looks like $ [0, 3, 6, 9, 12] $ , which is an arithmetic progression.

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