Gregor and the Pawn Game
Contest: Codeforces Round 736 (Div. 2)
Difficulty rating: 800
Tags: dfs and similar, dp, flows, graph matchings, graphs, greedy, implementation

There is a chessboard of size $$$n$$$ by $$$n$$$. The square in the $$$i$$$-th row from top and $$$j$$$-th column from the left is labelled $$$(i,j)$$$.

Currently, Gregor has some pawns in the $$$n$$$-th row. There are also enemy pawns in the $$$1$$$-st row. On one turn, Gregor moves one of his pawns. A pawn can move one square up (from $$$(i,j)$$$ to $$$(i-1,j)$$$) if there is no pawn in the destination square. Additionally, a pawn can move one square diagonally up (from $$$(i,j)$$$ to either $$$(i-1,j-1)$$$ or $$$(i-1,j+1)$$$) if and only if there is an enemy pawn in that square. The enemy pawn is also removed.

Gregor wants to know what is the maximum number of his pawns that can reach row $$$1$$$?

Note that only Gregor takes turns in this game, and the enemy pawns never move. Also, when Gregor's pawn reaches row $$$1$$$, it is stuck and cannot make any further moves.

Input format:
The first line of the input contains one integer $$$t$$$ ($$$1\le t\le 2\cdot 10^4$$$) — the number of test cases. Then $$$t$$$ test cases follow.

Each test case consists of three lines. The first line contains a single integer $$$n$$$ ($$$2\le n\le 2\cdot{10}^{5}$$$) — the size of the chessboard.

The second line consists of a string of binary digits of length $$$n$$$, where a $$$1$$$ in the $$$i$$$-th position corresponds to an enemy pawn in the $$$i$$$-th cell from the left, and $$$0$$$ corresponds to an empty cell.

The third line consists of a string of binary digits of length $$$n$$$, where a $$$1$$$ in the $$$i$$$-th position corresponds to a Gregor's pawn in the $$$i$$$-th cell from the left, and $$$0$$$ corresponds to an empty cell.

It is guaranteed that the sum of $$$n$$$ across all test cases is less than $$$2\cdot{10}^{5}$$$.

Output format:
For each test case, print one integer: the maximum number of Gregor's pawns which can reach the $$$1$$$-st row.

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

Note:
In the first example, Gregor can simply advance all $$$3$$$ of his pawns forward. Thus, the answer is $$$3$$$.

In the second example, Gregor can guarantee that all $$$4$$$ of his pawns reach the enemy row, by following the colored paths as demonstrated in the diagram below. Remember, only Gregor takes turns in this "game"!

In the third example, Gregor's only pawn is stuck behind the enemy pawn, and cannot reach the end.

In the fourth example, Gregor has no pawns, so the answer is clearly $$$0$$$.

Example input:
4
3
000
111
4
1111
1111
3
010
010
5
11001
00000

Example output:
3
4
0
0

Editorial:
The key insight is that due to the fact that there is only one row of enemy pawns, and those pawns never move, there are only $$$3$$$ possible columns where one of Gregor's pawns can end up in.
We can solve this problem greedily, going from column $$$1$$$ to column $$$N$$$. At the current column $$$j$$$, if Gregor has a pawn in this column, then we greedily consider 3 cases.
1. If there is an uncaptured enemy pawn in column $$$j-1$$$, mark that pawn has captured and increment the answer. Column $$$j-1$$$ will never be looked at again.
2. If there is no pawn in column $$$j$$$, just move Gregor's pawn, and increment the answer.
3. If there is an uncaptured enemy pawn in column $$$j+1$$$, mark that pawn has captured and increment the answer.
This greedy solution is guaranteed to produce the maximum possible answer.
The Time Complexity is $$$\mathcal{O}(N)$$$.