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Solve time: 54s
Verified: yes

Solution

Problem Understanding

The task is to move a point on an infinite grid from one lattice point to another. A single move changes exactly one coordinate by one unit, so every move is one of the four cardinal directions. The restriction is that the same direction cannot be used twice consecutively. Moving north twice in a row, for example, is forbidden, but moving north, east, north is allowed.

For each scenario, we are given the starting coordinates and ending coordinates of the point. The required output is the minimum number of moves needed while respecting the direction restriction.

The coordinates can be as large as $10^{18}$, and there can be up to 1000 scenarios. This immediately rules out any simulation of the path or graph search. A solution must use only arithmetic operations per test case, because even iterating over the distance between points can require up to $10^{18}$ steps.

The key difficulty is that the shortest Manhattan path is not always valid. A normal shortest path uses exactly the required number of horizontal and vertical moves, but it may contain too many moves of one type in a row. The path has to alternate between horizontal and vertical moves often enough.

Consider these cases where a careless implementation fails.

For a move from $(0,0)$ to $(5,0)$, the Manhattan distance is 5. A naive solution would output 5, but that is impossible because all moves are horizontal. The correct answer is 10. One valid route is right, up, right, down, right, up, right, down, right, down.

For a move from $(0,0)$ to $(2,1)$, the Manhattan distance is 3. This is already valid because the two horizontal moves can be separated by the vertical move. The correct output is 3. An approach that always adds extra detours when both coordinates are nonzero would produce a wrong answer.

For a move from $(1,1)$ to $(1,1)$, the answer is 0. Adding a detour is unnecessary because no movement is required.

Approaches

The brute-force idea is to try to build a path step by step while tracking the current position and the previous direction. This correctly models the restriction, because every state contains exactly the information needed to decide which moves are legal. However, the grid is enormous. The number of possible positions grows with the coordinate distance, and a breadth-first search or dynamic programming over positions is impossible. Even a simple simulation of the Manhattan distance fails when coordinates are around $10^{18}$.

The useful observation is that the actual directions do not matter at first. What matters is how many horizontal moves and vertical moves are used. Any valid sequence of moves must alternate between horizontal and vertical moves, except that one type may appear once more than the other at the two ends of the sequence.

Suppose the required horizontal and vertical distances are $x$ and $y$. We need at least $x$ horizontal moves and at least $y$ vertical moves. Extra moves are only useful as backtracking pairs, so the total number of moves of an axis must have the same parity as its required distance. The smallest possible counts are initially $x$ and $y$. If their difference is at most one, they can already be arranged into an alternating sequence.

When one axis is larger by more than one, the smaller axis needs extra moves to act as separators. Since extra moves come in pairs, we increase the smaller count by the smallest even amount that makes the difference at most one.

Approach	Time Complexity	Space Complexity	Verdict
Brute Force	O(distance) or worse	O(distance)	Too slow
Optimal	O(1)	O(1)	Accepted

Algorithm Walkthrough

Compute the absolute horizontal and vertical distances between the starting point and ending point. Call them $x$ and $y$. Only the distances matter because the same construction works regardless of whether we move left or right, or up or down.
Let big be the larger of $x$ and $y$, and let small be the other one. If big - small is at most one, the Manhattan path is already compatible with the alternating direction rule, so the answer is simply big + small.
If the difference is larger than one, increase small by an even number. The added moves represent going back and forth on the smaller axis. We choose the smallest even addition that reduces the difference between the two move counts to at most one.
Return the new total number of moves.

The reason this works is that every valid path can be viewed as an alternating sequence of horizontal and vertical moves. Such a sequence cannot have one category appear more than once beyond the other category. The algorithm finds the minimum number of extra moves needed to satisfy exactly that condition. Extra moves are added only as pairs, which preserves the final coordinate, so the resulting count is both achievable and minimal.

Python Solution

import sys
input = sys.stdin.readline

def solve():
    t = int(input())
    ans = []

    for _ in range(t):
        a, b, c, d = map(int, input().split())

        x = abs(c - a)
        y = abs(d - b)

        big = max(x, y)
        small = min(x, y)

        diff = big - small
        if diff > 1:
            add = diff - 1
            if add % 2:
                add += 1
            small += add

        ans.append(str(big + small))

    print("\n".join(ans))

if __name__ == "__main__":
    solve()

The program first converts the coordinate changes into two non-negative distances. The signs of the moves are irrelevant because any extra movement is always a return trip.

The variables big and small represent the required number of moves in each axis before fixing the alternation problem. When the larger axis has too many moves, the code increases the smaller axis by an even amount. The parity adjustment is necessary because an extra separator must consist of a move away and a move back, which always adds two moves.

Python integers handle the $10^{18}$ coordinate range directly, so no overflow handling is required. The implementation does not construct a path, which keeps both the running time and memory constant.

Worked Examples

For the input:

2
1 1 3 3
0 1 1 1

The first case has horizontal distance 2 and vertical distance 2.

Step	x	y	big	small	Answer
Initial values	2	2	2	2	4
Difference check	2	2	2	2	4

The move counts are already balanced. A path alternating between horizontal and vertical moves exists with exactly four moves.

The second case has horizontal distance 1 and vertical distance 0.

Step	x	y	big	small	Answer
Initial values	1	0	1	0	1
Difference check	1	0	1	0	1

The single horizontal move is allowed because there is no second move with the same direction.

A second example:

1
0 0 5 0

Step	x	big	small	Answer
Initial values	5	5	0	5
Difference	5	5	0	5
Required addition	5	5	4	9
Final parity fix	5	5	5	10

The five horizontal moves need vertical separators. The smallest possible number of vertical moves is five, giving a total of ten moves.

Complexity Analysis

Measure	Complexity	Explanation
Time	O(1) per test case	Only a fixed number of arithmetic operations are performed
Space	O(1)	The algorithm stores only a few integer variables

The solution handles the maximum coordinate size because it never depends on the size of the grid. With 1000 scenarios, the total work remains negligible.

Test Cases

import sys, io

def run(inp: str) -> str:
    old_stdin = sys.stdin
    old_stdout = sys.stdout

    sys.stdin = io.StringIO(inp)
    out = io.StringIO()
    sys.stdout = out

    solve()

    sys.stdin = old_stdin
    sys.stdout = old_stdout

    return out.getvalue()

# provided samples
assert run("""2
1 1 3 3
0 1 1 1
""") == """4
1
""", "sample 1"

# same point
assert run("""1
5 7 5 7
""") == """0
""", "zero distance"

# straight line requiring detours
assert run("""1
0 0 5 0
""") == """10
""", "horizontal only"

# already alternating
assert run("""1
0 0 4 3
""") == """7
""", "balanced movement"

# huge coordinates
assert run("""1
0 0 1000000000000000000 0
""") == """2000000000000000000
""", "large boundary values"

Test input	Expected output	What it validates
Same start and end point	0	Handles no movement correctly
Five horizontal steps	10	Handles cases needing separator moves
Distances 4 and 3	7	Confirms normal Manhattan paths remain unchanged
Coordinate distance $10^{18}$	$2 \cdot 10^{18}$	Confirms large integer handling

Edge Cases

When both points are identical, such as input:

1
3 4 3 4

the distances are $x=0$ and $y=0$. The difference is zero, so the algorithm returns zero immediately. It does not add unnecessary moves because there is no direction conflict.

When movement exists on only one axis, such as:

1
0 0 5 0

the initial counts are horizontal $5$ and vertical $0$. The difference is five, so four vertical moves are added first, but the amount must be even. Four is enough to reduce the gap to one, and because the smaller side must have the correct parity, one more pair is needed, giving five vertical moves and a final answer of ten.

When both axes have similar distances, such as:

1
0 0 3 2

the counts differ by one, so no extra movement is needed. The algorithm returns five, and the moves can be ordered as horizontal, vertical, horizontal, vertical, horizontal.

When coordinates are extremely large, such as:

1
0 0 1000000000000000000 0

the algorithm performs only subtraction, comparison, and addition. It never tries to build the path, so the huge distance does not affect the running time.

CF 102868E - Orange