【二分查找位运算】3145. 大数组元素的乘积

本文涉及知识点

二分查找算法合集
位运算、状态压缩、枚举子集汇总

LeetCode3145. 大数组元素的乘积

一个整数 x 的强数组指的是满足和为 x 的二的幂的最短有序数组。比方说，11 的强数组为 [1, 2, 8] 。
我们将每一个正整数 i （即1，2，3等等）的强数组连接得到数组 big_nums ，big_nums 开始部分为 [1, 2, 1, 2, 4, 1, 4, 2, 4, 1, 2, 4, 8, …] 。
给你一个二维整数数组 queries ，其中 queries[i] = [fromi, toi, modi] ，你需要计算 (big_nums[fromi] * big_nums[fromi + 1] * … * big_nums[toi]) % modi 。
请你返回一个整数数组 answer ，其中 answer[i] 是第 i 个查询的答案。

示例 1：

输入：queries = [[1,3,7]]

输出：[4]

解释：

只有一个查询。

big_nums[1…3] = [2,1,2] 。它们的乘积为 4 ，4 对 7 取余数得到 4 。

示例 2：

输入：queries = [[2,5,3],[7,7,4]]

输出：[2,2]

解释：

有两个查询。

第一个查询：big_nums[2…5] = [1,2,4,1] 。它们的乘积为 8 ，8 对 3 取余数得到 2 。

第二个查询：big_nums[7] = 2 ，2 对 4 取余数得到 2 。

提示：

1 <= queries.length <= 500
queries[i].length == 3
0 <= queries[i][0] <= queries[i][1] <= 10¹⁵
1 <= queries[i][2] <= 10⁵

二分查找、位运算

令 m = max(queries[i][1])<= 10¹⁵
令f(x)是强数组，即f(1) = {1} f(2) = {2} f(3} = {1,2} 。特殊的情况f(0)为空。
fl(x)等于f(x)的长度，即数组元素的个数。
big_nums 显然就是f(1) 、f(2)、f(3) $\cdots$ f( $\infty$ )的拼接。
big_nums[0…toi] 令f(1)到f(i2)的拼接，其中f(i2)可能部分元素在 big_nums[0…toi] 中，也可能全部在big_nums[0…toi] 中。
令big_nums[formi] 是f(i1)的元素。则：big_nums[formi…toi]则可以分成三部分：
一，f(i1)的后缀二，f(i1+1) $\cdots$ f(i2-1) 三，f(i2)的前缀。
如果i1等于i2，直接枚举i1，就可以了。
g(x) = $\sum_{i:0}^{x}fl(i)$
has = g(i1-1)
从低位到高位枚举i1的二进制1
has ++ 如果 has > formi vCnt[bit]++
v= g2(i2-1) - g2(i1)
将v加到has中。
从低位到高位枚举i2的二进制1
has ++ 如果 has <= toi+1 vCnt[bit]++

$\Pi_{i:0}^{60}(1 << i)^{vCnt[i]}$
注意：要求mod。

x的n次方

如果 n是奇数 xⁿ=x $\times$ (x $t im es$ x) ^n/2
如果n是偶数 xⁿ= (x $t im es$ x) ^n/2
无论那种情况 n都是/2，故复杂度是O(logn)

g2和g函数

0到x的第bit位有多少个1？以 1 << bit 为半周期halfUnit，前半个周期是0,后半个周期是1 。最后一个周期可能不完整 max(0,(x+1)%(2*halfUnit) -haflUnit)
g函数就是g2 返回数组之和。

二分查找

auto ll1 = NBinarySearch::FindFrist(0LL, llMax, [&](long long mid) {return TotalBitCount(mid) > fromi; });auto ll2 = NBinarySearch::FindFrist(0LL, llMax, [&](long long mid) {return TotalBitCount(mid) >= toi + 1; });

二分查找+ 位运算代码

核心代码

namespace NBinarySearch
{template<class INDEX_TYPE, class _Pr>INDEX_TYPE FindFrist(INDEX_TYPE left, INDEX_TYPE rightInclue, _Pr pr){while (rightInclue - left > 1){const auto mid = left + (rightInclue - left) / 2;if (pr(mid)){rightInclue = mid;}else{left = mid;}}return rightInclue;}template<class INDEX_TYPE, class _Pr>INDEX_TYPE FindEnd(INDEX_TYPE leftInclude, INDEX_TYPE right, _Pr pr){while (right - leftInclude > 1){const auto mid = leftInclude + (right - leftInclude) / 2;if (pr(mid)){leftInclude = mid;}else{right = mid;}}return leftInclude;}
}class C1ToXBitCount {
public:C1ToXBitCount(int iBitCount = 60):m_iBitCount(iBitCount){}long long TotalBitCount(long long x) {auto v = BitCount(x);return std::accumulate(v.begin(), v.end(), 0LL);}vector<long long> BitCount(long long x) {vector<long long> ret(m_iBitCount);auto cnt = x + 1;for (int bit = 0; bit < m_iBitCount; bit++) {long long halfUnit = (1LL << bit);const auto curBitCount = cnt / 2 / halfUnit * halfUnit + max(0LL, cnt % (2 * halfUnit) - halfUnit);ret[bit] = curBitCount;}return ret;}int  m_iBitCount = 0;
};
class Solution : public C1ToXBitCount {
public:vector<int> findProductsOfElements(vector<vector<long long>>& queries) {vector<int> vRet;for (const auto& v : queries) {vRet.emplace_back(Do(v[0], v[1], v[2]));}return vRet;}int Do(long long fromi, long long toi, int modi) {const long long llMax = 1e15;auto ll1 = NBinarySearch::FindFrist(0LL, llMax, [&](long long mid) {return TotalBitCount(mid) > fromi; });auto ll2 = NBinarySearch::FindFrist(0LL, llMax, [&](long long mid) {return TotalBitCount(mid) >= toi + 1; });long long has = TotalBitCount(ll1 - 1);vector<long long> vCnt(m_iBitCount);for (int i = 0; i < m_iBitCount; i++) {if ((1LL << i) & ll1) {has++;if ((has > fromi)&&(has <= toi + 1)) { vCnt[i]++; };}			}if (ll1+1 < ll2) {auto v1 = BitCount(ll1);auto v2 = BitCount(ll2 - 1);for (int i = 0; i < m_iBitCount; i++) {vCnt[i] += v2[i] - v1[i];has += v2[i] - v1[i];}}for (int i = 0; (i < m_iBitCount)&&(has < toi +1 ); i++) {if ((1LL << i) & ll2) {vCnt[i]++; has++;}}long long ret = 1;for (int i = 0; i < m_iBitCount; i++) {ret = ret * pow(1LL << i, vCnt[i], modi) % modi;}return ret;}int  pow(long long cur,long long n,int mod){cur %= mod;int ret = 1;while (n){if (n & 1){ret = (ret*cur)%mod;}cur = (cur * cur) % mod;n >>= 1;}return ret;}};

测试用例

template<class T>
void Assert( vector<T> v1,  vector<T> v2)
{if (v1.size() != v2.size()){assert(false);return;}for (int i = 0; i < v1.size(); i++){assert(v1[i] == v2[i]);}
}template<class T>
void Assert(const T& t1, const T& t2)
{assert(t1 == t2);
}int main()
{vector<vector<long long>> queries = { {1,3,7} };{Solution slu;queries = {{ 7,7,4}, { 2,5,3 } };auto res = slu.findProductsOfElements(queries);Assert({ 2,2 }, res);}{Solution slu;queries = { {1,3,7} };auto res = slu.findProductsOfElements(queries);Assert({ 4 }, res);}{Solution slu;queries = { 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};auto res = slu.findProductsOfElements(queries);Assert({ 69466,27076,2758,38294,19964,10490,8,4126,7564,13576,60878,84827,2928,793,81926,23438,9878,46888,77918,28378,3952,52396,17170,3847,15620,39396,29300,43833,6700,42449,47432,15005,52112,42486,4375,23326,63234,30104,30496,712,76070,65084,11444,2048,12672,128,55936,576,1808,2,31780,19469,20824,1695,11756,50956,55324,35126,25014,3218,43657,30968,42760,26992,9547,14741,29548,9848,4856,856,1728,8747,19342,55728,22576,5674,21225,29928,10518,17208,1087,320,18188,32988,20320,13387,73110,14170,28780,19684,67778,224,15278,14627,2608,10488,8900,19936,12796,45094,718,10692,4502,3698,578,54222,13502,21363,16459,63236,89932,50359,39668,62739,14372,8888,7016,38436,39968,13058,22964,67992,1164,9278,23948,21148,39472,56388,39376,12136,20644,9529,14828,12940,35128,13329,4112,22904,45815,68,13170,128,47858,14693,20094,4248,42109,37698,10375,26406,3040,927,1024,41644,26779,29556,72038,20477,2779,10760,5592,18349,350,55232,1024,44544,63239,42246,8564,7822,26464,24240,10199,25467,23185,27514,12224,56732,26500,39488,20992,49044,20372,7814,9862,6552,15432,24073,27649,31084,18256,17347,11674,29612,53544,328,5032,65478,512,15464,4796,4096,64,39646,20230,4552,69288,497,41932,3860,816,30496,26049,22334,71252,56904,1488,32,74174,2207,2610,11190,30807,15386,45460,19730,24590,18064,13529,47491,35962,19208,23296,15017,18748,16746,26269,7288,18302,24662,5482,7890,7070,19971,9563,11050,35988,68366,67343,30962,40096,14825,6362,212,256,26896,61147,32397,1114,18170,18776,5235,8057,36020,72852,37358,1097,13549,34007,1241,33312,59119,38617,757,31963,57384,38548,7456,32680,36964,380,55930,25112,60856,27094,21447,9800,35158,17672,11360,21908,6518,16914,5776,7906,43438,50164,9076,266,64514,4960,39374,20696,24010,21048,37608,33464,2740,15524,1664,3035,33856,33435,22006,16200,55418,44800,30052,42817,6290,44080,8192,17827,20833,72832,3396,24544,14120,84014,47951,18200,8108,39442,15872,41926,1843,19534,2308,63744,64969,2508,8762,11669,21872,51474,5900,16366,21271,77379,28568,32344,966,40484,79009,30700,47616,268,5988,15697,1220,880,46429,4096,20685,2048,40312,3846,2048,12928,39096,49593,67722,47062,15920,24088,21664,60020,21232,20429,56767,33684,4872,32255,56458,512,1936,5777,31154,16088,48808,22622,512,45152,2598,47792,1126,15560,6957,27778,13080,17049,57122,22118,18028,2707,16286,11528,11316,1356,9340,1460,92396,1,6722,68682,60859,8192,2580,988,2468,21952,31298,49783,21292,20353,6928,16358,2084,22483,40590,22032,23905,23033,56340,20054,39008,7800,407,512,1024,8914,29944,712,936,18776,9160,11842,22462,10012,78080,3709,31063,27932,40672,19366,6832,12748,11485,13136,14704,12900,8942,15868,42982,13085,20342,194,128,3532,8869,3712,44056,24220,28381,124,80202,8,27226,6901,62464,48803,13651,42473,6475,1152,31532,396,24970,32800,45109,1001,1024,8702,6278,18384,46787,32,17987,34108,313}, res);}
}

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有效学习：明确的目标及时的反馈拉伸区（难度合适），可以先学简单的课程，请移步CSDN学院，听白银讲师（也就是鄙人）的讲解。
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我想对大家说的话
《喜缺全书算法册》以原理、正确性证明、总结为主。
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