【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

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 <= 10⁴
1 <= m == andValues.length <= min(n, 10)
1 <= nums[i] < 10⁵
0 <= andValues[j] < 10⁵

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] = $\Large \min _{y:max(left,0)}^{r-1}pre[y]$ +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);//}}

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我想对大家说的话
闻缺陷则喜是一个美好的愿望，早发现问题，早修改问题，给老板节约钱。
子墨子言之：事无终始，无务多业。也就是我们常说的专业的人做专业的事。
如果程序是一条龙，那算法就是他的是睛