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Problem Statement for "ThreeWatchtowers"

Problem Statement

You wish to build three watchtowers to overlook several points of interest. The points of interest are given in the int[]s x and y, where each element of x and the corresponding element of y represent the coordinates of a single point of interest.

Each of the three watchtowers has a field of view that is the same in all directions. The viewing distance of each of the three watchtowers is given in the int[] view, so the i-th watchtower can see all points which are located not further than view[i] units from it. Assuming optimal placement, return the maximum number of points of interest that are within view of at least one of the watchtowers.

Definition

Class:
ThreeWatchtowers
Method:
maximumCoverage
Parameters:
int[], int[], int[]
Returns:
int
Method signature:
int maximumCoverage(int[] x, int[] y, int[] view)
(be sure your method is public)

Notes

  • Even though points of interest are restricted to a given area, the watchtowers themselves may be located outside of that area, and do not necessarily have to be placed on lattice points.
  • A watchtower can see a point exactly on the edge of its field of view. Thus, a watchtower at (0, 0) with a radius of 2 could see a point at (0, 2).

Constraints

  • x and y will each contain between 1 and 20 elements, inclusive.
  • x and y will each contain the same number of elements.
  • Each element of x and y will be between 0 and 20, inclusive.
  • view will contain exactly 3 elements.
  • Each element of view will be between 1 and 20, inclusive.

Examples

  1. {5, 10, 5, 20, 4, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 6, 7}

    {5, 10, 5, 5, 6, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 1, 2, 3, 3, 2}

    {20, 20, 20}

    Returns: 20

  2. {0, 10, 20}

    {15, 16, 20}

    {1, 1, 1}

    Returns: 3

    Even though the watchtowers can't see very far, with only three points of interest, it is trivial to cover them all.

  3. {0, 1, 10, 20, 15, 14}

    {1, 0, 10, 20, 18, 12}

    {2, 2, 2}

    Returns: 4

    The first two points are close enough together to be covered by a single watchtower. The other points are too spread out, so the other two towers can each only see a single point.

  4. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

    {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

    {1, 1, 13}

    Returns: 10

    The first two watchtowers are not really needed, since the last one has a wide enough range to see all 10 points at once.

  5. {0, 5, 6, 16, 18, 20}

    {0, 6, 7, 20, 20, 20}

    {1, 3, 5}

    Returns: 6

    The first watchtower covers the first point. The second watchtower can cover the next two points. The last three points can be covered by the final tower.

  6. {0, 0, 1, 1, 2, 0}

    {10, 20, 2, 0, 1, 1}

    {1, 1, 1}

    Returns: 6

  7. {0, 0, 0, 0}

    {0, 10, 18, 20}

    {1, 1, 1}

    Returns: 4

  8. {0, 0, 1, 1, 2, 0}

    {10, 20, 2, 0, 1, 1}

    {1, 1, 2}

    Returns: 6

  9. {0,20,1,20,20}

    {10,10,0,1,10}

    {10,1,1}

    Returns: 5

  10. {0, 0, 6, 0, 20}

    {0, 8, 0, 20, 0}

    {1, 1, 5}

    Returns: 5

  11. {11,1,19,20,0}

    {10,1,1,20,20}

    {19,1,1}

    Returns: 5

  12. {10,19,18,0,20}

    {10,20,10,0,0}

    {19,1,1}

    Returns: 5

  13. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}

    {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}

    {20, 20, 20}

    Returns: 20

  14. {1,0}

    {0,0}

    {1,1,1}

    Returns: 2

  15. {1, 2, 0, 3, 1}

    {0, 3, 2, 1, 0}

    {1, 1, 1}

    Returns: 4

  16. {4,13,0,15,13}

    {6,13,16,7,12}

    {3,2,2}

    Returns: 4

  17. {1,4,7,5,4}

    {7,5,2,7,1}

    {1,2,1}

    Returns: 4

  18. {18,4,5,14}

    {12,11,7,13}

    {12,14,18}

    Returns: 4

  19. {17,15,11,3,20,18,10,10}

    {17,15,13,18,7,12,2,12}

    {15,18,7}

    Returns: 8

  20. {16,3,6,16,13}

    {3,18,13,12,0}

    {8,7,7}

    Returns: 5

  21. {20,18,1,20,4,2,10,4,17,11,13,19,14,3,14,7,20,15,15}

    {7,0,3,5,10,12,6,0,12,9,19,5,20,1,14,0,12,13,19}

    {18,19,10}

    Returns: 19

  22. {12,7}

    {5,15}

    {15,16,4}

    Returns: 2

  23. {9}

    {7}

    {9,14,13}

    Returns: 1

  24. {1,0,11,0,1,0,20,0,9,1,14,8,16,17}

    {2,1,3,17,2,10,20,6,3,5,7,11,15,18}

    {11,5,3}

    Returns: 14

  25. {16,4,19,0,7,10,14,5,11,7,17,1,14,15,13,16,7}

    {5,18,0,16,14,5,0,1,14,8,11,9,15,20,2,7,4}

    {2,5,5}

    Returns: 12

  26. {15,16,0,19,4,3}

    {7,12,13,5,12,15}

    {16,12,16}

    Returns: 6

  27. {6,2,17,17,5,0,2,2,9,15,4,10,13,5,17,6,7,16}

    {5,6,13,9,17,5,9,12,9,13,10,16,20,15,12,16,5,9}

    {20,1,13}

    Returns: 18

  28. {18,17,19,9,9,17}

    {2,9,18,8,19,16}

    {19,11,3}

    Returns: 6

  29. {6,19,8,9,17,7,7,3,8,3,16,19,8}

    {2,0,1,10,11,0,10,5,12,3,13,11,17}

    {10,17,16}

    Returns: 13

  30. {6,13}

    {10,10}

    {1,12,12}

    Returns: 2

  31. {0,1,20,8,20,13,7,0,5,10,13,0,1,20,19,3,19,15,13}

    {3,9,17,20,15,8,4,7,9,18,7,11,3,20,7,14,13,7,0}

    {12,7,10}

    Returns: 19

  32. {11,8,11,8,3,4,11,15,9}

    {12,7,4,7,3,9,4,2,18}

    {9,5,11}

    Returns: 9

  33. {5,19,14,1,9,10,17,16,12,15,9,19,0,1}

    {15,7,20,8,6,6,16,7,14,0,8,11,1,20}

    {5,12,5}

    Returns: 14

  34. {18,15,8,17}

    {17,8,18,20}

    {15,8,16}

    Returns: 4

  35. {0,9,10,7,5,16,8,8,5,13,14,20,8,3}

    {10,18,16,17,9,6,18,13,20,1,0,4,12,11}

    {17,4,14}

    Returns: 14

  36. {1,20,4,16,10,15,1,2,5,11,5,13,9,19,14,16,13}

    {8,0,15,5,12,16,20,10,6,14,4,20,2,9,10,14,20}

    {13,7,13}

    Returns: 17

  37. {11,10,7}

    {9,1,14}

    {11,16,19}

    Returns: 3

  38. {8,14,15,1,2,9,1,13,17,1,19,13,5,1,9,19,8}

    {18,10,13,14,18,3,15,1,14,20,17,14,12,10,3,10,2}

    {6,1,9}

    Returns: 17

  39. {2,6,11}

    {20,3,8}

    {12,20,18}

    Returns: 3

  40. {16,2,14,7}

    {15,19,4,1}

    {19,16,15}

    Returns: 4

  41. {20,6,13,19,10,0,10,3,5,10,4,1,4,7}

    {2,4,1,15,1,10,6,2,19,3,17,3,19,11}

    {10,2,2}

    Returns: 13

  42. {8,16,14,6,20,18,4,5,13,11,13,1,5,13,12,15,15,4}

    {5,20,11,4,20,14,20,10,13,8,5,20,18,2,1,20,18,9}

    {9,3,3}

    Returns: 18

  43. {11,3,13,11,7,5,8,12,12,6,16,17,1,18,14,13,4,17,12}

    {15,15,18,17,0,18,16,10,10,6,1,0,16,1,4,11,11,19,12}

    {5,10,14}

    Returns: 19

  44. {18}

    {2}

    {7,10,16}

    Returns: 1

  45. {11,2,0,15,11,5,16,5,20,1,18,13}

    {6,3,15,10,5,13,13,6,11,0,20,20}

    {2,4,17}

    Returns: 12

  46. {17,9,1}

    {5,1,3}

    {7,12,3}

    Returns: 3

  47. {10,9,7,16,5,20,17,17,0}

    {16,1,6,4,20,14,0,6,17}

    {14,14,19}

    Returns: 9

  48. {8,0,5,16,0,13,2,11,14,6,5,15,3,13,1,15,18,4,19,5}

    {9,7,0,11,4,13,5,13,13,13,9,6,11,8,1,9,5,2,9,6}

    {1,1,1}

    Returns: 4

  49. {7,2,20,6,1,1,13,11,3,4,19,4,17,5,19,9,18,13,16,5}

    {8,9,11,11,8,4,19,17,2,14,17,5,9,6,7,12,14,7,18,7}

    {1,1,1}

    Returns: 5

  50. {2,14,15,17,2,6,20,14,8,15,11,8,5,13,19,12,15,13,1,1}

    {13,0,3,7,8,8,20,7,0,14,2,1,12,13,18,12,5,19,11,14}

    {1,1,2}

    Returns: 7

  51. {7,8,13,8,15,7,6,0,1,12,16,20,20,20,16,14,16,16,3,11}

    {15,2,14,11,7,9,17,15,11,3,11,1,9,1,9,20,17,10,10,1}

    {1,1,1}

    Returns: 6

  52. {4,12,20,9,7,4,9,15,14,8,14,4,12,13,0,18,16,16,9,10}

    {14,1,20,7,7,5,0,1,1,13,19,4,2,8,20,9,15,14,20,10}

    {2,1,1}

    Returns: 8

  53. {4,13,13,5,0,1,18,16,20,17,17,1,14,4,19,7,20,18,12,4}

    {18,17,2,1,15,4,20,4,5,15,10,14,5,0,14,6,16,1,10,14}

    {4,1,1}

    Returns: 9

  54. {2,8,5,11,14,10,19,15,20,17,14,8,2,15,10,16,11,13,20,2}

    {6,17,0,17,1,4,5,7,15,14,1,3,20,19,3,20,17,5,1,8}

    {4,1,1}

    Returns: 10

  55. {10,19,10,9,2,8,14,0,1,8,5,13,3,10,8,16,3,15,1,13}

    {10,6,19,19,18,19,15,20,7,1,10,19,10,15,12,16,3,1,4,16}

    {1,4,2}

    Returns: 11

  56. {15,8,16,7,10,1,15,11,12,13,18,16,6,14,18,17,8,17,16,1}

    {14,5,12,4,18,15,20,4,5,10,13,12,18,10,16,11,10,16,6,1}

    {4,1,1}

    Returns: 13

  57. {11,8,0,8,17,10,5,1,8,14,5,16,18,6,13,12,15,8,9,14}

    {15,13,7,12,2,6,7,3,3,20,12,1,4,3,20,13,0,3,0,13}

    {1,3,4}

    Returns: 12

  58. {6,16,19,7,18,8,16,15,4,3,20,0,12,5,13,19,18,8,3,9}

    {18,5,9,1,20,2,6,5,16,11,2,17,6,10,14,7,9,4,0,9}

    {1,8,1}

    Returns: 15

  59. {10,15,11,15,6,14,0,13,4,14,10,8,0,20,9,3,3,7,19,6}

    {10,13,17,13,1,10,9,0,13,17,14,4,13,7,19,17,18,12,7,11}

    {5,3,1}

    Returns: 14

  60. {9,12,10,20,18,14,12,8,20,7,4,6,4,20,9,2,8,7,14,12}

    {7,11,7,19,18,8,14,8,18,8,0,17,12,1,12,14,4,6,0,15}

    {6,2,1}

    Returns: 16

  61. {20,9,12,2,3,20,0,1,4,18,12,16,8,0,10,10,10,11,12,15}

    {20,15,15,15,13,0,2,8,6,9,0,7,9,12,17,5,13,16,17,3}

    {1,9,1}

    Returns: 17

  62. {17,2,13,11,0,7,0,10,6,9,0,20,17,11,7,2,14,9,10,1}

    {13,9,5,2,6,0,17,2,2,2,3,2,14,2,6,7,18,14,19,1}

    {3,1,8}

    Returns: 18

  63. {14,13,9,18,0,13,0,1,5,1,10,15,18,2,17,9,16,1,14,1}

    {12,13,15,15,10,20,11,10,18,13,11,16,9,3,17,1,19,19,19,13}

    {1,1,10}

    Returns: 20

  64. {1,20,13,11,3,19,20,8,15,6,14,15,8,2,5,4,8,1,5,19}

    {0,0,7,14,17,3,19,3,16,18,8,12,1,8,12,7,9,12,7,11}

    {6,7,5}

    Returns: 19

  65. {10, 18, 16, 9, 17, 20, 4, 3, 11, 9}

    {4, 17, 6, 14, 14, 14, 12, 12, 13, 16}

    {1, 1, 2}

    Returns: 7

  66. {0,9,7,20,20}

    {0,3,7,0,20}

    {5,1,1}

    Returns: 5

  67. {0,9,4,20,20}

    {0,3,8,0,20}

    {5,1,1}

    Returns: 5

  68. {0,8,4,20,20}

    {0,0,8,0,20}

    {5,1,1}

    Returns: 5

  69. {0,6,3,20,20}

    {0,0,9,0,20}

    {5,1,1}

    Returns: 5

  70. {0, 1, 10, 20, 15, 14 }

    {1, 0, 10, 20, 18, 12 }

    {2, 2, 2 }

    Returns: 4

  71. {0, 10, 20 }

    {15, 16, 20 }

    {1, 1, 1 }

    Returns: 3

  72. {1, 2, 0, 3, 1 }

    {0, 3, 2, 1, 0 }

    {1, 1, 1 }

    Returns: 4

  73. {10, 18, 16, 9, 17, 20, 4, 3, 11, 9 }

    {4, 17, 6, 14, 14, 14, 12, 12, 13, 16 }

    {1, 1, 2 }

    Returns: 7

  74. {1, 3, 5, 7, 9 }

    {2, 2, 2, 2, 2 }

    {1, 1, 1 }

    Returns: 5

  75. {0, 1, 3, 4, 5, 7, 8, 9, 11 }

    {0, 3, 0, 8, 11, 8, 1, 4, 1 }

    {2, 2, 2 }

    Returns: 9

  76. {2, 5, 4, 9, 12, 10, 3, 8, 4, 6, 3, 4, 12, 19, 20, 18, 15, 13, 16, 1 }

    {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 }

    {2, 3, 3 }

    Returns: 13

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