[算法] 求解和值最大子段及绝对值最小子段

求和值最大子段：不断对已有子段加值，当和值小于0时舍弃子段；由遍历决定复杂度O(n)；

求和值绝对值最小子段：求前n子段和值，然后求最小差值；求最小差值时使用了排序后遍历的方法，由排序决定复杂度O(nlogn)；这里求最小差值的问题中，在某些限制条件下（数组波动不大）可以用桶排序进一步降低时间复杂度，见求最小差值，但因有条件限制不稳定，故不选取；

 

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# -*- coding: utf-8 -*-   array = (4, -3, 2, 5, -9, 6) #array = (4, -3, 1) #array = (-4, -3, -2, -5, -9, -6)    # 求解和值最大子段 # 算法关键是将和值对整体不利的子序列舍弃，修减问题树 # 复杂度O(n) def big_sub():     try_sum = 0     try_start = 0        start = 0     end = 0     sum = 0     big = array[0]     big_index = 0        for i in range(0, len(array)):            if try_sum >= 0:             try_sum += array[i]             if (try_sum > sum):                 sum = try_sum                 end = i                 start = try_start         else: # try_sum < 0             try_sum = array[i]             try_start = i            if array[i] > big:             big = array[i]             big_index = i        if sum == 0:         sum = big         start = end = big_index        print array[start:end+1],     print sum    # 求解和值绝对值最小子段 # 算法关键是求解前n子段和后进行排序，找数列最小差值 # 因为有排序过程存在，所以时间复杂度达到O(nlogn)    # 快排 def split(tuples, start, end):     middle = tuples[end]     index = start-1     for i in range(start, end+1):         if tuples[i][1] <= middle[1]:             index += 1             temp = tuples[index]             tuples[index] = tuples[i]             tuples[i] = temp     return index    def sort_tuples(tuples, start, end):     if start < end:         index = split(tuples, start, end)         sort_tuples(tuples, start, index-1)         sort_tuples(tuples, index+1, end)    def small_sub():     sum = 0     sums = []     for i in range(0, len(array)):         sum += array[i]         sums.append((i, sum))        # 排序     sort_tuples(sums, 0, len(sums)-1)        # 找最小差值     small_diff = -1     small_diff_index = 0     for i in range(0, len(array)-1):         diff = sums[i+1][1] - sums[i][1]            if (small_diff < 0) or (diff < small_diff):             small_diff = diff             small_diff_index = i        if (sums[small_diff_index][0] < sums[small_diff_index+1][0]):         start = sums[small_diff_index][0]         end = sums[small_diff_index+1][0]         minus = False     else:         end = sums[small_diff_index][0]         start = sums[small_diff_index+1][0]         minus = True        print array[start+1:end+1],     if minus:         print -small_diff     else:         print small_diff    if __name__ == "__main__":     big_sub()     small_sub()

 

注：为处理全负数组，在求最大和子段时，用了一个空间big保存最大元素值；