原題鏈接 The Water Bowls Time Limit: 1000MS Memory Limit: 65536K Total Submissions: 6210 Accepted: 2443 Description

The cows have a line of 20 water bowls from which they drink. The bowls can be either right-side-up (PRoperly oriented to serve refreshing cool water) or upside-down (a position which holds no water). They want all 20 water bowls to be right-side-up and thus use their wide snouts to flip bowls.

Their snouts, though, are so wide that they flip not only one bowl but also the bowls on either side of that bowl (a total of three or – in the case of either end bowl – two bowls).

Given the initial state of the bowls (1=undrinkable, 0=drinkable – it even looks like a bowl), what is the minimum number of bowl flips necessary to turn all the bowls right-side-up? Input

Line 1: A single line with 20 space-separated integers Output

Line 1: The minimum number of bowl flips necessary to flip all the bowls right-side-up (i.e., to 0). For the inputs given, it will always be possible to find some combination of flips that will manipulate the bowls to 20 0’s. Sample Input

0 0 1 1 1 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 Sample Output

3 Hint

Explanation of the sample:

Flip bowls 4, 9, and 11 to make them all drinkable: 0 0 1 1 1 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 [initial state] 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 [after flipping bowl 4] 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 [after flipping bowl 9] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [after flipping bowl 11] Source

USACO 2006 January Bronze 題意：有20個碗排成一排，有些碗口朝上，有些碗口朝下。每次可以反轉其中的一個碗，但是在反轉該碗時，該碗左右兩邊的碗也跟著被反轉（如果該碗為邊界上的碗，則只有一側的碗被反轉）。求最少需要反轉幾次，可以使得所有碗口均朝上。 思路：對于第0只碗可以選擇反轉或者不翻轉，如果反轉第i只碗，那么第i+1，第i+2只碗也會反轉，但是我們只是記錄下來第i只碗是否反轉即可，由于 的奇偶決定了i處是否需要反轉，而 決定了i+1處是否需要反轉，其實我們發現二者就是+f[i]和-f[i-k+1]的關系，所以我們維護的其實就是一個區間的和，這樣的話我們就可以把復雜度從O(nk)降低到O(n)