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QOJ
ID | Problem | Submitter | Result | Time | Memory | Language | File size | Submit time | Judge time |
---|---|---|---|---|---|---|---|---|---|
#177715 | #6417. Classical Summation Problem | pikachu_coder | AC ✓ | 397ms | 11060kb | C++14 | 2.5kb | 2023-09-13 11:03:11 | 2023-09-13 11:03:13 |
Judging History
answer
/*
3 2
Ev = sum(value of outcome i/possibility of outcome i) for all i
Ev of city number i = sum((value of city i*p(i))/2^k) where p(i) is the probability of choosing city i as the median, all p(i) sum to one!
Because the probability distributions summing to 1 are symmetrical, the result is the center.
*/
#include <iostream>
#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define vi vector<int>
#define vl vector<ll>
#define vii vector<vector<int>>
#define vll vector<vector<ll>>
#define pii pair<int, int>
#define pil pair<int, ll>
#include <ext/pb_ds/assoc_container.hpp>
using namespace __gnu_pbds;
template<typename T>
using ordered_set = tree<T, null_type, less<T>,
rb_tree_tag, tree_order_statistics_node_update>;
//ordered_set<int> st; ordered set
//gp_hash_table<int, int> table; faster hash table
const ll mod = 998244353LL;
const int maxn = 1000001;
vl facts(maxn, 1LL);
ll pw(ll a, ll b){ //a^b
if(b == 0) return 1LL;
ll half = pw(a, b/2);
if(b%2 == 0) return (half*half)%mod;
return (((half*half)%mod)*a)%mod;
}
ll dvd(ll a, ll b){
return (a*pw(b, mod-2LL))%mod;
}
ll choose(int a, int b){ //a choose b, a!/(b!*(a-b)!), const factor of ~30 so use wisely
if(b > a) return 0LL;
ll num = facts[a];
ll den = (facts[b] * facts[a-b])%mod;
return dvd(num, den);
}
ll n; ll k; //n cities, k friends
//all friends meet at the median
int main() {
ios::sync_with_stdio(false);
cin.tie(0);
for(int a = 1; a < maxn; a++) facts[a] = (facts[a-1]*(ll)a)%mod;
cin >> n >> k;
if(k%2 == 0){ //given that there is an even number of friends, the floor between the two edge friends becomes the median -->
//subtract expected shift left distance - ((n+1)*n^k - e(d))/2
//how much does road i...i+1 contribute to the answer?
ll sm = 0LL; ll k2 = k/2LL;
for(int i = 1; i < n; i++){
//k/2 to the left, k/2 to the right, 3rd city left 4th city right
ll e = (choose(k, k2) * pw(i, k2))%mod;
e = (e * pw(n-i, k2))%mod;
sm = (sm + e)%mod;
}
ll rt = ((n+1LL)*pw(n, k))%mod;
cout << dvd((rt-sm+mod)%mod, 2LL);
}
else{ //given median at whole number position i, floor(k/2) friends meet at either end OR live at position i along with the middle friend --> all arrangements are unique to the i (a single arrangement cannot represent two different medians)
cout << (dvd(n+1LL, 2LL) * pw(n, k))%mod << "\n";
}
}
Details
Tip: Click on the bar to expand more detailed information
Test #1:
score: 100
Accepted
time: 6ms
memory: 10872kb
input:
3 2
output:
14
result:
ok 1 number(s): "14"
Test #2:
score: 0
Accepted
time: 6ms
memory: 10924kb
input:
5 3
output:
375
result:
ok 1 number(s): "375"
Test #3:
score: 0
Accepted
time: 6ms
memory: 10988kb
input:
2 2
output:
5
result:
ok 1 number(s): "5"
Test #4:
score: 0
Accepted
time: 6ms
memory: 10592kb
input:
10 9
output:
508778235
result:
ok 1 number(s): "508778235"
Test #5:
score: 0
Accepted
time: 6ms
memory: 10964kb
input:
69 3
output:
11497815
result:
ok 1 number(s): "11497815"
Test #6:
score: 0
Accepted
time: 3ms
memory: 10876kb
input:
994 515
output:
33689623
result:
ok 1 number(s): "33689623"
Test #7:
score: 0
Accepted
time: 8ms
memory: 10992kb
input:
4476 6182
output:
114894183
result:
ok 1 number(s): "114894183"
Test #8:
score: 0
Accepted
time: 6ms
memory: 10976kb
input:
58957 12755
output:
932388891
result:
ok 1 number(s): "932388891"
Test #9:
score: 0
Accepted
time: 77ms
memory: 10896kb
input:
218138 28238
output:
392861201
result:
ok 1 number(s): "392861201"
Test #10:
score: 0
Accepted
time: 236ms
memory: 10992kb
input:
644125 316810
output:
420621854
result:
ok 1 number(s): "420621854"
Test #11:
score: 0
Accepted
time: 251ms
memory: 10872kb
input:
612914 505428
output:
973686286
result:
ok 1 number(s): "973686286"
Test #12:
score: 0
Accepted
time: 6ms
memory: 10956kb
input:
998216 938963
output:
251335926
result:
ok 1 number(s): "251335926"
Test #13:
score: 0
Accepted
time: 7ms
memory: 10648kb
input:
990516 996933
output:
549551960
result:
ok 1 number(s): "549551960"
Test #14:
score: 0
Accepted
time: 397ms
memory: 10976kb
input:
999019 999012
output:
637189128
result:
ok 1 number(s): "637189128"
Test #15:
score: 0
Accepted
time: 370ms
memory: 11060kb
input:
999928 999950
output:
185229465
result:
ok 1 number(s): "185229465"
Test #16:
score: 0
Accepted
time: 3ms
memory: 10924kb
input:
999999 999999
output:
384164916
result:
ok 1 number(s): "384164916"
Test #17:
score: 0
Accepted
time: 369ms
memory: 10604kb
input:
999999 1000000
output:
696165930
result:
ok 1 number(s): "696165930"
Test #18:
score: 0
Accepted
time: 6ms
memory: 10888kb
input:
1000000 999999
output:
219071706
result:
ok 1 number(s): "219071706"
Test #19:
score: 0
Accepted
time: 360ms
memory: 10920kb
input:
1000000 1000000
output:
128206597
result:
ok 1 number(s): "128206597"
Test #20:
score: 0
Accepted
time: 3ms
memory: 10960kb
input:
2 10
output:
1410
result:
ok 1 number(s): "1410"
Test #21:
score: 0
Accepted
time: 3ms
memory: 10900kb
input:
84 16
output:
297627153
result:
ok 1 number(s): "297627153"
Test #22:
score: 0
Accepted
time: 6ms
memory: 10892kb
input:
643 800
output:
489237163
result:
ok 1 number(s): "489237163"
Test #23:
score: 0
Accepted
time: 6ms
memory: 10804kb
input:
9903 880
output:
595167333
result:
ok 1 number(s): "595167333"
Test #24:
score: 0
Accepted
time: 36ms
memory: 10968kb
input:
97446 89750
output:
410205549
result:
ok 1 number(s): "410205549"
Test #25:
score: 0
Accepted
time: 74ms
memory: 10964kb
input:
186460 646474
output:
32638530
result:
ok 1 number(s): "32638530"
Test #26:
score: 0
Accepted
time: 198ms
memory: 10976kb
input:
508940 244684
output:
598321755
result:
ok 1 number(s): "598321755"
Test #27:
score: 0
Accepted
time: 217ms
memory: 10908kb
input:
583646 557758
output:
858695621
result:
ok 1 number(s): "858695621"
Test #28:
score: 0
Accepted
time: 367ms
memory: 10928kb
input:
969610 992608
output:
256683498
result:
ok 1 number(s): "256683498"
Test #29:
score: 0
Accepted
time: 375ms
memory: 10648kb
input:
995106 996434
output:
411791999
result:
ok 1 number(s): "411791999"
Test #30:
score: 0
Accepted
time: 370ms
memory: 10860kb
input:
999961 999872
output:
61222370
result:
ok 1 number(s): "61222370"
Test #31:
score: 0
Accepted
time: 386ms
memory: 10920kb
input:
999977 999908
output:
831096762
result:
ok 1 number(s): "831096762"
Test #32:
score: 0
Accepted
time: 388ms
memory: 11056kb
input:
999992 999998
output:
562977678
result:
ok 1 number(s): "562977678"
Test #33:
score: 0
Accepted
time: 220ms
memory: 10904kb
input:
1000000 2
output:
118436113
result:
ok 1 number(s): "118436113"
Test #34:
score: 0
Accepted
time: 7ms
memory: 10924kb
input:
2 1000000
output:
298823641
result:
ok 1 number(s): "298823641"