[zoj3593]扩展欧几里得+三分

题意：给一个数A，有6种操作，+a,-a,+b,-b,+(a+b),-(a+b)，每次选择一种，用最少的次数变成B。

思路：由于不同的操作先后顺序对最后的结果没有影响，并且加一个数与减一个相同的数不能同时有，如果有，把它们都去掉结果更优。所以不妨设+a操作进行了x次（x为负表示-a操作进行了|x|次，y同理），+b操作进行了y次，那么问题转化为ax + by = |A-B|的解，使得结果最小，其中如果x,y异号，结果为|x|+|y|，同号则为max(|x|,|y|，因为+a,+b或-a,-b可以合并)。令c=|A-B|，如果c%gcd(a,b)!=0，这个显然是没有解的，直接输出-1。否则令a=a/g,b=b/g,c=c/g，其中gcd(a,b)=1，令ax+by=1的一个解为(x0,y0)，则方程的通解可以表示为(x0+kb,y0-ka)，下面进行关键的一步：由于ax+bx=c可以转化为c个ax+bx=1相加，所以ax+bx=c的解可以表示为(cx0+Kb,cy0-Ka)（K取任意整数)，这里a,b,c,x0,y0都是已知的，K是个变量，K确定后那么答案也就确定了，由于答案是K的单峰函数所以可以三分一下，求得最小答案。由于K的范围不确定，需要用大整数=_=（醉了）！具体看代码。

 #pragma comment(linker, "/STACK:10240000,10240000")

 #include <iostream>
 #include <cstdio>
 #include <algorithm>
 #include <cstdlib>
 #include <cstring>
 #include <map>
 #include <queue>
 #include <deque>
 #include <cmath>
 #include <vector>
 #include <ctime>
 #include <cctype>
 #include <set>
 #include <bitset>
 #include <functional>
 #include <numeric>
 #include <stdexcept>
 #include <utility>

 using namespace std;

 #define mem0(a) memset(a, 0, sizeof(a))
 #define mem_1(a) memset(a, -1, sizeof(a))
 #define lson l, m, rt << 1
 #define rson m + 1, r, rt << 1 | 1
 #define define_m int m = (l + r) >> 1
 #define rep_up0(a, b) for (int a = 0; a < (b); a++)
 #define rep_up1(a, b) for (int a = 1; a <= (b); a++)
 #define rep_down0(a, b) for (int a = b - 1; a >= 0; a--)
 #define rep_down1(a, b) for (int a = b; a > 0; a--)
 #define all(a) (a).begin(), (a).end()
 #define lowbit(x) ((x) & (-(x)))
 #define constructInt4(name, a, b, c, d) name(int a = 0, int b = 0, int c = 0, int d = 0): a(a), b(b), c(c), d(d) {}
 #define constructInt3(name, a, b, c) name(int a = 0, int b = 0, int c = 0): a(a), b(b), c(c) {}
 #define constructInt2(name, a, b) name(int a = 0, int b = 0): a(a), b(b) {}
 #define pchr(a) putchar(a)
 #define pstr(a) printf("%s", a)
 #define sstr(a) scanf("%s", a)
 #define sint(a) scanf("%d", &a)
 #define sint2(a, b) scanf("%d%d", &a, &b)
 #define sint3(a, b, c) scanf("%d%d%d", &a, &b, &c)
 #define pint(a) printf("%d\n", a)
 #define test_print1(a) cout << "var1 = " << a << endl
 #define test_print2(a, b) cout << "var1 = " << a << ", var2 = " << b << endl
 #define test_print3(a, b, c) cout << "var1 = " << a << ", var2 = " << b << ", var3 = " << c << endl

 typedef long long LL;
 typedef pair<int, int> pii;
 typedef vector<int> vi;

 const int dx[] = {, , -, , , , -, -};
 const int dy[] = {-, , , , , -, , - };
 const int maxn = ;
 const int md = ;
 const int inf = 1e9 + ;
 const LL inf_L = 1e18 + ;
 const double pi = acos(-1.0);
 const double eps = 1e-;

 template<class T>T gcd(T a, T b){return b==?a:gcd(b,a%b);}
 template<class T>bool max_update(T &a,const T &b){if(b>a){a = b; return true;}return false;}
 template<class T>bool min_update(T &a,const T &b){if(b<a){a = b; return true;}return false;}
 template<class T>T condition(bool f, T a, T b){return f?a:b;}
 template<class T>void copy_arr(T a[], T b[], int n){rep_up0(i,n)a[i]=b[i];}
 int make_id(int x, int y, int n) { return x * n + y; }

 const int maxI = 1e8;
 const int Len = ;

 struct BigInt {
     vi num;
     bool symbol;
     BigInt() { num.clear(); symbol = ; }
     BigInt(int x) { symbol = ; if (x < ) { symbol = ; x = -x; } num.push_back(x % maxI); if (x >= maxI) num.push_back(x / maxI); }
     BigInt(bool s, vi x) { symbol = s;  num = x; }
     BigInt(char s[]) {
         int len = strlen(s), x = , sum = , p = s[] == '-';
         symbol = p;
         for (int i = len - ; i >= p; i--) {
             sum += (s[i] - '') * x;
             x *= ;
             if (x == 1e8 || i == p) {
                 num.push_back(sum);
                 sum = ;
                 x = ;
             }
         }
         while (num.back() ==  && num.size() > ) num.pop_back();
     }

     void push(int x) { num.push_back(x); }

     BigInt abs() const { return BigInt(false, num); }

     bool smaller(const vi &a, const vi &b) const {
         if (a.size() != b.size()) return a.size() < b.size();
         for (int i = a.size() - ; i >= ; i--) {
             if (a[i] != b[i]) return a[i] < b[i];
         }
         return ;
     }

     bool operator < (const BigInt &p) const {
         if (symbol && !p.symbol) return true;
         if (!symbol && p.symbol) return false;
         if (symbol && p.symbol) return smaller(p.num, num);
         return smaller(num, p.num);
     }

     bool operator > (const BigInt &p) const {
         return p < *this;
     }

     bool operator == (const BigInt &p) const {
         return !(p < *this) && !(*this < p);
     }

     bool operator >= (const BigInt &p) const {
         return !(*this < p);
     }

     bool operator <= (const BigInt &p) const {
         return !(p < *this);
     }

     vi add(const vi &a, const vi &b) const {
         vi c;
         c.clear();
         int x = ;
         for (int i = ; i < a.size(); i++) {
             x += a[i];
             if (i < b.size()) x += b[i];
             c.push_back(x % maxI);
             x /= maxI;
         }
         for (int i = a.size(); i < b.size(); i++) {
             x += b[i];
             c.push_back(x % maxI);
             x /= maxI;
         }
         if (x) c.push_back(x);
         while (c.back() ==  && c.size() > ) c.pop_back();
         return c;
     }

     vi sub(const vi &a, const vi &b) const {
         vi c;
         c.clear();
         int x = ;
         for (int i = ; i < b.size(); i++) {
             x += maxI + a[i] - b[i] - ;
             c.push_back(x % maxI);
             x /= maxI;
         }
         for (int i = b.size(); i < a.size(); i++) {
             x += maxI + a[i] - ;
             c.push_back(x % maxI);
             x /= maxI;
         }
         while (c.back() ==  && c.size() > ) c.pop_back();
         return c;
     }

     vi mul(const vi &a, const vi &b) const {
         vi c;
         c.resize(a.size() + b.size());
         for (int i = ; i < a.size(); i++) {
             for (int j = ; j < b.size(); j++) {
                 LL tmp = (LL)a[i] * b[j] + c[i + j];
                 c[i + j + ] += tmp / maxI;
                 c[i + j] = tmp % maxI;
             }
         }
         while (c.back() ==  && c.size() > ) c.pop_back();
         return c;
     }

     vi div(const vi &a, const vi &b) const {
         vi c(a.size()), x(, ), y(, ), z(, ), t(, );
         y.push_back();
         for (int i = a.size() - ; i >= ; i--) {
             z[] = a[i];
             x = add(mul(x, y), z);
             if (smaller(x, b)) continue;
             int l = , r = maxI - ;
             while (l < r) {
                 int m = (l + r + ) >> ;
                 t[] = m;
                 if (smaller(x, mul(b, t))) r = m - ;
                 else l = m;
             }
             c[i] = l;
             t[] = l;
             x = sub(x, mul(b, t));
         }
         while (c.back() ==  && c.size() > ) c.pop_back();
         return c;
     }

     BigInt operator + (const BigInt &p) const{
         if (!symbol && !p.symbol) return BigInt(false, add(num, p.num));
         if (!symbol && p.symbol) return *this >= p.abs()? BigInt(false, sub(num, p.num)) : BigInt(true, sub(p.num, num));
         if (symbol && !p.symbol) return (*this).abs() > p? BigInt(true, sub(num, p.num)) : BigInt(false, sub(p.num, num));
         return BigInt(true, add(num, p.num));
     }

     BigInt operator - (const BigInt &p) const {
         return *this + BigInt(!p.symbol, p.num);
     }

     BigInt operator * (const BigInt &p) const {
         BigInt res(symbol ^ p.symbol, mul(num, p.num));
         if (res.symbol && res.num.size() ==  && res.num[] == ) res.symbol = false;
         return res;
     }

     BigInt operator / (const BigInt &p) const {
         if (p == BigInt()) return p;
         BigInt res(symbol ^ p.symbol, div(num, p.num));
         if (res.symbol && res.num.size() ==  && res.num[] == ) res.symbol = false;
         return res;
     }

     BigInt operator % (const BigInt &p) const {
         return *this - *this / p * p;
     }

     void show() const {
         if (symbol) putchar('-');
         printf("%d", num[num.size() - ]);
         for (int i = num.size() - ; i >= ; i--) {
             printf("%08d", num[i]);
         }
         putchar('\n');
     }

     int TotalDigit() const {
         int x = num[num.size() - ] / , t = ;
         while (x) {
             x /= ;
             t++;
         }
         return t + (num.size() - ) * Len;
     }

 };
 typedef BigInt bi;
 bi A, B, a, b;
 bi x, y, d, t;
 char s[];
 void gcd(bi a, bi b, bi &d, bi &x, bi &y) {
     if (b == ) {
         d = a;
         x = ;
         y = ;
     }
     else {
         gcd(b, a % b, d, y, x);
         y = y - x * (a / b);
     }
 }

 bi f(bi k) {
     bi p = t * x + k * b;
     bi q = t * y - k * a;
     if (p >  && q >  || p <  && q < ) return max(p.abs(), q.abs());
     return p.abs() + q.abs();
 }

 int main() {
     //freopen("in.txt", "r", stdin);
     int T;
     cin >> T;
     while (T --) {
         cin >> s;
         A = bi(s);
         cin >> s;
         B = bi(s);
         cin >> s;
         a = bi(s);
         cin >> s;
         b = bi(s);
         t = (A - B).abs();
         bi g = gcd(a, b);
         if (!(t % g == )) {
             puts("-1");
             continue;
         }
         a = a / g;
         b = b / g;
         t = t / g;

         gcd(a, b, d, x, y);

         bi l = "-100000000000000000", r = "";
         while (r - l > ) {
             bi m1 = l + (r - l) / , m2 = r - (r - l) / ;
             if (f(m1) > f(m2)) l = m1 + ;
             else r = m2;
         }
         bi ans = f(l);
         min_update(ans, f(l + ));
         min_update(ans, f(l + ));
         ans.show();
     }
 }