Overlapping Disks

Overlapping Disks

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DE English FR PL UK ZH-HANS

Given a list of disks on the two-dimensional plane represented as tuples (x, y, r) so that x, y is the center point and r is the radius of that disk, count how many pairs of disks intersect.

Two disks (x1, y1, r1) and (x2, y2, r2) intersect if and only if they satisfy the Pythagorean inequality (x2-x1)**2+(y2-y1)**2<=(r1+r2)**2.

Note how this precise formula runs on pure integer arithmetic whenever its arguments are integers, so no square roots or any other irrational numbers gum up the works with all that decimal noise. (This formula also uses the operator <= to count two kissing disks as an intersecting pair).

For this problem, crudely looping through all possible pairs of disks would work, but also become horrendously inefficient for large lists. However, a sweep line algorithm can solve this problem not just effectively, but also efficiently (a crucial but often overlooked “eff-ing” distinction) by looking at a far fewer pairs of disks.

Here is a scheme for [(0, 0, 3), (6, 0, 3), (6, 6, 3), (0, 6, 3)]:

example

Input: List of tuples (tuple) of integers (int).

Output: Integer (int).

Examples:

assert overlapping_disks([(0, 0, 3), (6, 0, 3), (6, 6, 3), (0, 6, 3)]) == 4
assert overlapping_disks([(4, -1, 3), (-3, 3, 2), (-3, 4, 2), (3, 1, 4)]) == 2
assert (
    overlapping_disks([(-10, 6, 2), (6, -4, 5), (6, 3, 5), (-9, -8, 1), (1, -5, 3)])
    == 2
)
assert (
    overlapping_disks(
        [
            (2, 2, 1),
            (3, 3, 1),
            (4, 4, 2),
            (5, 5, 2),
            (6, 6, 3),
            (7, 7, 3),
            (8, 8, 4),
            (9, 9, 4),
            (10, 10, 5),
            (11, 11, 5),
            (12, 12, 6),
            (13, 13, 6),
            (14, 14, 7),
            (15, 15, 7),
            (16, 16, 8),
            (17, 17, 8),
            (18, 18, 9),
            (19, 19, 9),
            (20, 20, 10),
            (21, 21, 10),
            (22, 22, 11),
            (23, 23, 11),
            (24, 24, 12),
            (25, 25, 12),
            (26, 26, 13),
            (27, 27, 13),
            (28, 28, 14),
            (29, 29, 14),
            (30, 30, 15),
            (31, 31, 15),
            (32, 32, 16),
            (33, 33, 16),
            (34, 34, 17),
            (35, 35, 17),
            (36, 36, 18),
            (37, 37, 18),
            (38, 38, 19),
            (39, 39, 19),
            (40, 40, 20),
            (41, 41, 20),
            (42, 42, 21),
            (43, 43, 21),
            (44, 44, 22),
            (45, 45, 22),
            (46, 46, 23),
            (47, 47, 23),
            (48, 48, 24),
            (49, 49, 24),
            (50, 50, 25),
            (51, 51, 25),
            (52, 52, 26),
            (53, 53, 26),
            (54, 54, 27),
            (55, 55, 27),
            (56, 56, 28),
            (57, 57, 28),
            (58, 58, 29),
            (59, 59, 29),
            (60, 60, 30),
            (61, 61, 30),
            (62, 62, 31),
            (63, 63, 31),
            (64, 64, 32),
            (65, 65, 32),
            (66, 66, 33),
            (67, 67, 33),
            (68, 68, 34),
            (69, 69, 34),
            (70, 70, 35),
            (71, 71, 35),
            (72, 72, 36),
            (73, 73, 36),
            (74, 74, 37),
            (75, 75, 37),
            (76, 76, 38),
            (77, 77, 38),
            (78, 78, 39),
            (79, 79, 39),
            (80, 80, 40),
            (81, 81, 40),
            (82, 82, 41),
            (83, 83, 41),
            (84, 84, 42),
            (85, 85, 42),
            (86, 86, 43),
            (87, 87, 43),
            (88, 88, 44),
            (89, 89, 44),
            (90, 90, 45),
            (91, 91, 45),
            (92, 92, 46),
            (93, 93, 46),
            (94, 94, 47),
            (95, 95, 47),
            (96, 96, 48),
            (97, 97, 48),
            (98, 98, 49),
            (99, 99, 49),
            (100, 100, 50),
        ]
    )
    == 2563
)
If you feel yourself in need of a hint, click this line.

Sweep through the space from left to right for all relevant x-coordinate values and maintain the set of active disks at the moment. Each individual disk (x, y, r) enters the active set when the vertical sweep line reaches the x-coordinate x-r, and leaves the active set when the sweep line reaches x+r. At the time when a disk enters the active set, you need to look for its intersections only in this active set.

The mission was taken from Python CCPS 109. It is taught for Ryerson Chang School of Continuing Education by Ilkka Kokkarinen

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