Wi-Fi

2000ms

262144K

Description:

You work as a system administrator in a dormitory, which has $$$n$$$ rooms one after another along a straight hallway. Rooms are numbered from $$$1$$$ to $$$n$$$.

You have to connect all $$$n$$$ rooms to the Internet.

You can connect each room to the Internet directly, the cost of such connection for the $$$i$$$-th room is $$$i$$$ coins.

Some rooms also have a spot for a router. The cost of placing a router in the $$$i$$$-th room is also $$$i$$$ coins. You cannot place a router in a room which does not have a spot for it. When you place a router in the room $$$i$$$, you connect all rooms with the numbers from $$$max(1,~i - k)$$$ to $$$min(n,~i + k)$$$ inclusive to the Internet, where $$$k$$$ is the range of router. The value of $$$k$$$ is the same for all routers.

Calculate the minimum total cost of connecting all $$$n$$$ rooms to the Internet. You can assume that the number of rooms which have a spot for a router is not greater than the number of routers you have.

Input:

The first line of the input contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n, k \le 2 \cdot 10^5$$$) — the number of rooms and the range of each router.

The second line of the input contains one string $$$s$$$ of length $$$n$$$, consisting only of zeros and ones. If the $$$i$$$-th character of the string equals to '1' then there is a spot for a router in the $$$i$$$-th room. If the $$$i$$$-th character of the string equals to '0' then you cannot place a router in the $$$i$$$-th room.

Output:

Print one integer — the minimum total cost of connecting all $$$n$$$ rooms to the Internet.

Sample Input:

5 2
00100

Sample Output:

Sample Input:

6 1
000000

Sample Output:

Sample Input:

4 1
0011

Sample Output:

Sample Input:

12 6
000010000100

Sample Output:

Note:

In the first example it is enough to place the router in the room $$$3$$$, then all rooms will be connected to the Internet. The total cost of connection is $$$3$$$.

In the second example you can place routers nowhere, so you need to connect all rooms directly. Thus, the total cost of connection of all rooms is $$$1 + 2 + 3 + 4 + 5 + 6 = 21$$$.

In the third example you need to connect the room $$$1$$$ directly and place the router in the room $$$3$$$. Thus, the total cost of connection of all rooms is $$$1 + 3 = 4$$$.

In the fourth example you need to place routers in rooms $$$5$$$ and $$$10$$$. Then all rooms will be connected to the Internet. The total cost of connection is $$$5 + 10 = 15$$$.

Informação

Provedor Codeforces

Código CF1216F