Summing Pieces


Problem Statement :


Consider an array, A, of length n. We can split A into contiguous segments called pieces and store them as another array, B. For example, if A = [1,2,3], we have the following arrays of pieces:

 B = [(1),(2),(3)] contains three 1-element pieces.
 B = [(1,2),(3)] contains two pieces, one having 2 elements and the other having 1 element.
 B = [(1),(2,3)] contains two pieces, one having 1 element and the other having 2 elements.
 B = [(1,2,3)] contains one 3-element piece.
We consider the value of a piece in some array B to be (sum of all numbers in the piece) * (length of piece), and we consider the total value of some array B to be the sum of the values for all pieces in that B. For example, the total value of B = [1,2,4),(5,1),(2)] is (1+2+4)*3+(5+1)*2+(2)*1 = 35.

Given A, find the total values for all possible B's, sum them together, and print this sum modulo (10^9 +  7) on a new line.

Input Format

The first line contains a single integer, n, denoting the size of array A.
The second line contains n space-separated integers describing the respective values in A (i.e.,a0,a1,...,an-1).

Constraints
1 <= n <= 10^6
1 <= ai <= 10^9

Output Format

Print a single integer denoting the sum of the total values for all piece arrays (B's) of A, modulo (10^9 + 7).


Solution :



title-img


                            Solution in C :

In C++ :





//It’s never too late to become what you might have been....
 #include <iostream>
#include<bits/stdc++.h>
using namespace std;
#define ll long long int
#define inf 1000000000000
#define mod 1000000007
#define pb push_back
#define mp make_pair
#define all(v) v.begin(),v.end()
#define S second
#define F first
#define boost1 ios::sync_with_stdio(false);
#define boost2 cin.tie(0);
#define mem(a,val) memset(a,val,sizeof a)
#define endl "\n"
#define maxn 1000001

ll power[maxn],sub[maxn],pre[maxn],suff[maxn],a[maxn];

ll ways(ll x)
{
	if(x==0)
	return 1;
	return power[x-1];
}
int main()
{
	boost1;boost2;	
	ll i,j,n,q,x,y,sum=0,ans=0;
	power[0]=1;
	for(i=1;i<=1000000;i++)
	power[i]=(2*power[i-1])%mod;
	cin>>n;
	for(i=1;i<=n;i++)
	{
		cin>>a[i];
		sum+=a[i];
		sum%=mod;
	}	
	for(i=1;i<=n;i++)
	pre[i]=(pre[i-1]+a[i])%mod;
	for(i=n;i>=1;i--)
	suff[i]=(suff[i+1]+a[i])%mod;
	
	sub[1]=sum;
	for(i=2;i<=n;i++)
	{
		sub[i]=sub[i-1];
		sub[i]=(sub[i]+suff[i])%mod;
		sub[i]=(sub[i]-suff[n-i+2]+mod)%mod;
	}	

	for(i=1;i<=n-2;i++)
	{
		x=sub[i];
		x=(x-pre[i]+mod)%mod;
		x=(x-suff[n-i+1]+mod)%mod;
		ans=(ans+(((i*power[n-i-2])%mod)*x)%mod)%mod;
	}
	for(i=1;i<=n;i++)
	ans=(ans+(((i*pre[i])%mod)*ways(n-i))%mod)%mod;
	for(i=n;i>1;i--)
	ans=(ans+((((n-i+1)*suff[i])%mod)*ways(i-1))%mod)%mod;
	cout<<ans;
	return 0;
}








In Java :





import java.io.*;
import java.util.*;
import java.text.*;
import java.math.*;
import java.util.regex.*;

public class Solution {

    public static void main(String[] args) {
        Scanner in = new Scanner(System.in);
        
        int n = in.nextInt();
        long sum = 0;
        long[] powers2 = new long[n+1];
        powers2[0] = 1;
        for(int i=1; i<=n; i++)
            powers2[i] = (powers2[i-1] << 1) % 1000000007;
        
        for(int i=1; i<=n; i++){
            long left = ((powers2[i] - 1) * powers2[n-i]) % 1000000007;
            long right = ((powers2[1+n-i]-1) * powers2[i-1]) % 1000000007;
            long v = left + right - powers2[n-1];
            sum = (sum + (v * in.nextLong())) % 1000000007;
        }
        
        System.out.println(sum);
        
    }
}








In C :





#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>

typedef long long int LL;

#define din(n) scanf("%d",&n)
#define dout(n) printf("%d\n",n)
#define llin(n) scanf("%lld",&n)
#define llout(n) printf("%lld\n",n)
#define strin(n) scanf(" %s",n)
#define strout(n) printf("%s\n",n)

int arr[1000005];
int dp[1000005];
int m = 1e9 + 7;
int mod=1e9+7;

int mult(int a,int b)
{
	LL tmp = (LL)a*(LL)b ;
	tmp = tmp%m;
	return (int)tmp;
}

int add(int a, int b)
{
	LL tmp = (LL)a + (LL)b;
	tmp = tmp%m;
	return (int)tmp;
}

int max(int a, int b)
{
	if(a>b) return a; return b;
}

long long po(int x, int y)
{
	int pro=1;
	while(y>0)
	{
		if(mod==1)
			return(0);
		else if(y&1 != 0)
			pro = mult(pro, x);
		x = mult(x,x);
		y=y>>1;
	}
	return pro;
}

int main()
{
	int n; din(n);
	for(int i=0;i<n;i++) 
	{
		din(arr[i]);
	}
	dp[0] = po(2,n) - 1;
	dp[n-1] = po(2,n) - 1;
	int len1 = n-2, len2 = 0;
	for(int i=1;i<=n/2;i++)
	{
		dp[i] = add(dp[i-1], (po(2, len1--)-po(2,len2++))%m ); // mod
		dp[n-i-1] = dp[i];
	}
	int ans = 0;
	for(int i=0;i<n;i++)
	{
		//printf("%d ", dp[i]);
		ans = add(ans, mult(arr[i], dp[i]));
	}
	//printf("\n");
	dout(ans);
	return(0);
}








In Python3 :





import math

n = int(input())
A = list(map(int,input().split()))
T = [0]*n
MOD = 10**9 +7

def pow_mod(x, y):
    number = 1
    while y:
        if y & 1:
            number = number * x % MOD
        y >>= 1
        x = x * x % MOD
    return number

mem = pow_mod(2,n) + pow_mod(2,n-1)
ans = 0
k = 0
for i in range(1,math.ceil(n/2)+1):
    temp = mem - pow_mod(2,n-i) - pow_mod(2,i-1) 
    T[i-1] = temp
    T[n-i] = temp

# print(T)

for a in A:
    # print(k)
    ans = (ans + T[k]*a)%MOD
    k+=1
    
print(ans)
                        




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