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## Village Road Network SOLUTIONS OCTOBER CHALLENGE 2020

In the nation of Chefland there are villages at co-ordinates (x,y) whenever x and y are non-zero odd co-prime integers. x can be either positive or negative, but y is always positive.

In each village (x,y) there is a crossroads, with the roads marked North, South, East, West.

Road East leads to (x+2y,y).

Road West leads to (x−2y,y).

Road North leads to either (x,y+2x) or (−x,−(y+2x)), chosen so that the new y co-ordinate is positive.

Road South leads to either (x,y−2x) or (−x,−(y−2x)), chosen so that the new y co-ordinate is positive.

Some citizens of Chefland prefer these equivalent definitions for the North and South roads:

Road North leads to {(x,y+2x)(−x,−(y+2x))when y+2x>0when y+2x<0

Road South leads to {(x,y−2x)(−x,−(y−2x))when y−2x>0when y−2x<0

The cost to travel a road between two adjacent villages is 1 pound. Determine the minimum cost to travel between villages at (x1,y1) and (x2,y2).

Notes:

the road network is is defined so that the villages form an undirected graph.

there are two roads joining (−1,1) and (1,1), but otherwise there is a single road between any adjacent pair of villages.

the road network does not form a simple grid – that would be too easy!

Input:

First line will contain T, number of testcases. Then the testcases follow.

Each testcase contains of a single line of input, four integers x1,y1,x2,y2.

Output:

For each testcase, output in a single line answer, the minimum cost in pounds to travel from (x1,y1) to (x2,y2).

Constraints

T≤1000

The co-ordinates (x1,y1) and (x2,y2) of the villages satisfy

−1015<x1,x2<1015 and

0<y1,y2<1015

Subtasks

10 points : −10<x1,x2<10 and 0<y1,y2<10

10 points : original constraints, but the villages are selected so that the cost for each testcase is <6

30 points: −106<x1,x2<106 and 0<y1,y2<106

50 points: original constraints

Sample Input:

5

7 5 7 5

1 1 -1 1

3 7 5 9

113 167 361765 170831

24844359 74532371 14673633991 3057007029

Sample Output:

0

1

6

78

278031

Explanation:

In the first test case, the start and end village co-ordinates are both (7,5), so the cost is 0.

In the second test case:

From (1,1) go West to (1,−1) in one step. So the cost is 1.

Note that the South road from (1,1) also goes to (1,−1).

In the third test case:

From (3,7) go South to (3,1).

From (3,1) go West to (1,1).

From (1,1) go West to (−1,1).

From (−1,1) go West to (−3,1).

From (−3,1) go West to (−5,1).

From (−5,1) go North to (5,9).

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