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【线段树 区间位运算模板】3117划分数组得到最小的值之和

本文涉及知识点

线段树 区间位运算模板

LeetCode3117. 划分数组得到最小的值之和

给你两个数组 nums 和 andValues，长度分别为 n 和 m。
数组的 值 等于该数组的 最后一个 元素。
你需要将 nums 划分为 m 个 不相交的连续 子数组，对于第 ith 个子数组 [li, ri]，子数组元素的按位AND运算结果等于 andValues[i]，换句话说，对所有的 1 <= i <= m，nums[li] & nums[li + 1] & … & nums[ri] == andValues[i] ，其中 & 表示按位AND运算符。
返回将 nums 划分为 m 个子数组所能得到的可能的 最小 子数组 值 之和。如果无法完成这样的划分，则返回 -1 。
示例 1：
输入： nums = [1,4,3,3,2], andValues = [0,3,3,2]
输出： 12
解释：
唯一可能的划分方法为：
[1,4] 因为 1 & 4 == 0
[3] 因为单元素子数组的按位 AND 结果就是该元素本身
[3] 因为单元素子数组的按位 AND 结果就是该元素本身
[2] 因为单元素子数组的按位 AND 结果就是该元素本身
这些子数组的值之和为 4 + 3 + 3 + 2 = 12
示例 2：

输入： nums = [2,3,5,7,7,7,5], andValues = [0,7,5]

输出： 17

解释：

划分 nums 的三种方式为：

[[2,3,5],[7,7,7],[5]] 其中子数组的值之和为 5 + 7 + 5 = 17
[[2,3,5,7],[7,7],[5]] 其中子数组的值之和为 7 + 7 + 5 = 19
[[2,3,5,7,7],[7],[5]] 其中子数组的值之和为 7 + 7 + 5 = 19
子数组值之和的最小可能值为 17

示例 3：

输入： nums = [1,2,3,4], andValues = [2]

输出： -1

解释：

整个数组 nums 的按位 AND 结果为 0。由于无法将 nums 划分为单个子数组使得元素的按位 AND 结果为 2，因此返回 -1。
提示：
1 <= n == nums.length <= 10
1 <= m == andValues.length <= min(n, 10)
1 <= nums[i] < 10
0 <= andValues[j] < 10

线段树

求区间位运算，可以用封装的模板。
滚动线段数。
pre[i] 记录 将nums[0…i]划分成r-1个数组的最小值之和。
dp[i]记录将nums[0…i]划分成r个数组的最小值之和。
pre的初始值：
如果nums[0…i]的与值为andValues[0]，则值为nums[i]，否则为非法。
dp[i]的值
如果nums[x…i]的与值为andValues[r-1]，假定x ∈ \in ∈(left,right]。
如果r 为0，则非法；否则
dp[i] = min ⁡ y : m a x ( l e f t , 0 ) r − 1 p r e [ y ] \Large \min _{y:max(left,0)}^{r-1}pre[y] miny:max(left,0)r−1​pre[y]+nums[i]

代码

核心代码

template<class TSave, class TRecord >
class CRangUpdateLineTree 
{
protected:
	virtual void OnQuery(const TSave& save, const int& iSaveLeft, const int& iSaveRight) = 0;
	virtual void OnUpdate(TSave& save, const int& iSaveLeft, const int& iSaveRight, const TRecord& update) = 0;
	virtual void OnUpdateParent(TSave& par, const TSave& left, const TSave& r, const int& iSaveLeft, const int& iSaveRight) = 0;
	virtual void OnUpdateRecord(TRecord& old, const TRecord& newRecord) = 0;
};


template<class TSave, class TRecord >
class CTreeRangeLineTree : public CRangUpdateLineTree<TSave, TRecord>
{
protected:
	struct CTreeNode
	{
		int Cnt()const { return m_iMaxIndex - m_iMinIndex + 1; }
		int m_iMinIndex;
		int m_iMaxIndex;
		TRecord record;
		TSave data;
		CTreeNode* m_lChild = nullptr, * m_rChild = nullptr;
	};
	CTreeNode* m_root;
	TSave m_tDefault;
	TRecord m_tRecordDef;
public:
	CTreeRangeLineTree(int iMinIndex, int iMaxIndex, TSave tDefault,TRecord tRecordDef) {
		m_tDefault = tDefault;
		m_tRecordDef = tRecordDef;
		m_root = CreateNode(iMinIndex, iMaxIndex);
	}
	void Update(int iLeftIndex, int iRightIndex, TRecord value)
	{
		Update(m_root, iLeftIndex, iRightIndex, value);
	}
	TSave QueryAll() {
		return m_root->data;
	}
	void Query(int leftIndex, int leftRight) {
		Query(m_root, leftIndex, leftRight);
	}
protected:
	void Query(CTreeNode* node, int iQueryLeft, int iQueryRight) {
		if ((node->m_iMinIndex >= iQueryLeft) && (node->m_iMaxIndex <= iQueryRight)) {
			this->OnQuery(node->data,node->m_iMinIndex,node->m_iMaxIndex);
			return;
		}
		if (1 == node->Cnt()) {//没有子节点
			return;
		}
		CreateChilds(node);
		Fresh(node);
		const int mid = node->m_iMinIndex + (node->m_iMaxIndex - node->m_iMinIndex) / 2;
		if (mid >= iQueryLeft) {
			Query(node->m_lChild, iQueryLeft, iQueryRight);
		}
		if (mid + 1 <= iQueryRight) {
			Query(node->m_rChild, iQueryLeft, iQueryRight);
		}
	}
	void Update(CTreeNode* node, int iOpeLeft, int iOpeRight, TRecord value)
	{
		const int& iSaveLeft = node->m_iMinIndex;
		const int& iSaveRight = node->m_iMaxIndex;
		if ((iOpeLeft <= iSaveLeft) && (iOpeRight >= iSaveRight))
		{
			this->OnUpdate(node->data, iSaveLeft, iSaveRight, value);
			this->OnUpdateRecord(node->record, value);
			return;
		}
		if (1 == node->Cnt()) {//没有子节点
			return;
		}
		CreateChilds(node);
		Fresh(node);
		const int mid = node->m_iMinIndex + (node->m_iMaxIndex - node->m_iMinIndex) / 2;
		if (mid >= iOpeLeft) {
			this->Update(node->m_lChild, iOpeLeft, iOpeRight, value);
		}
		if (mid + 1 <= iOpeRight) {
			this->Update(node->m_rChild, iOpeLeft, iOpeRight, value);
		}
		// 如果有后代，至少两个后代
		this->OnUpdateParent(node->data, node->m_lChild->data, node->m_rChild->data,node->m_iMinIndex,node->m_iMaxIndex);
	}
	void CreateChilds(CTreeNode* node) {
		if (nullptr != node->m_lChild) { return; }
		const int iSaveLeft = node->m_iMinIndex;
		const int iSaveRight = node->m_iMaxIndex;
		const int mid = iSaveLeft + (iSaveRight - iSaveLeft) / 2;
		node->m_lChild = CreateNode(iSaveLeft, mid);
		node->m_rChild = CreateNode(mid + 1, iSaveRight);
	}
	CTreeNode* CreateNode(int iMinIndex, int iMaxIndex) {
		CTreeNode* node = new CTreeNode;
		node->m_iMinIndex = iMinIndex;
		node->m_iMaxIndex = iMaxIndex;
		node->data = m_tDefault;
		node->record = m_tRecordDef;
		return node;
	}
	void Fresh(CTreeNode* node)
	{
		if (m_tRecordDef == node->record)
		{
			return;
		}
		CreateChilds(node);
		Update(node->m_lChild, node->m_lChild->m_iMinIndex, node->m_lChild->m_iMaxIndex, node->record);
		Update(node->m_rChild, node->m_rChild->m_iMinIndex, node->m_rChild->m_iMaxIndex, node->record);
		node->record = m_tRecordDef;
	}
};

template<class TSave, class TRecord >
class CVectorRangeUpdateLineTree : public CRangUpdateLineTree<TSave, TRecord>
{
public:
	CVectorRangeUpdateLineTree(int iEleSize,TSave tDefault, TRecord tRecordNull):m_iEleSize(iEleSize)
		,m_save(iEleSize*4,tDefault), m_record(iEleSize * 4, tRecordNull){
		m_recordNull = tRecordNull;		
	}
	void Update(int iLeftIndex, int iRightIndex, TRecord value)
	{
		Update(1, 0, m_iEleSize - 1, iLeftIndex, iRightIndex, value);
	}
	void Query(int leftIndex, int rightIndex) {
		Query(1, 0, m_iEleSize - 1, leftIndex, rightIndex);
	}
	//void Init() {
	//	Init(1, 0, m_iEleSize - 1);
	//}
	TSave QueryAll() {
		return m_save[1];
	}
	void swap(CVectorRangeUpdateLineTree<TSave, TRecord>& other) {
		m_save.swap(other.m_save);
		m_record.swap(other.m_record);
		std::swap(m_recordNull, other.m_recordNull);
		assert(m_iEleSize == other.m_iEleSize);
	}
protected:
	//void Init(int iNodeNO, int iSaveLeft, int iSaveRight)
	//{
	//	if (iSaveLeft == iSaveRight) {
	//		this->OnInit(m_save[iNodeNO], iSaveLeft);
	//		return;
	//	}
	//	const int mid = iSaveLeft + (iSaveRight - iSaveLeft) / 2;
	//	Init(iNodeNO * 2, iSaveLeft, mid);
	//	Init(iNodeNO * 2 + 1, mid + 1, iSaveRight);
	//	this->OnUpdateParent(m_save[iNodeNO], m_save[iNodeNO * 2], m_save[iNodeNO * 2 + 1], iSaveLeft, iSaveRight);
	//}
	void Query(int iNodeNO, int iSaveLeft, int iSaveRight, int iQueryLeft, int iQueryRight) {
		if ((iSaveLeft >= iQueryLeft) && (iSaveRight <= iQueryRight)) {
			this->OnQuery(m_save[iNodeNO],iSaveLeft,iSaveRight);
			return;
		}
		if (iSaveLeft == iSaveRight) {//没有子节点
			return;
		}
		Fresh(iNodeNO, iSaveLeft, iSaveRight);
		const int mid = iSaveLeft + (iSaveRight - iSaveLeft) / 2;
		if (mid >= iQueryLeft) {
			Query(iNodeNO * 2, iSaveLeft, mid, iQueryLeft, iQueryRight);
		}
		if (mid + 1 <= iQueryRight) {
			Query(iNodeNO * 2 + 1, mid + 1, iSaveRight, iQueryLeft, iQueryRight);
		}
	}
	void Update(int iNode, int iSaveLeft, int iSaveRight, int iOpeLeft, int iOpeRight, TRecord value)
	{
		if ((iOpeLeft <= iSaveLeft) && (iOpeRight >= iSaveRight))
		{
			this->OnUpdate(m_save[iNode], iSaveLeft, iSaveRight, value);
			this->OnUpdateRecord(m_record[iNode], value);
			return;
		}
		if (iSaveLeft == iSaveRight) {
			return;//没有子节点
		}
		Fresh(iNode, iSaveLeft, iSaveRight);
		const int iMid = iSaveLeft + (iSaveRight - iSaveLeft) / 2;
		if (iMid >= iOpeLeft)
		{
			Update(iNode * 2, iSaveLeft, iMid, iOpeLeft, iOpeRight, value);
		}
		if (iMid + 1 <= iOpeRight)
		{
			Update(iNode * 2 + 1, iMid + 1, iSaveRight, iOpeLeft, iOpeRight, value);
		}
		// 如果有后代，至少两个后代
		this->OnUpdateParent(m_save[iNode], m_save[iNode * 2], m_save[iNode * 2 + 1], iSaveLeft, iSaveRight);
	}
	void Fresh(int iNode, int iDataLeft, int iDataRight)
	{
		if (m_recordNull == m_record[iNode])
		{
			return;
		}
		const int iMid = iDataLeft + (iDataRight - iDataLeft) / 2;
		Update(iNode * 2, iDataLeft, iMid, iDataLeft, iMid, m_record[iNode]);
		Update(iNode * 2 + 1, iMid + 1, iDataRight, iMid + 1, iDataRight, m_record[iNode]);
		m_record[iNode] = m_recordNull;
	}
	vector<TSave> m_save;
	vector<TRecord> m_record;
	TRecord m_recordNull;
	const int m_iEleSize;
};



template<class TSave = int , class TRecord = int >
class CMyLineTree : public CVectorRangeUpdateLineTree<TSave, TRecord>
{
public:
	CMyLineTree(int iSize,int iNotMay) :CVectorRangeUpdateLineTree<TSave, TRecord>(iSize,iNotMay,iNotMay){

	}
	void Query(int leftIndex, int leftRight) {
		m_iQuery = CVectorRangeUpdateLineTree<TSave, TRecord>::m_recordNull;
		CVectorRangeUpdateLineTree<TSave, TRecord>::Query(leftIndex, leftRight);
	}
	int m_iQuery;
protected:
	virtual void OnQuery(const TSave& save, const int& iSaveLeft, const int& iSaveRight)	{
		m_iQuery = min(m_iQuery, save);
	}
	virtual void OnUpdate(TSave& save, const int& iSaveLeft, const int& iSaveRight, const TRecord& update) {
		save = min(save,update);
	}
	virtual void OnUpdateParent(TSave& par, const TSave& left, const TSave& r, const int& iSaveLeft, const int& iSaveRight) {
		par = min(left, r);
	}
	virtual void OnUpdateRecord(TRecord& old, const TRecord& newRecord) {
		old = min(newRecord,old);
	}
};
class Solution {
public:
	int minimumValueSum(vector<int>& nums, const vector<int>& andValues) {

		vector<vector<pair<int, int>>> vNumIndex(nums.size());
		for (int i = 0; i < nums.size(); i++) {
			if (i) {
				for (const auto& [preNum, preIndex] : vNumIndex[i - 1]) {
					const int iNew = preNum & nums[i];
					if (vNumIndex[i].empty() || (vNumIndex[i].back().first != iNew)) {
						vNumIndex[i].emplace_back(make_pair(iNew, preIndex));
					}
					else {
						vNumIndex[i].back().second = preIndex;
					}
				}
			}
			if (vNumIndex[i].empty() || (vNumIndex[i].back().first != nums[i])) {
				vNumIndex[i].emplace_back(make_pair(nums[i], i));
			}
			else {
				vNumIndex[i].back().second = i;
			}
		}

		m_r = andValues.size();
		m_c = nums.size();
		CMyLineTree pre(m_c, m_iNotMay);
		for (int i = 0; i < m_c; i++) {
			if (andValues.front() == vNumIndex[i].front().first) {
				pre.Update(i, i, nums[i]);
			}
		}	
		for (int r = 1; r < m_r; r++)
		{
			CMyLineTree dp(m_c, m_iNotMay);
			for (int cur = 1; cur < m_c; cur++)
			{
				for (int j = vNumIndex[cur].size() - 1; j >= 0; j--) {
					if (andValues[r] == vNumIndex[cur][j].first) {
						const int left = j ? vNumIndex[cur][j - 1].second : -1;
						const int r = vNumIndex[cur][j].second;
						if (0 == r) { continue; }
						pre.Query(max(0,left), r-1);
						dp.Update(cur, cur, pre.m_iQuery + nums[cur]);
						break;
					}
				}
			}
			pre.swap(dp);
		}	
		pre.Query(m_c-1,m_c-1);
		return (pre.m_iQuery >= 1'000'000) ? -1 : pre.m_iQuery;
	}
	int m_r, m_c;
	const int m_iNotMay = 1'000'000'000;
};

测试用例

template<class T>
void Assert(const T& t1, const T& t2)
{

	assert(t1 == t2);
}

template<class T>
void Assert(const vector<T>& v1, const vector<T>& v2)
{
	if (v1.size() != v2.size())
	{
		assert(false);
		return;
	}
	for (int i = 0; i < v1.size(); i++)
	{
		Assert(v1[i], v2[i]);
	}

}

int main()
{
	vector<int>  nums, andValues;
	int k;

	{
		Solution sln;
		nums = { 1, 9, 8, 8 }, andValues = { 1,8 };
		auto res = sln.minimumValueSum(nums, andValues);
		Assert(9, res);
	}
	{
		Solution sln;
		nums = { 1, 3, 2, 4, 7, 5, 3 }, andValues = { 0, 5, 3 };
		auto res = sln.minimumValueSum(nums, andValues);
		Assert(12, res);
	}
	{
		Solution sln;
		nums = { 1, 4, 3, 3, 2 }, andValues = { 0, 3, 3, 2 };
		auto res = sln.minimumValueSum(nums, andValues);
		Assert(12, res);
	}

	//vector<int>  nums = { 3,6,9 };
	//int k;
	//
	//{
	//	Solution sln;
	//	nums = { 3,6,9 }, k = 3;
	//	auto res = sln.findKthSmallest(nums, k);
	//	Assert(9LL, res);
	//}

}

【线段树 区间位运算模板】3117划分数组得到最小的值之和-LMLPHP

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测试环境

操作系统：win7 开发环境： VS2019 C++17
或者 操作系统：win10 开发环境： VS2022 C++17
如无特殊说明，本算法用**C++**实现。

【线段树 区间位运算模板】3117划分数组得到最小的值之和-LMLPHP

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