ZOJ---3998：Yet Another Data Structure Problem【线段树+离散对数】

题面:

ZOJ---3998

题意:

给定一个序列，再给出三种操作：（1）区间每个数乘V（2）区间每个数变为原来的K次幂（3）询问区间乘积

分析：

由于只查询区间元素的乘积，直接开两个lazy+降幂是可以写的，但转化离散对数会优化掉快速幂的log

离散对数（摘自百度百科）：

当模m有原根时，设l为模m的一个原根，则当  时：

 为x以整数l为底，模  时的离散对数值，可以通过BSGS算法快速求K

存在和一般对数一样的性质：

 通过上面的性质，就可以将乘法转换为加法，将幂转换为乘法；由于原序列每个值小于1000，那么就可以先将[1，1000]的每个数在模 m意义下的离散对数值K预处理出来；1e9+7的原根为5，通过BSGS算法求解即可；用线段树维护区间和，每次只需查询区间和，再还原回去，5^sum就是答案

PS：1000次BSGS跑得非常慢，打表即可

代码:

#include <bits/stdc++.h>

using namespace std;
typedef long long LL;
const int mod = 1e9+7;
const int MOD = 1e9+6;
const int maxn = 2e5+15;
const int maxm = 1e3+13;
int T,n,q,op,l,r,v;
LL tr[maxn<<2],add[maxn<<2],mul[maxn<<2],res;
LL val[maxm] = {0,0,381838282,884237698,763676564,1,266075974,936544532,145514840,768475390,381838283,649233168,647914256,144547565,318382808,884237699,527353122,895459693,150313666,701153241,763676565,820782224,31071444,82047235,29752532,2,526385847,652713082,700221090,347375814,266075975,198670980,909191404,533470860,277297969,936544533,532151948,289190395,82991517,28785257,145514841,802264261,202620500,24616936,412909726,768475391,463885517,730741137,411590814,873089058,381838284,779697385,908224129,616590335,34551358,649233169,82059366,585390933,729214096,178404868,647914257,234516540,580509262,705019916,291029680,144547566,915309142,351358075,659136251,966284933,318382809,557546595,913990230,559624399,671028677,884237700,464829799,585777694,410623539,414776906,527353123,536950774,184102537,809672376,584458782,895459694,406455218,231613506,794748008,689565125,150313667,81092091,845723799,82908672,112579413,701153242,793429096,993378125,254927334,417708552,763676566,495914368,161535661,991051336,290062405,820782225,998428617,326036185,416389640,525353314,31071445,173428087,463897648,429750477,967229215,82047236,111052372,913022955,560243150,832004219,29752533,298466330,616354822,686501953,962347544,3,86858192,155622902,672867962,908854634,526385848,26958408,297147418,637697767,733196357,652713083,40974527,540374100,348123209,293399235,700221091,614978829,939384877,793780733,295828506,347375815,941462681,757326750,52866953,109607515,266075976,468364065,846668081,663935077,967615976,198670981,792461821,967234449,796615188,500828027,909191405,18591761,918789056,238291131,565940819,533470861,191510652,98758628,966297064,289095130,277297970,469628625,788293500,328851128,613451788,936544534,176586284,62642560,71403401,492996069,532151949,546888614,462930373,118754232,227562075,289190396,464746954,544692855,494417695,589257608,82991518,366164793,175267372,23539458,375216401,28785258,636765616,829772542,799546834,912024915,145514842,235595767,877752650,283920340,543373943,802264262,372889612,850522625,671900687,350386403,202620501,437098732,380266893,441784287,707874467,24616937,798227922,135215506,907191596,443862091,412909727,40007252,555266369,768794522,845735930,768475392,811588759,342799273,349067491,566661897,463885518,470015386,492890654,924677391,294861231,730741138,942081432,299014598,213842495,732754158,411590815,733768720,680304612,421188466,998193104,873089059,68340229,845700806,344185820,693910068,381838285,241119035,468696474,731280403,537461184,779697386,54706238,17661842,290692910,225734921,908224130,115851198,408796690,872301485,678985700,616590336,19536043,573802817,115034633,298959763,34551359,462560014,422812809,965329789,922212382,649233170,729961491,31035300,675237517,967146370,82059367,345578910,996817111,82704431,321223153,585390934,175619009,738808787,677666788,790919380,729214097,877615817,323300957,912101836,139165026,178404869,434705235,301946244,491445797,226594800,647914258,961161468,850202347,380152060,228506357,234516541,45773353,235180417,349454252,875289028,580509263,800796434,174300097,489302579,349072725,705019917,178453464,145590444,882666309,996608982,291029681,210273877,400430043,596612928,300627332,144547567,620129413,409591006,947779101,667285663,915309143,602268191,573348934,57665779,480596910,351358076,348135340,319223134,670933412,313988169,659136252,847904148,851466907,809633584,170131776,966284934,710689410,312731484,995290070,329982019,318382810,797260647,558424566,533507235,444480842,557546596,453241683,716241911,874834351,366249874,913990231,402306476,928726896,182704022,844768655,559624400,500592514,368582295,609400357,570739645,671028678,553134861,846585236,624185975,926531137,884237701,876255977,491923379,971095890,18654643,464829800,39860594,748003075,970055489,557105654,585777695,405377740,793092326,757054683,720569060,410623540,977506928,18603892,911196106,211610818,414776907,181385110,648104085,293863191,521935459,527353124,574882870,617434049,343218545,259590926,536950775,665758622,938423563,925212225,324474452,184102538,424611792,754727894,114949394,232360901,809672377,53738963,177636927,732224685,247548773,584458783,485092561,818937014,499216521,762105175,895459695,823622569,171061066,89712743,678018425,406455219,22572234,180066198,946054688,517053788,231613507,289029872,783200476,825700373,340871020,794748009,641564442,421845534,129669796,937104651,689565126,150632798,993845213,227574206,977541982,150313668,451497423,193427035,352601757,724637555,81092092,730905773,761838722,948500179,548172769,845723800,475803029,851853668,919505556,874728936,82908673,306515667,410804030,676699513,287902601,112579414,851472141,323919708,673850104,680852880,701153243,595680777,385065719,114592434,146327964,793429097,433737960,115606996,902829459,62142888,993378126,803026748,727424537,380031380,122528823,254927335,687842587,450178511,242835501,227539082,417708553,726024102,494091121,75748344,615613462,763676567,982996326,622957317,192803239,850534756,495914369,113118679,173332822,919299466,734145295,161535662,496168925,436544520,353866317,399500124,991051337,672531192,379974299,607573203,213088820,290062406,486914916,497689480,636072233,790634972,820782226,254139761,94130667,60823976,164094470,998428618,946880258,401374325,946811826,955641099,326036186,496872915,377233761,680798045,522322220,416389641,794831137,844398296,431126306,804651091,525353315,347168065,659991722,304050658,2991924,31071446,48529049,111799767,351321432,412873582,173428088,57075793,377246546,348984646,169164501,463897649,428930547,727417192,427073348,378655387,429750478,464542713,473495300,703061435,835310771,967229216,678916854,557457291,250402485,120647063,82047237,59505064,298408322,172757656,907777156,111052373,746216902,259454093,265823497,705139239,913022956,293940112,195024716,521003308,899824221,560243151,714010234,816543517,708296745,683784526,832004220,873284079,796262607,608433082,696897442,29752534,748641546,342999744,119833459,232040623,298466331,761990342,986766785,610344639,168158032,616354823,875288702,427611635,307293942,617018699,686501954,731292534,368755462,257127304,314280758,962347545,734760317,182634710,626109651,556138379,4,871140861,234624095,730911007,184650082,86858193,385696585,560291746,321336424,527428726,155622903,264504585,17636617,378447258,326021979,672867963,81412879,592112159,158329748,782268325,908854635,978451210,614024899,682465614,827638036,526385849,19453198,1967689,443555346,791429288,26958409,329617377,328099783,49123939,342531174,297147419,491028875,984106473,924244950,955187216,637697768,439504061,429423049,862435192,653032214,733196358,883749708,729973622,247787358,701061416,652713084,52771688,936414888,695826451,929922651,40974528,227036965,229742424,656202916,233305183,540374101,191471860,450899589,551970058,761137900,348123210,957940143,92527686,354253078,694569766,293399236,377128346,697723948,711820301,808915083,700221092,421457424,179098923,990343636,940262848,614978830,915345517,432458894,826319124,696576885,939384878,183252290,835079965,280718215,98080187,793780734,256672627,616991850,748088156,544104739,295828507,927595862,784144758,618006412,310565172,347375816,564542304,978569639,226606931,305426158,941462682,920076629,882430796,921332824,750420577,757326751,991238639,591237,952577927,114605847,52866954,729938498,934973143,695101199,228423512,109607516,6024251,578147760,308369413,262580711,266075977,709275566,258094253,125356727,873761661,468364066,352934166,539586166,400492925,615518095,846668082,319462292,421698876,461897840,129841351,663935078,351893765,322952433,938943936,518957057,967615977,901899540,787216022,791579360,174930602,198670982,138892959,109972613,102407336,503417496,792461822,206779757,359345204,88890,400442174,967234450,293034382,270431985,593449100,756539177,796615189,366295003,563223392,379064105,29942361,500828028,675701473,941189873,903773741,626200824,909191406,458040509,956721152,208857561,999272331,18591762,725056827,183197455,641429208,378740433,918789057,171504504,47596898,346797706,320261839,238291132,307050501,725770177,706312734,849567481,565940820,607079689,806450074,891734457,136566170,533470862,496787676,689646277,614199183,613043530,191510653,915272998,435577245,768548745,559475209,98758629,114062961,851384062,629387055,164451860,966297065,694751628,866930843,229816602,200775290,289095131,881054803,235010856,143943451,966942129,277297971,371237630,205460845,516096998,552899348,469628626,471551025,241952849,59856701,239248300,788293501,623046479,404410516,937618645,561904480,328851129,327892964,675157072,898892070,64010068,613451789,495905640,670868154,761853509,165038752,936544535,207538649,403193844,722709302,796339528,176586285,903509641,23402718,991122233,803683816,62642561,511508078,310672493,318942927,92167428,71403402,186183936,532471080,431894372,375683489,492996070,609412488,110832492,359380258,546046794,532151950,512050022,833335705,845399160,575265317,546888615,734440039,668881664,106475831,264389752,462930374,700685266,112744049,458905538,143676998,118754233,330338455,963502940,930011051,492514271,227562076,119418109,857641311,702094160,233691944,289190397,301343832,759526720,256567212,798432364,464746955,574242293,688353949,685034126,792642312,544692856,58537789,889171523,669740883,373540271,494417696,630233002,233310417,884311496,705757990,589257609,55688380,964453890,62691156,704171964,82991519,29828136,977519059,772408781,766904001,366164794,496430716,880846674,528166246,476918626,175267373,397341960,815576242,94511569,497445278,23539459,284667735,815035131,443981170,480850620,375216402,962094621,184865024,229943761,109262813,28785259,761869662,545661558,504367105,338798287,636765617,293828698,69680863,331110272,832016793,829772543,624673783,551523355,609377364,106664171,799546835,728364560,107862378,486505883,875929403,912024916,457586626,65098646,997451744,941903477,145514843,730325259,364834602,73864555,4795593,235595768,574641521,317743570,232373032,990413208,877752651,203460826,494956961};
LL qpow(LL a,LL x){
    LL res = 1LL;
    while(x){
        if(x&1) res = res * a % mod;
        a = a * a % mod;
        x >>= 1;
    }
    return res;
}
/*LL BsGs(LL a,LL n,LL p)                     //打表代码
{
    map<LL,LL> mp;
    int k = ceil(sqrt(p*1.0));
    for(int j = 1; j <= k ; ++j){
       n =  n * a % p;
       mp[n] = j;
    }
    LL u = qpow(a,k),v = 1LL;
    for(int i = 1; i <= k + 1; ++i){
         v = v * u % p;
         if(mp[v]) return i * k - mp[v];
    }
    return -1;
}
void init(){
     for(int i = 1;i < maxm; ++i){
        printf("%lld,", BsGs(5LL,i*1LL,mod));
     }
}*/
inline void pushup(int x){
    tr[x] = (tr[x<<1] + tr[x<<1|1]) % MOD;
}
inline void pushdown(int x,int l,int r){
    if(mul[x] != 1||add[x]){
        int mid = (l+r)>>1;
        tr[x<<1] = (tr[x<<1]*mul[x]+1LL*(mid-l+1)*add[x])%MOD;
        tr[x<<1|1] = (tr[x<<1|1]*mul[x]+1LL*(r-mid)*add[x])%MOD;
        add[x<<1] = (add[x<<1]*mul[x]+add[x])%MOD;
        add[x<<1|1] = (add[x<<1|1]*mul[x]+add[x])%MOD;
        mul[x<<1] = (mul[x<<1]*mul[x])%MOD;
        mul[x<<1|1] = (mul[x<<1|1]*mul[x])%MOD;
        mul[x] = 1LL;
        add[x] = 0; mul[x] = 1LL;
    }
}
void build(int l,int r,int x){
    add[x] = 0; mul[x] = 1LL;
    if(l == r){
        cin >> v;
        tr[x] = val[v];
        return ;
    }
    int mid = (l+r) >> 1;
    build(l,mid,x<<1);
    build(mid+1,r,x<<1|1);
    pushup(x);
}
void query(int l,int r,int L,int R,int x,LL &sum){
    if(l > R || r < L) return ;
    if(l >= L && r <= R){
        sum = (sum+tr[x])%MOD;
        return ;
    }
    pushdown(x,l,r);
    int mid = (l+r)>>1;
    query(l,mid,L,R,x<<1,sum);
    query(mid+1,r,L,R,x<<1|1,sum);
}
void updata(int l,int r,int L,int R,int x,int op,LL v){
    if(l > R || r < L) return ;
    if(l >= L && r <= R){
        if(op == 1){
            tr[x] = (tr[x]+1LL*(r-l+1)*val[v])%MOD;
            add[x] = (add[x]+val[v])%MOD;
        }
        else{
            tr[x] = (tr[x]*v)%MOD;
            add[x] = add[x]*v%MOD;
            mul[x] = (mul[x]*v)%MOD;
        }
        return ;
    }
    pushdown(x,l,r);
    int mid = (l+r)>>1;
    updata(l,mid,L,R,x<<1,op,v);
    updata(mid+1,r,L,R,x<<1|1,op,v);
    pushup(x);
}
int main(){
    cin >> T;
    while(T--){
        cin >> n >> q;
        build(1,n,1);
        while(q--){
            cin >> op >> l >> r;
            if(op == 3){
                LL x = 0LL;
                query(1,n,l,r,1,x);
                cout << qpow(5LL,x) << '\n';
            }
            else{
                cin >> v;
                updata(1,n,l,r,1,op,v); 
            }
        }
    }
    return 0;
}