Educational Game (Codeforces 178A3)
Tags: Greedy, greedy

The Smart Beaver from ABBYY began to develop a new educational game for children. The rules of the game are fairly simple and are described below.The playing field is a sequence of n non-negative integers ai numbered from 1 to n. The goal of the game is to make numbers a1, a2, ..., ak (i.e. some prefix of the sequence) equal to zero for some fixed k (k < n), and this should be done in the smallest possible number of moves.One move is choosing an integer i (1 ≤ i ≤ n) such that ai > 0 and an integer t (t ≥ 0) such that i + 2t ≤ n. After the values of i and t have been selected, the value of ai is decreased by 1, and the value of ai + 2t is increased by 1. For example, let n = 4 and a = (1, 0, 1, 2), then it is possible to make move i = 3, t = 0 and get a = (1, 0, 0, 3) or to make move i = 1, t = 1 and get a = (0, 0, 2, 2) (the only possible other move is i = 1, t = 0).You are given n and the initial sequence ai. The task is to calculate the minimum number of moves needed to make the first k elements of the original sequence equal to zero for each possible k (1 ≤ k < n).

Input: The first input line contains a single integer n. The second line contains n integers ai (0 ≤ ai ≤ 104), separated by single spaces.The input limitations for getting 20 points are:   1 ≤ n ≤ 300 The input limitations for getting 50 points are:   1 ≤ n ≤ 2000 The input limitations for getting 100 points are:   1 ≤ n ≤ 105 

Output: Print exactly n - 1 lines: the k-th output line must contain the minimum number of moves needed to make the first k elements of the original sequence ai equal to zero.Please do not use the %lld specifier to read or write 64-bit integers in С++. It is preferred to use the cin, cout streams, or the %I64d specifier.


Sample Inputs:
4
1 0 1 2

8
1 2 3 4 5 6 7 8

Sample Outputs:
1
1
3

1
3
6
10
16
24
40

Solution (Python):
import os
CODEFORCES = 'INSIDE_EMACS' not in os.environ
if not CODEFORCES:
    import inspect
    PATH = inspect.getfile(lambda : None)
    from time import perf_counter
    tic = perf_counter()
    import sys
    orig_stdin = sys.stdin
    sys.stdin = open(PATH[:-2] + 'in')
import bisect
II = lambda : int(input())
F = lambda : map(int, input().split())
FL = lambda : [*F()]
(n, a) = (II(), FL())
if n > 1:
    diff = 1 << (n - 1).bit_length() - 1
    total = 0
    for i in range(n - 1):
        j = i + diff
        while j >= n:
            diff //= 2
            j = i + diff
        total += a[i]
        a[j] += a[i]
        print(total)
if not CODEFORCES:
    toc = perf_counter()
    print(toc - tic, 'seconds')
    sys.stdin.close()
    sys.stdin = orig_stdin

Solution (Python):
from math import log2
n = int(input())
w = list(map(int, input().split()))
c = 0
for i in range(n - 1):
    d = i + 2 ** int(log2(n - i - 1))
    if d <= n - 1:
        w[d] += w[i]
    c += w[i]
    print(c)

Solution (Python):
def main():
    from math import floor, log2
    n = int(input())
    numbers = [0] + [int(_) for _ in input().split()]
    moves = [0] * (n + 1)
    for i in range(1, n):
        t = floor(log2(n - i))
        j = i + 2 ** t
        moves[i] = numbers[i]
        numbers[j] += numbers[i]
        numbers[i] = 0
    for i in range(2, n):
        moves[i] += moves[i - 1]
    print('\n'.join([str(moves[i]) for i in range(1, n)]))
if __name__ == '__main__':
    main()