結果

問題 No.2266 Fractions (hard)
ユーザー chineristACchineristAC
提出日時 2023-04-08 01:00:41
言語 PyPy3
(7.3.15)
結果
AC  
実行時間 5,792 ms / 6,000 ms
コード長 6,812 bytes
コンパイル時間 253 ms
コンパイル使用メモリ 82,380 KB
実行使用メモリ 82,804 KB
最終ジャッジ日時 2024-04-14 12:03:57
合計ジャッジ時間 41,464 ms
ジャッジサーバーID
（参考情報） 		judge5 / judge3
このコードへのチャレンジ
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テストケース

テストケース表示
入力 結果 実行時間
実行使用メモリ
testcase_00 AC 183 ms
79,464 KB
testcase_01 AC 1,908 ms
79,912 KB
testcase_02 AC 240 ms
78,156 KB
testcase_03 AC 64 ms
69,800 KB
testcase_04 AC 73 ms
73,360 KB
testcase_05 AC 178 ms
79,272 KB
testcase_06 AC 462 ms
81,008 KB
testcase_07 AC 340 ms
80,372 KB
testcase_08 AC 303 ms
80,768 KB
testcase_09 AC 99 ms
78,352 KB
testcase_10 AC 147 ms
79,284 KB
testcase_11 AC 66 ms
71,640 KB
testcase_12 AC 242 ms
79,768 KB
testcase_13 AC 2,417 ms
80,928 KB
testcase_14 AC 165 ms
78,212 KB
testcase_15 AC 178 ms
78,080 KB
testcase_16 AC 2,631 ms
81,276 KB
testcase_17 AC 1,621 ms
80,632 KB
testcase_18 AC 3,204 ms
82,804 KB
testcase_19 AC 1,961 ms
80,700 KB
testcase_20 AC 155 ms
78,000 KB
testcase_21 AC 2,435 ms
80,708 KB
testcase_22 AC 1,564 ms
80,372 KB
testcase_23 AC 193 ms
78,148 KB
testcase_24 AC 194 ms
78,132 KB
testcase_25 AC 3,582 ms
80,932 KB
testcase_26 AC 193 ms
78,092 KB
testcase_27 AC 1,789 ms
80,424 KB
testcase_28 AC 192 ms
77,944 KB
testcase_29 AC 193 ms
77,968 KB
testcase_30 AC 193 ms
78,128 KB
testcase_31 AC 194 ms
78,124 KB
testcase_32 AC 1,640 ms
80,424 KB
testcase_33 AC 62 ms
68,684 KB
testcase_34 AC 385 ms
78,756 KB
testcase_35 AC 389 ms
78,656 KB
testcase_36 AC 5,792 ms
82,468 KB
testcase_37 AC 3,679 ms
80,548 KB
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ソースコード

diff # 

import sys,random,bisect
from collections import deque,defaultdict
from heapq import heapify,heappop,heappush
from itertools import permutations
from math import gcd,log

input = lambda :sys.stdin.readline().rstrip()
mi = lambda :map(int,input().split())
li = lambda :list(mi())

from collections import deque
class Dinic:
    def __init__(self, N):
        self.N = N
        self.G = [[] for i in range(N)]

    def add_edge(self, fr, to, cap):
        forward = [to, cap, None]
        forward[2] = backward = [fr, 0, forward]
        self.G[fr].append(forward)
        self.G[to].append(backward)

    def add_multi_edge(self, v1, v2, cap1, cap2):
        edge1 = [v2, cap1, None]
        edge1[2] = edge2 = [v1, cap2, edge1]
        self.G[v1].append(edge1)
        self.G[v2].append(edge2)

    def bfs(self, s, t):
        self.level = level = [None]*self.N
        deq = deque([s])
        level[s] = 0
        G = self.G
        while deq:
            v = deq.popleft()
            lv = level[v] + 1
            for w, cap, _ in G[v]:
                if cap and level[w] is None:
                    level[w] = lv
                    deq.append(w)
        return level[t] is not None

    def dfs(self, v, t, f):
        if v == t:
            return f
        level = self.level
        for e in self.it[v]:
            w, cap, rev = e
            if cap and level[v] < level[w]:
                d = self.dfs(w, t, min(f, cap))
                if d:
                    e[1] -= d
                    rev[1] += d
                    return d
        return 0

    def flow(self, s, t):
        flow = 0
        INF = 10**9 + 7
        G = self.G
        while self.bfs(s, t):
            *self.it, = map(iter, self.G)
            f = INF
            while f:
                f = self.dfs(s, t, INF)
                flow += f
        return flow

def floor_sum(n: int, m: int, a: int, b: int) -> int:
    ans = 0

    if a >= m:
        ans += (n - 1) * n * (a // m) // 2
        a %= m

    if b >= m:
        ans += n * (b // m)
        b %= m

    y_max = (a * n + b) // m
    x_max = y_max * m - b

    if y_max == 0:
        return ans

    ans += (n - (x_max + a - 1) // a) * y_max
    ans += floor_sum(y_max, a, m, (a - x_max % a) % a)

    return ans

M = 10**4 + 1
mebius = [1] * (M+1)
prime = [True] * (M+1)
for p in range(2,M+1):
    if not prime[p]:
        continue
    for n in range(p,M+1,p):
        if n%(p*p) == 0:
            mebius[n] = 0
        mebius[n] *= -1
        prime[n] = False



mebius_cum = [mebius[n] for n in range(M+1)]
mebius[0] = 0
for n in range(1,M):
    mebius_cum[n+1] += mebius_cum[n]

for N in range(1,100):
    tmp = 0
    for n in range(1,N+1):
        tmp += mebius_cum[N//n]
    assert tmp == 1


def solve(N,K):
    lower = (0,1)
    upper = (1,0)

    calc = 0

    B = int(N**.5)

    big_mertens = [0] * (B+10)
    for n in range(1,B+10)[::-1]:
        """
        Mertens(N/n)を計算

        
        sum M(N/ni) for i in 1...N/n = 1
        を利用
        """
        tmp = 0
        T = N//n
        """
        sum M(T/i) for i in 2...N/n 
        を計算
        """
        TB = int(T**.5)
        for i in range(2,TB+1):
            if M < T//i :
                tmp += big_mertens[n*i]
            else:
                tmp += mebius_cum[T//i]
        
        for Q in range(1,T//(TB+1)+1):
            """
            q <= N/(n*i) < q+1
            N//(q+1)*i+1 <= i <= N//n*q
            """

            L = max(N//((Q+1)*n)+1,2)
            R = min(N//(Q*n),N//n)
            if L<=R:
                tmp += (R-L+1) * mebius_cum[Q]
                
        big_mertens[n] = 1 - tmp
    
    
    


    def cond(p,q):
        nonlocal calc
        calc += 1
        #assert p <= N  and q <= N
        """
        ・qx<=py
        ・1 <= x,y <= N
        ・gcd(x,y)=1
        をみたすx,yを数える

        gcd(x,y)がkの倍数とすると

        sum((N//k)-(qi)//p for i in range(1,(N//k)+1))
        -> floor_sum

        +メビウス変換
        """

        res = 0
        B = int(N**.5)
        for k in range(1,B+1):
            t = N//k
            tmp = 0

            ok = max(-1,(p*t+1-q)//q)
            ng = min(t,((t+1)*p+1-q)//q+1)
            while ng-ok>1:
                mid = (ok+ng)//2
                if (q*mid+q-1)//p <= t:
                    ok = mid
                else:
                    ng = mid
            
            
            tmp = t * ng - floor_sum(ng,p,q,q-1)

            
            
            res += tmp * mebius[k]

        
        
        
        for t in range(1,N//(B+1)+1):
            
            k_l = N//(t+1)+1
            k_r = N//t

            tmp = 0
            ok = max(-1,(p*t+1-q)//q)
            ng = min(t,((t+1)*p+1-q)//q+1)
            while ng-ok>1:
                mid = (ok+ng)//2
                if (q*mid+q-1)//p <= t:
                    ok = mid
                else:
                    ng = mid
            
            
            tmp = t * ng - floor_sum(ng,p,q,q-1)
            c = 0
            if k_r <= M:
                c += mebius_cum[k_r]
            else:
                c += big_mertens[t]
            if k_l-1 <= M:
                c -= mebius_cum[k_l-1]
            else:
                c -= big_mertens[t+1]
            res += tmp * c

        
        return res

    
    if cond(N,1) < K:
        print(-1)
        return 

    while True:
        a,b = lower
        c,d = upper
        p,q = (a+c,b+d)


        check = cond(p,q)
        if  check < K:
            """
            右に潜る
            """
            ok = 1
            if d!=0:
                ng = min((N-b)//d,(N-a)//c) + 1
            else:
                ng = (N-a)//c + 1
            while ng-ok>1:
                mid = (ok+ng)//2
                p,q = a+c*mid,b+d*mid
                xxx = cond(p,q)
                if  xxx < K:
                    ok = mid
                elif K < xxx:
                    ng = mid
                else:
                    p,q = a+c*mid,b+d*mid
                    print("{}/{}".format(p,q))
                    return 
            lower = (a+c*ok,b+d*ok)
        elif K < check:
            """
            左に潜る
            """
            ok = 1
            if a!=0:
                ng = min((N-d)//b,(N-c)//a) + 1
            else:
                ng = (N-d)//b + 1
            
            
            
            while ng-ok>1:
                mid = (ok+ng)//2
                p,q = a*mid+c,b*mid+d
                if cond(p,q) > K:
                    ok = mid
                else:
                    ng = mid
            upper = (a*ok+c,b*ok+d)
        else:
            print("{}/{}".format(p,q))
            return 

for _ in range(1):
    N,K = mi()
    solve(N,K)
0