清华计算几何-算法LowBound和ConvexHull(凸包)-GrahamScan

news2024/11/15 11:13:43

算法复杂度最低界限LowBound

算法求解复杂度是否存在一个最低界限，有时候想尽一切办法优化一个算法，去优化其复杂度，比如

清华计算几何-ConvexHull(凸包)-求极点InTriangle/ToLeft Test-CSDN博客

清华计算几何-ConvexHull(凸包)-求极边_计算几何中的toleft测试-CSDN博客

清华计算几何-ConvexHull(凸包)-JarivsMarch-CSDN博客

算法复杂度依次到: O(n4), O(n3), O(n2)

一直优化下去能最低到哪个复杂度，有时候不好直接估算出。有个比较好的办法是从经典算法(比如排序)中，找到此算法和经典算法的等价转换。如果此算法和经典算法等价，则此算法的LowBound也是经典算法的LowBound。

算法等价转换

算法等价转换就是: 假设A为经典算法, 已经知道其算法复杂度。存在另外一个未知算法复杂度的算法B，如今要估算算法B的lowBond. 如果A算法的输入经过O(n)以内复杂度转换算法变为B算法输入，B算法计算的输出能在O(n)复杂度以内转换为A算法的输入，则称A算法和B算法等价。

估算ConvexHull算法的LowBound

从前面可以隐隐看出点集求解ConvexHull的流程和排序类似，建立起排序和凸包的点集合之间O(n)的转换关系（点输入和点输出），排序的算法Lowbound就是凸包算法的Lowbound。

排序的LowBound是O(nlogn), 而等价下凸包算法也是O(nlogn)

凸包算法-GrahamScan

GrahamScan算法流程

Presorting(预排序)

[1]找到LowertToLeft点P

[2]找到和P点相连最右边的点(逆时针CCW)

Scan(扫描)

GrahamScan算法Backtrack执行案例

GrahamScan算法代码实现

算法核心代码

#include <iostream>
#include <vector>
#include <stack>
#include <list>
#include <algorithm>

#include "BasicCompute.h"

using namespace std;


template<typename type>
class CustomStack
{
private:
	vector<type> datas;

public:
	CustomStack()
	{

	}

	void Push(const type& value)
	{
		datas.push_back(value);
	}

	type Pop()
	{
		int num = GetNum();
		if (num == 0)
			throw "size is zero, do not allow pop";

		int value = datas[num - 1];
		datas.pop_back();
		return value;
	}

	int GetNum()
	{
		return datas.size();
	}

	type GetTopValue(int topIndex = 0)
	{
		return datas[GetNum() - topIndex - 1];
	}

	void GetVectorData(vector<type>& outDatas)
	{
		outDatas = datas;
	}
};


void GrahamScan_GetConvexPointSet(const vector<Point>& inPoints, vector<int>& convecHullPointIndexs)
{
	if (inPoints.size() <= 3)
		return;

	convecHullPointIndexs.empty();

	int ltfIndex = FindLowestThenLeftmost(inPoints);

	// quick sort by to left
	vector<int> tArray;
	for (int index = 0; index < inPoints.size(); index++)
	{
		if (index != ltfIndex)
			tArray.push_back(index);
	}

	auto CompareFunc = [&](int a, int b)
	{
		return IsLeft(inPoints[ltfIndex], inPoints[a], inPoints[b]);
	};
	
	sort(tArray.begin(), tArray.end(), CompareFunc);

	// init stack s and stack t
	CustomStack<int> s;
	s.Push(ltfIndex);
	s.Push(tArray[0]);


	CustomStack<int> t;
	for (int index = tArray.size() - 1; index >= 1; index--)
	{
		t.Push(tArray[index]);
	}

	while (t.GetNum() != 0)
	{
		int sTopIndex = s.GetTopValue(0);
		int sTopSecondIndex = s.GetTopValue(1);
		int tTopIndex = t.GetTopValue(0);
		if (IsLeft(inPoints[sTopSecondIndex], inPoints[sTopIndex], inPoints[tTopIndex]))
		{
			s.Push(t.Pop());
		}
		else
		{
			s.Pop();
		}
	}

	s.GetVectorData(convecHullPointIndexs);
}

测试代码


#include <iostream>
#include <vector>
#include "ExtremityEdgeConvex.h"
#include "JarvisMarch.h"
#include "GrahamScan.h"


using namespace std;



int main()
{
    std::cout << "Hello World!\n";

	// point set contruct
	vector<Point> inPoints =
	{
		{0, 0},
		{-1, -1},
		{5, 2},
		{4, 5},
		{3, 3},
		{-1, 3},
		{2, 2},
		{-3, 2},
	};


	vector<int> convecHullPointIndexs;
	GrahamScan_GetConvexPointSet(inPoints, convecHullPointIndexs);
	for (int index = 0; index < convecHullPointIndexs.size(); index++)
	{
		int pointIndex = convecHullPointIndexs[index];
		printf("(%f, %f)\n", inPoints[pointIndex].x, inPoints[pointIndex].y);
	}
}