題目鏈接:CF750E New Year and Old Subsequence
E. New Year and Old Subsequence
time limit per test
3 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output
A string t is called nice if a string "2017" occurs in t as a subsequence but a string "2016" doesn't occur in t as a subsequence. For example, strings "203434107" and "9220617" are nice, while strings "20016", "1234" and "20167" aren't nice.
The ugliness of a string is the minimum possible number of characters to remove, in order to obtain a nice string. If it's impossible to make a string nice by removing characters, its ugliness is - 1.
Limak has a string s of length n, with characters indexed 1 through n. He asks you q queries. In the i-th query you should compute and print the ugliness of a substring (continuous subsequence) of s starting at the index a__i and ending at the index b__i (inclusive).
Input
The first line of the input contains two integers n and q (4 ≤ n ≤ 200 000, 1 ≤ q ≤ 200 000) — the length of the string s and the number of queries respectively.
The second line contains a string s of length n. Every character is one of digits '0'–'9'.
The i-th of next q lines contains two integers a__i and b__i (1 ≤ a__i ≤ b__i ≤ n), describing a substring in the i-th query.
Output
For each query print the ugliness of the given substring.
Examples
Input
Copy
8 3
20166766
1 8
1 7
2 8
Output
Copy
4
3
-1
Input
Copy
15 5
012016662091670
3 4
1 14
4 15
1 13
10 15
Output
Copy
-1
2
1
-1
-1
Input
Copy
4 2
1234
2 4
1 2
Output
Copy
-1
-1
Note
In the first sample:
- In the first query, ugliness("20166766") = 4 because all four sixes must be removed.
- In the second query, ugliness("2016676") = 3 because all three sixes must be removed.
- In the third query, ugliness("0166766") = - 1 because it's impossible to remove some digits to get a nice string.
In the second sample:
- In the second query, ugliness("01201666209167") = 2. It's optimal to remove the first digit '2' and the last digit '6', what gives a string "010166620917", which is nice.
- In the third query, ugliness("016662091670") = 1. It's optimal to remove the last digit '6', what gives a nice string "01666209170".
題意: 在區間\([l,r]\)中刪去最少的字符數使得在這個區間內不含有子序列\("2016"\)且含有子序列\("2017"\),如果無法滿足條件,輸出\(-1\).
題解: 首先看到題目中要求區間的某個值,想到用某種數據結構來維護這個值.
然而這個最小值似乎是要用\(DP\)來求的?
那么這里就有點動態\(DP\)的意思了:我們將某個位置的狀態加入矩陣中,再對這個序列開一棵線段樹,線段樹中的節點維護矩陣的狀態.
我們將\("2017"\)拆成5份,分別是\(\empty,2,20,201,2017\),用\(0 \to 4\)表示這\(5\)個狀態.
我們設轉移矩陣$$D=
\left[
\begin{matrix}
a_{0,0} & ... & a_{0,4} \
a_{i,j-1} & a_{i,j} & a_{i,j+1} \
a_{4,0} & ... & a_{4,4}
\end{matrix}
\right] \tag{3}
同理,對於該位為\(0,1,7\)都是一樣的.
但是如果該位為\(6\)呢?顯然我們是不能讓串中出現\("2016"\)的,所以是一定要刪除這個字符的,所以它的轉移矩陣為$$D=
\left[
\begin{matrix}
0 & \inf & \inf & \inf & \inf \
\inf & 0 & \inf & \inf & \inf \
\inf & \inf & 0 & \inf & \inf \
\inf & \inf & \inf & 1 & \inf \
\inf & \inf & \inf & \inf & 1
\end{matrix}
\right] \tag{3}