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22092_AdditionChains

2022-5-16 18:21| 发布者: Hocassian| 查看: 30| 评论: 0|原作者: 肇庆学院ACM合集

摘要:

C:\Users\Administrator\Downloads\2019-10-12-10-14-4-89506103365700-Problem List-采集的数据-后羿采集器.html

Pro.ID

22092

Title

Addition Chains

Title链接

http://10.20.2.8/oj/exercise/problem?problem_id=22092

AC

Submit

Ratio

0.00%

时间&空间限制

Time Limit: 1000/500 MS (Java/Others) Memory Limit: 65536/65536 K (Java/Others)

描述

An addition chain for n is an integer sequence with the following four properties:

a₀ = 1
a_m = n
a₀ < a₁ < a₂ < ... < a_m-1 < a_m
For each k (1 ≤ k ≤ m) there exist two (not necessarily different) integers i and j (0 ≤ i, j ≤ k-1) with a_k = a_i + a_j

You are given an integer n. Your job is to construct an addition chain for n with minimal length. If there is more than one such sequence, any one is acceptable.

For example, <1,2,3,5> and <1,2,4,5> are both valid solutions when you are asked for an addition chain for 5.

输入

The input will contain one or more test cases. Each test case consists of one line containing one integer n ( 1 ≤ n ≤ 100 ). Input is terminated by a value of zero (0) for n.

输出

Description

An addition chain for n is an integer sequence with the following four properties:

a₀ = 1
a_m = n
a₀ < a₁ < a₂ < ... < a_m-1 < a_m
For each k (1 ≤ k ≤ m) there exist two (not necessarily different) integers i and j (0 ≤ i, j ≤ k-1) with a_k = a_i + a_j

You are given an integer n. Your job is to construct an addition chain for n with minimal length. If there is more than one such sequence, any one is acceptable.

For example, <1,2,3,5> and <1,2,4,5> are both valid solutions when you are asked for an addition chain for 5.

Input

The input will contain one or more test cases. Each test case consists of one line containing one integer n ( 1 ≤ n ≤ 100 ). Input is terminated by a value of zero (0) for n.

Output

For each test case, print one line containing the required integer sequence. Separate the numbers by one blank.

Sample Input

5
7
12
15
77
0

Sample Output

1 2 4 5
1 2 4 6 7
1 2 4 8 12
1 2 4 5 10 15
1 2 4 8 9 17 34 68 77

Hint

The problem is a little time-critical, so use proper break conditions where necessary to reduce the search space.

Source

Ulm Local 1997