PROBLEM LINK:
Practice
Contest: Division 1
Contest: Division 2
Contest: Division 3
Author: Lavish Gupta
Tester: Danny Mittal
Editorialist: Nishank Suresh
DIFFICULTY:
Medium
PREREQUISITES:
Linearity of expectation and/or observation
PROBLEM:
N players with distinct ratings compete in a tournament, the i-th of whom has rating A_i. Each round, two people are uniformly randomly chosen from those remaining, and the one among them with higher rating wins; with the loser being eliminated.
Find the expected rating of the player who finishes second.
EXPLANATION:
Let’s order the players by decreasing rating, A_1 > A_{2} > \dots > A_{N-1} > A_N.
What’s the probability that the player with rating A_N finishes second?
Answer
This is only possible if A_N is not chosen in any match except the last, because they are guaranteed to lose any match they play.
- For the first match, there are \binom{N}{2} possible matches, of which \binom{N-1}{2} don’t include A_N.
- For the second match, \binom{N-1}{2} matches of which \binom{N-2}{2} don’t include A_N.
\vdots
Multiplying all these together, we see that the probability comes out to be
which cancels out to just give us
Now, what about the case when A_N doesn’t finish second?
The expected value then is simply the expected value of the process run on the set \{A_1, A_2, \dots, A_{N-2}, A_{N-1}\}.
This is because if A_N is guaranteed to lose any match they are chosen to play in - and thus A_N plays exactly one match. Thus, without loss of generality, we can assume that A_N is chosen to be in the very first match and immediately loses, leaving us with only the first N-1 elements contributing to the expected value.
This observation allows us to formulate a dynamic programming solution.
Let E_i denote the expected rating of the second player when considering only A_1, A_2, \dots, A_i. Our final answer will be E_N.
As a base case, we define E_2 = A_2 (because there is only one choice of second player).
Now let’s look at how to compute E_i for i\geq 3.
Our earlier arguments tell us that the probability that A_i is the second player is exactly \dfrac{1}{\binom{i}{2}} (given that we’re only considering the first i players), and if A_i is not the second player the expectation is E_{i-1}.
Linearity of expectation then gives us the recurrence
which is easily computed by simply iterating i from 3 to N.
An alternate “solution”
It is possible to ‘solve’ this problem by other methods as well, for example by writing a brute-force solution for small values of N and guessing the multipliers.
It turns out that the solution is (given that A_i have been sorted in increasing order)
TIME COMPLEXITY:
\mathcal{O}(N\log N) per test.
CODE:
Setter (C++)
#define ll long long
#define dd long double
#define pub push_back
#define pob pop_back
#define puf push_front
#define pof pop_front
#define mp make_pair
#define mt make_tuple
#define fo(i , n) for(ll i = 0 ; i < n ; i++)
#define tll tuple<ll ,ll , ll>
#define pll pair<ll ,ll>
#include<bits/stdc++.h>
/*#include<iomanip>
#include<cmath>
#include<cstdio>
#include<utility>
#include<iostream>
#include<vector>
#include<string>
#include<algorithm>
#include<map>
#include<queue>
#include<set>
#include<stack>
#include<bitset>*/
dd pi = acos(-1) ;
ll z = 1000000007 ;
ll inf = 100000000000000000 ;
ll p1 = 37 ;
ll p2 = 53 ;
ll mod1 = 202976689 ;
ll mod2 = 203034253 ;
ll fact[100] ;
ll gdp(ll a , ll b){return (a - (a%b)) ;}
ll ld(ll a , ll b){if(a < 0) return -1*gdp(abs(a) , b) ; if(a%b == 0) return a ; return (a + (b - a%b)) ;} // least number >=a divisible by b
ll gd(ll a , ll b){if(a < 0) return(-1 * ld(abs(a) , b)) ; return (a - (a%b)) ;} // greatest number <= a divisible by b
ll gcd(ll a , ll b){ if(b > a) return gcd(b , a) ; if(b == 0) return a ; return gcd(b , a%b) ;}
ll e_gcd(ll a , ll b , ll &x , ll &y){ if(b > a) return e_gcd(b , a , y , x) ; if(b == 0){x = 1 ; y = 0 ; return a ;}
ll x1 , y1 , g; g = e_gcd(b , a%b , x1 , y1) ; x = y1 ; y = (x1 - ((a/b) * y1)) ; return g ;}
ll power(ll a ,ll b , ll p){if(b == 0) return 1 ; ll c = power(a , b/2 , p) ; if(b%2 == 0) return ((c*c)%p) ; else return ((((c*c)%p)*a)%p) ;}
ll inverse(ll a ,ll n){return power(a , n-2 , n) ;}
ll max(ll a , ll b){if(a > b) return a ; return b ;}
ll min(ll a , ll b){if(a < b) return a ; return b ;}
ll left(ll i){return ((2*i)+1) ;}
ll right(ll i){return ((2*i) + 2) ;}
ll ncr(ll n , ll r){if(n < r|| (n < 0) || (r < 0)) return 0 ; return ((((fact[n] * inverse(fact[r] , z)) % z) * inverse(fact[n-r] , z)) % z);}
void swap(ll&a , ll&b){ll c = a ; a = b ; b = c ; return ;}
//ios_base::sync_with_stdio(0);
//cin.tie(0); cout.tie(0);
using namespace std ;
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace __gnu_pbds;
#define ordered_set tree<int, null_type,less<int>, rb_tree_tag,tree_order_statistics_node_update>
// ordered_set s ; s.order_of_key(val) no. of elements strictly less than val
// s.find_by_order(i) itertor to ith element (0 indexed)
//__builtin_popcount(n) -> returns number of set bits in n
ll seed;
mt19937 rnd(seed=chrono::steady_clock::now().time_since_epoch().count()); // include bits
const int N = 1000000 ;
void solve()
{
ll n ;
cin >> n ;
ll arr[n] ;
fo(i , n)
cin >> arr[i] ;
sort(arr , arr + n);
reverse(arr , arr + n) ;
ll ans[n] ;
ans[0] = arr[0] ;
ans[1] = arr[1] ;
for(int i = 2 ; i < n ; i++)
{
ll val = (i * (i+1))/2 ;
val %= z ;
val = inverse(val , z) ;
ans[i] = (1-val)*(ans[i-1]) + (val * arr[i]) ;
ans[i] = ((ans[i]%z) + z)%z ;
}
cout << ans[n-1] << '\n' ;
return ;
}
int main()
{
ios_base::sync_with_stdio(0);
cin.tie(0); cout.tie(0);
#ifndef ONLINE_JUDGE
freopen("inputf.txt" , "r" , stdin) ;
freopen("outputf.txt" , "w" , stdout) ;
freopen("error.txt" , "w" , stderr) ;
#endif
ll t = 1;
cin >> t ;
fo(i , t)
solve() ;
cerr << "Time : " << 1000 * ((double)clock()) / (double)CLOCKS_PER_SEC << "ms\n";
return 0;
}
Tester (Kotlin)
import java.io.BufferedInputStream
const val MOD = 1000000007L
const val BILLION = 1000000000
fun main(omkar: Array<String>) {
val jin = FastScanner()
var nSum = 0
repeat(jin.nextInt(500)) {
val n = jin.nextInt(500000)
nSum += n
if (nSum > 500000) {
throw IllegalArgumentException("limit on sum n exceeded")
}
val ratings = Array(n) { jin.nextInt(BILLION, it == n - 1).toLong() }
ratings.sort()
if (ratings.toSet().size != n) {
throw IllegalArgumentException("ratings aren't distinct")
}
var answer = 0L
var coefficient = (((n.toLong() * (n - 1).toLong()) / 2L) % MOD) pow -1
for (k in 1 until n) {
//println("k = $k, coefficient = $coefficient")
answer += coefficient * ratings[k - 1]
answer %= MOD
coefficient *= (n + 2 - k).toLong()
coefficient %= MOD
coefficient *= (n - k).toLong() pow -1
coefficient %= MOD
}
println(answer)
}
jin.assureInputDone()
}
const val MOD_TOTIENT = MOD.toInt() - 1
infix fun Long.pow(power: Int): Long {
var e = power
e %= MOD_TOTIENT
if (e < 0) {
e += MOD_TOTIENT
}
if (e == 0 && this == 0L) {
return this
}
var b = this % MOD
var res = 1L
while (e > 0) {
if (e and 1 != 0) {
res *= b
res %= MOD
}
b *= b
b %= MOD
e = e shr 1
}
return res
}
class FastScanner {
private val BS = 1 shl 16
private val NC = 0.toChar()
private val buf = ByteArray(BS)
private var bId = 0
private var size = 0
private var c = NC
private var `in`: BufferedInputStream? = null
constructor() {
`in` = BufferedInputStream(System.`in`, BS)
}
private val char: Char
private get() {
while (bId == size) {
size = try {
`in`!!.read(buf)
} catch (e: Exception) {
return NC
}
if (size == -1) return NC
bId = 0
}
return buf[bId++].toChar()
}
fun assureInputDone() {
if (char != NC) {
throw IllegalArgumentException("excessive input")
}
}
fun nextInt(endsLine: Boolean): Int {
var neg = false
c = char
if (c !in '0'..'9' && c != '-' && c != ' ' && c != '\n') {
throw IllegalArgumentException("found character other than digit, negative sign, space, and newline")
}
if (c == '-') {
neg = true
c = char
}
var res = 0
while (c in '0'..'9') {
res = (res shl 3) + (res shl 1) + (c - '0')
c = char
}
if (endsLine) {
if (c != '\n') {
throw IllegalArgumentException("found character other than newline")
}
} else {
if (c != ' ') {
throw IllegalArgumentException("found character other than space")
}
}
return if (neg) -res else res
}
fun nextInt(from: Int, to: Int, endsLine: Boolean = true): Int {
val res = nextInt(endsLine)
if (res !in from..to) {
throw IllegalArgumentException("$res not in range $from..$to")
}
return res
}
fun nextInt(to: Int, endsLine: Boolean = true) = nextInt(1, to, endsLine)
}
Editorialist (Python)
mod = int(10**9 + 7)
for _ in range(int(input())):
n = int(input())
a = list(map(int, input().split()))
a.sort()
ans = 0
for i in range(n-1):
mul = 2 * (n+1)
mul *= pow((n-1)*(n-i)*(n-i+1), mod-2, mod)
ans += mul*a[i]
ans %= mod
print(ans)