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2013. Detect Squares

Description

You are given a stream of points on the X-Y plane. Design an algorithm that:

  • Adds new points from the stream into a data structure. Duplicate points are allowed and should be treated as different points.
  • Given a query point, counts the number of ways to choose three points from the data structure such that the three points and the query point form an axis-aligned square with positive area.

An axis-aligned square is a square whose edges are all the same length and are either parallel or perpendicular to the x-axis and y-axis.

Implement the DetectSquares class:

  • DetectSquares() Initializes the object with an empty data structure.
  • void add(int[] point) Adds a new point point = [x, y] to the data structure.
  • int count(int[] point) Counts the number of ways to form axis-aligned squares with point point = [x, y] as described above.

 

Example 1:

Input
["DetectSquares", "add", "add", "add", "count", "count", "add", "count"]
[[], [[3, 10]], [[11, 2]], [[3, 2]], [[11, 10]], [[14, 8]], [[11, 2]], [[11, 10]]]
Output
[null, null, null, null, 1, 0, null, 2]

Explanation
DetectSquares detectSquares = new DetectSquares();
detectSquares.add([3, 10]);
detectSquares.add([11, 2]);
detectSquares.add([3, 2]);
detectSquares.count([11, 10]); // return 1. You can choose:
// - The first, second, and third points
detectSquares.count([14, 8]); // return 0. The query point cannot form a square with any points in the data structure.
detectSquares.add([11, 2]); // Adding duplicate points is allowed.
detectSquares.count([11, 10]); // return 2. You can choose:
// - The first, second, and third points
// - The first, third, and fourth points

 

Constraints:

  • point.length == 2
  • 0 <= x, y <= 1000
  • At most 3000 calls in total will be made to add and count.

Solutions

Solution 1: Hash Table

We can use a hash table $cnt$ to maintain all the information of the points, where $cnt[x][y]$ represents the count of point $(x, y)$.

When calling the $add(x, y)$ method, we increase the value of $cnt[x][y]$ by $1$.

When calling the $count(x_1, y_1)$ method, we need to get three other points to form an axis-aligned square. We can enumerate the point $(x_2, y_1)$ that is parallel to the $x$-axis and at a distance $d$ from $(x_1, y_1)$. If such a point exists, based on these two points, we can determine the other two points as $(x_1, y_1 + d)$ and $(x_2, y_1 + d)$, or $(x_1, y_1 - d)$ and $(x_2, y_1 - d)$. We can add up the number of schemes for these two situations.

In terms of time complexity, the time complexity of calling the $add(x, y)$ method is $O(1)$, and the time complexity of calling the $count(x_1, y_1)$ method is $O(n)$; the space complexity is $O(n)$. Here, $n$ is the number of points in the data stream.

  • class DetectSquares {
    private Map<Integer, Map<Integer, Integer>> cnt = new HashMap<>();

    public DetectSquares() {
    }

    public void add(int[] point) {
        int x = point[0], y = point[1];
        cnt.computeIfAbsent(x, k -> new HashMap<>()).merge(y, 1, Integer::sum);
    }

    public int count(int[] point) {
        int x1 = point[0], y1 = point[1];
        if (!cnt.containsKey(x1)) {
            return 0;
        }
        int ans = 0;
        for (var e : cnt.entrySet()) {
            int x2 = e.getKey();
            if (x2 != x1) {
                int d = x2 - x1;
                var cnt1 = cnt.get(x1);
                var cnt2 = e.getValue();
                ans += cnt2.getOrDefault(y1, 0) * cnt1.getOrDefault(y1 + d, 0)
                    * cnt2.getOrDefault(y1 + d, 0);
                ans += cnt2.getOrDefault(y1, 0) * cnt1.getOrDefault(y1 - d, 0)
                    * cnt2.getOrDefault(y1 - d, 0);
            }
        }
        return ans;
    }
}

/**
 * Your DetectSquares object will be instantiated and called as such:
 * DetectSquares obj = new DetectSquares();
 * obj.add(point);
 * int param_2 = obj.count(point);
 */

  • class DetectSquares {
public:
    DetectSquares() {
    }

    void add(vector<int> point) {
        int x = point[0], y = point[1];
        ++cnt[x][y];
    }

    int count(vector<int> point) {
        int x1 = point[0], y1 = point[1];
        if (!cnt.count(x1)) {
            return 0;
        }
        int ans = 0;
        for (auto& [x2, cnt2] : cnt) {
            if (x2 != x1) {
                int d = x2 - x1;
                auto& cnt1 = cnt[x1];
                ans += cnt2[y1] * cnt1[y1 + d] * cnt2[y1 + d];
                ans += cnt2[y1] * cnt1[y1 - d] * cnt2[y1 - d];
            }
        }
        return ans;
    }

private:
    unordered_map<int, unordered_map<int, int>> cnt;
};

/**
 * Your DetectSquares object will be instantiated and called as such:
 * DetectSquares* obj = new DetectSquares();
 * obj->add(point);
 * int param_2 = obj->count(point);
 */

  • class DetectSquares:
    def __init__(self):
        self.cnt = defaultdict(Counter)

    def add(self, point: List[int]) -> None:
        x, y = point
        self.cnt[x][y] += 1

    def count(self, point: List[int]) -> int:
        x1, y1 = point
        if x1 not in self.cnt:
            return 0
        ans = 0
        for x2 in self.cnt.keys():
            if x2 != x1:
                d = x2 - x1
                ans += self.cnt[x2][y1] * self.cnt[x1][y1 + d] * self.cnt[x2][y1 + d]
                ans += self.cnt[x2][y1] * self.cnt[x1][y1 - d] * self.cnt[x2][y1 - d]
        return ans


# Your DetectSquares object will be instantiated and called as such:
# obj = DetectSquares()
# obj.add(point)
# param_2 = obj.count(point)


  • type DetectSquares struct {
	cnt map[int]map[int]int
}

func Constructor() DetectSquares {
	return DetectSquares{map[int]map[int]int{} }
}

func (this *DetectSquares) Add(point []int) {
	x, y := point[0], point[1]
	if _, ok := this.cnt[x]; !ok {
		this.cnt[x] = map[int]int{}
	}
	this.cnt[x][y]++
}

func (this *DetectSquares) Count(point []int) (ans int) {
	x1, y1 := point[0], point[1]
	if cnt1, ok := this.cnt[x1]; ok {
		for x2, cnt2 := range this.cnt {
			if x2 != x1 {
				d := x2 - x1
				ans += cnt2[y1] * cnt1[y1+d] * cnt2[y1+d]
				ans += cnt2[y1] * cnt1[y1-d] * cnt2[y1-d]
			}
		}
	}
	return
}

/**
 * Your DetectSquares object will be instantiated and called as such:
 * obj := Constructor();
 * obj.Add(point);
 * param_2 := obj.Count(point);
 */

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