如何找到包含所有点或与y = x轴对称的点的最小矩形？

Question

I have a set of points (x,y). How can I find the smallest rectangle (edges are perpendicular to the axis of the coordinate system) that contains these points or points symmetrical to the y=x axis? Not all the points have to be mirror images at the same time, so it is possible to have some normal and some symmetrical points to the original ones.

Answer 1

You could solve this using the brute-force O(n*2^n) approach: define your points as some class/structure which, in addition to the (x,y) coordinates of your point, also holds a boolean mirrored state. For n number of points, loop over all the 2^n possible set of states of this set of points. For each set of states, calculate the circumference ( 2x+2y ) of the bounding rectangle.

Below follows this solution implemented in Swift. If you choose to use this brute-force approach, hopefully you can translate it into your language of choice.

---

Point container and help functions:

struct Point {
    private var xOrig : Double
    private var yOrig : Double
    private var mirrored = false
    var x : Double { return mirrored ? yOrig : xOrig }
    var y : Double { return mirrored ? xOrig : yOrig }

    init(_ x: Double, _ y: Double) {
        xOrig = x
        yOrig = y
    }

    mutating func setState(state: Bool) {
        mirrored = state
    }
}

func boundingBoxCircumference(points: [Point]) -> Double {
    let xMax = points.maxElement { $0.x < $1.x }?.x ?? 0.0
    let xMin = points.minElement { $0.x < $1.x }?.x ?? 0.0
    let yMax = points.maxElement { $0.y < $1.y }?.y ?? 0.0
    let yMin = points.minElement { $0.y < $1.y }?.y ?? 0.0

    return 2*((xMax - xMin) + (yMax - yMin))
}

func setMirroredStates(inout points: [Point], states: [Bool]) {
    for i in 0..<points.count {
        points[i].setState(states[i])
    }
}

Main function and an example

func findSmallestBox(var points: [Point]) -> Double {
    var smallestBox = Double.infinity
    for i in 0..<Int(pow(2,Double(points.count))) {
        smallestBox = min(smallestBox, boundingBoxCircumference(points))

        var binString = String(i, radix: 2)
        if let a = Optional(points.count - binString.characters.count) where a > 0 {
            binString = "".stringByPaddingToLength(a, withString: "0", startingAtIndex: 0) + binString
        }
        let pointStates = Array(binString.characters).map { String($0) == "1" }

        setMirroredStates(&points, states: pointStates)
    }
    return smallestBox
}

var myPoints : [Point] = [Point(1,2), Point(-4,3), Point(7,-5), Point(5,4), Point(-5, -1)]
print(findSmallestBox(myPoints)) // 34