Posts tagged as “geometry”

花花酱 LeetCode 1266. Minimum Time Visiting All Points

By zxi on November 23, 2019

On a plane there are n points with integer coordinates points[i] = [xi, yi]. Your task is to find the minimum time in seconds to visit all points.

You can move according to the next rules:

In one second always you can either move vertically, horizontally by one unit or diagonally (it means to move one unit vertically and one unit horizontally in one second).
You have to visit the points in the same order as they appear in the array.

Example 1:

Input: points = [[1,1],[3,4],[-1,0]]
Output: 7
Explanation: One optimal path is [1,1] -> [2,2] -> [3,3] -> [3,4] -> [2,3] -> [1,2] -> [0,1] -> [-1,0]   
Time from [1,1] to [3,4] = 3 seconds 
Time from [3,4] to [-1,0] = 4 seconds
Total time = 7 seconds

Example 2:

Input: points = [[3,2],[-2,2]]
Output: 5

Constraints:

points.length == n
1 <= n <= 100
points[i].length == 2
-1000 <= points[i][0], points[i][1] <= 1000

Solution: Geometry + Greedy

dx = abs(x1 – x2)
dy = abs(y1 – y2)

go diagonally first for min(dx, dy) steps, and then go straight line for abs(dx – dy) steps.

Time complexity: O(n)
Space complexity: O(1)

C++

// Author: Huahua

class Solution {

public:

int minTimeToVisitAllPoints(vector<vector<int>>& points) {

int ans = 0;

for (size_t i = 1; i < points.size(); ++i) {

int dx = abs(points[i - 1][0] - points[i][0]);

int dy = abs(points[i - 1][1] - points[i][1]);

ans += min(dx, dy) + abs(dx - dy);

}

return ans;

}

};

花花酱 LeetCode 391. Perfect Rectangle

By zxi on July 11, 2019

Given N axis-aligned rectangles where N > 0, determine if they all together form an exact cover of a rectangular region.

Each rectangle is represented as a bottom-left point and a top-right point. For example, a unit square is represented as [1,1,2,2]. (coordinate of bottom-left point is (1, 1) and top-right point is (2, 2)).

Example 1:

rectangles = [

[1,1,3,3],

[3,1,4,2],

[3,2,4,4],

[1,3,2,4],

[2,3,3,4]

]

Return true. All 5 rectangles together form an exact cover of a rectangular region.

Example 2:

rectangles = [

[1,1,2,3],

[1,3,2,4],

[3,1,4,2],

[3,2,4,4]

]

Return false. Because there is a gap between the two rectangular regions.

Example 3:

rectangles = [

[1,1,3,3],

[3,1,4,2],

[1,3,2,4],

[3,2,4,4]

]

Return false. Because there is a gap in the top center.

Example 4:

rectangles = [

[1,1,3,3],

[3,1,4,2],

[1,3,2,4],

[2,2,4,4]

]

Return false. Because two of the rectangles overlap with each other.

Solution: Counting corner points

C++

// Author: Huahua

class Solution {

public:

bool isRectangleCover(vector<vector<int>>& rectangles) {

set<pair<int, int>> corners;

int area = 0;

for (const auto& rect : rectangles) {

pair<int, int> p1{rect[0], rect[1]};

pair<int, int> p2{rect[2], rect[1]};

pair<int, int> p3{rect[2], rect[3]};

pair<int, int> p4{rect[0], rect[3]};

for (const auto& p : {p1, p2, p3, p4}) {

const auto& ret = corners.insert(p);

if (!ret.second) corners.erase(ret.first);

}

area += (p3.first - p1.first) * (p3.second - p1.second);

}

if (corners.size() != 4) return false;

const auto& p1 = *begin(corners);

const auto& p3 = *rbegin(corners);

return area == (p3.first - p1.first) * (p3.second - p1.second);

}

};

花花酱 LeetCode 986. Interval List Intersections

By zxi on February 2, 2019

Given two lists of closed intervals, each list of intervals is pairwise disjoint and in sorted order.

Return the intersection of these two interval lists.

(Formally, a closed interval [a, b] (with a <= b) denotes the set of real numbers x with a <= x <= b. The intersection of two closed intervals is a set of real numbers that is either empty, or can be represented as a closed interval. For example, the intersection of [1, 3] and [2, 4] is [2, 3].)

Example 1:

Input: A = [[0,2],[5,10],[13,23],[24,25]], B = [[1,5],[8,12],[15,24],[25,26]]
Output: [[1,2],[5,5],[8,10],[15,23],[24,24],[25,25]]
Reminder: The inputs and the desired output are lists of Interval objects, and not arrays or lists.

Note:

0 <= A.length < 1000
0 <= B.length < 1000
0 <= A[i].start, A[i].end, B[i].start, B[i].end < 10^9

Solution: Two pointers

Time complexity: O(m + n)
Space complexity: O(1)

C++

// Author: Huahua, running time: 36 ms, 925.7 KB

class Solution {

public:

vector<Interval> intervalIntersection(vector<Interval>& A, vector<Interval>& B) {

size_t i = 0;

size_t j = 0;

vector<Interval> ans;

while (i < A.size() && j < B.size()) {

const int s = max(A[i].start, B[j].start);

const int e = min(A[i].end, B[j].end);

if (s <= e) ans.emplace_back(s, e);

if (A[i].end < B[j].end)

++i;

else

++j;

}

return ans;

}

};

Python3

# Author: Huahua, running time: 96 ms, 7.6 MB

class Solution:

def intervalIntersection(self, A: 'List[Interval]', B: 'List[Interval]') -> 'List[Interval]':

i, j, ans = 0, 0, []

while i < len(A) and j < len(B):

s = max(A[i].start, B[j].start)

e = min(A[i].end, B[j].end)

if s <= e:

ans.append(Interval(s, e))

if A[i].end < B[j].end:

i += 1

else:

j += 1

return ans

花花酱 LeetCode 973. K Closest Points to Origin

By zxi on January 13, 2019

We have a list of points on the plane. Find the K closest points to the origin (0, 0).

(Here, the distance between two points on a plane is the Euclidean distance.)

You may return the answer in any order. The answer is guaranteed to be unique (except for the order that it is in.)

Example 1:

Input: points = [[1,3],[-2,2]], K = 1 
Output: [[-2,2]] 
Explanation:  The distance between (1, 3) and the origin is sqrt(10). The distance between (-2, 2) and the origin is sqrt(8). Since sqrt(8) < sqrt(10), (-2, 2) is closer to the origin. We only want the closest K = 1 points from the origin, so the answer is just [[-2,2]].

Example 2:

Input: points = [[3,3],[5,-1],[-2,4]], K = 2 
Output: [[3,3],[-2,4]] (The answer [[-2,4],[3,3]] would also be accepted.)

Note:

1 <= K <= points.length <= 10000
-10000 < points[i][0] < 10000
-10000 < points[i][1] < 10000

Solution: Sort

Time complexity: O(nlogn)
Space complexity: O(n)

C++

// Author: Huahua, Running time: 180 ms

class Solution {

public:

vector<vector<int>> kClosest(vector<vector<int>>& points, int K) {

vector<long> s(points.size());

for (int i = 0; i < points.size(); ++i)

s[i] = (static_cast<long>(points[i][0] * points[i][0] +

points[i][1] * points[i][1]) << 32) | i;

std::sort(begin(s), end(s));

vector<vector<int>> ans(K);

for (int i = 0; i < K; ++i)

ans[i] = points[static_cast<int>(s[i])];

return ans;

}

};

Python3

# Author: Huahua, running time: 528 ms

class Solution:

def kClosest(self, points, K):

return sorted(points, key = lambda p: p[0]**2 + p[1]**2)[0:K]

花花酱 LeetCode 963. Minimum Area Rectangle II

By zxi on December 26, 2018

Given a set of points in the xy-plane, determine the minimum area of any rectangle formed from these points, with sides not necessarily parallel to the x and y axes.

If there isn’t any rectangle, return 0.

Example 1:

Input: [[1,2],[2,1],[1,0],[0,1]]
Output: 2.00000
Explanation: The minimum area rectangle occurs at [1,2],[2,1],[1,0],[0,1], with an area of 2.

Example 2:

Input: [[0,1],[2,1],[1,1],[1,0],[2,0]]
Output: 1.00000
Explanation: The minimum area rectangle occurs at [1,0],[1,1],[2,1],[2,0], with an area of 1.

Example 3:

Input: [[0,3],[1,2],[3,1],[1,3],[2,1]]
Output: 0
Explanation: There is no possible rectangle to form from these points.

Example 4:

Input: [[3,1],[1,1],[0,1],[2,1],[3,3],[3,2],[0,2],[2,3]]
Output: 2.00000
Explanation: The minimum area rectangle occurs at [2,1],[2,3],[3,3],[3,1], with an area of 2.

Note:

1 <= points.length <= 50
0 <= points[i][0] <= 40000
0 <= points[i][1] <= 40000
All points are distinct.
Answers within 10^-5 of the actual value will be accepted as correct.

Solution: HashTable

Iterate all possible triangles and check the opposite points that creating a quadrilateral.

Time complexity: O(n^3)
Space complexity: O(n)

C++

class Solution {

public:

double minAreaFreeRect(vector<vector<int>>& points) {

constexpr double INF_AREA = 1e100;

const auto n = points.size();

unordered_set<int> s;

for (const auto& p : points)

s.insert(p[0] << 16 | p[1]);

double min_area = INF_AREA;

for (int i = 0; i < n; ++i)

for (int j = 0; j < n; ++j) {

if (i == j) continue;

for (int k = 0; k < n; ++k) {

if (i == k || j == k) continue;

const auto& p1 = points[i];

const auto& p2 = points[j];

const auto& p3 = points[k];

int dot = (p2[0] - p1[0]) * (p3[0] - p1[0]) +

(p2[1] - p1[1]) * (p3[1] - p1[1]);

if (dot != 0) continue;

int p4_x = p2[0] + p3[0] - p1[0];

int p4_y = p2[1] + p3[1] - p1[1];

if (!s.count(p4_x << 16 | p4_y)) continue;

double a = pow(p2[0] - p1[0], 2) + pow(p2[1] - p1[1], 2);

double b = pow(p3[0] - p1[0], 2) + pow(p3[1] - p1[1], 2);

double area = a * b;

min_area = min(min_area, area);

}

return min_area < INF_AREA ? sqrt(min_area) : 0;

}

};