東京大学 新領域創成科学研究科 メディカル情報生命専攻 2018年8月実施 問題12
Author
Description
Let
As a first step, we calculate
(1) Write a formula for
(2) Write a formula for
(3) Write an algorithm that calculates the following value and outputs a pair
The running time of your algorithm should be
设
最大化。
第一步,我们计算
(1) 写出
(2) 写出
(3) 写出一个算法计算以下值并输出满足要求的
算法的运行时间应为
Kai
(1)
The formula for
(2)
To derive
This formula captures the choice between starting a new subarray at
(3)
The algorithm calculates the maximum subarray sum and identifies the starting and ending indices of the subarray that achieves this sum. Here is the algorithm in Python:
def max_subarray_sum(sequence):
max_sum = current_sum = sequence[1]
start = end = temp_start = 1
for i in range(2, len(sequence) + 1):
if current_sum > 0:
current_sum += sequence[i]
else:
current_sum = sequence[i]
temp_start = i
if current_sum > max_sum:
max_sum = current_sum
start = temp_start
end = i
return start, end, max_sum
# Example usage
sequence = [-2, 1, -3, 4, -1, 2, 1, -5, 4]
start, end, max_sum = max_subarray_sum(sequence)
print(f"Maximum subarray sum: {max_sum}")
print(f"Start index: {start}, End index: {end}")
Explanation
-
The algorithm initializes the variables
max_sum
,current_sum
,start
,end
, andtemp_start
to keep track of the maximum subarray sum and its indices. -
It iterates through the sequence, updating the
current_sum
based on the formula. -
If the
current_sum
is greater than themax_sum
, it updates themax_sum
,start
, andend
indices. -
The algorithm returns the
start
,end
, andmax_sum
values.
Complexity Analysis
-
Time Complexity: The algorithm has a time complexity of
since it iterates through the sequence once. -
Space Complexity: The space complexity is
since the algorithm uses a constant amount of space for variables.
Knowledge
动态规划 最大子序列和
难点思路
难点在于理解如何通过比较当前元素和当前子序列和来决定是否开始一个新的子序列或继续当前子序列。理解动态规划的状态转移方程
解题技巧和信息
- 动态规划是一种通过分解问题并利用子问题解的技巧。对于本问题,关键在于理解如何通过前一步的解来更新当前解。
- 对于最大子序列和问题,Kadane's Algorithm 是一个经典解法,其时间复杂度为
,适合处理大规模数据。
重点词汇
- subarray 子数组
- maximum subarray sum 最大子数组和
- dynamic programming 动态规划
- sequence 序列
参考资料
- Introduction to Algorithms by Cormen, Leiserson, Rivest, and Stein, Chapter 4: Divide-and-Conquer (Maximum Subarray Problem)
- The Art of Computer Programming, Volume 3: Sorting and Searching by Donald E. Knuth, Section 5.3.2: Maximum Subarray Problem