連續子向量的最大和問題（Python實現）

問題：

輸入：一個向量x，包含n個數。

輸出：向量的連續子向量的最大和，以及下標。

 

首先給出運行結果：

[1, 2, 3, 4]
low= 0 ;high= 4 ;max_sum= 10

[-11, -2, -3, -4]
NULL vector: max_sum = 0

[-1, -2, -3, 4]
low= 3 ;high= 4 ;max_sum= 4

[1, -2, -3, -4]
low= 0 ;high= 1 ;max_sum= 1

[-1, 2, 3, -4]
low= 1 ;high= 3 ;max_sum= 5

[31, -41, 59, 26, -53, 58, 97, -93, -23, 84]
low= 2 ;high= 7 ;max_sum= 187

算法分析：

用有限狀態機來解這道題，時間複雜度是O(n). 注：顯而易見，若是使用暴力算法的話時間複雜度爲O(n^3)。

狀態：start, skip, state1, state2, state3, exit

判斷條件：元素正負，下表是否在範圍內。

根據判斷條件在狀態件切換，而且記錄下表和最大和。

 

Python實現的源碼

#!/usr/bin/python
# identify the subvector in a given vector that has the maximum sum

# Global Vars
max_sum = 0
high = 0
low = 0
minus_sum = 0
cur_sum = 0
cur_low = 0
vector = [1, 2, 3, 4]
#vector = [-1, -1, -1, -1]
#vector = [-1, -1, -1, 1]

def check_range(pos):
    global vector
    if pos >= 0 and pos < len(vector):
        return True
    else:
        return False

def do_start(pos):
    global vector
    global low
    if not check_range(pos):
        do_exit()
        return
    if vector[pos] > 0:
        low = pos
        do_state1(pos)
        return
    else:
        do_skip(pos)
        return

def do_exit():
    global low
    global high
    global cur_sum
    global minu_sum
    global cur_low
    global max_sum
    if low >= high:
        print 'NULL vector: max_sum = 0'
    else:
        print 'low=', low, ';high=', high, ';max_sum=', max_sum
    cur_sum = 0
    max_sum = 0
    cur_low = 0
    minus_sum = 0
    low = 0
    high = 0
    return


def do_skip(pos):
    global vector
    global low
    if not check_range(pos):
        do_exit()
        return
    while vector[pos] <= 0:
        pos += 1
        if not check_range(pos):
            do_exit()
            return
    if vector[pos] > 0:
        low = pos
        do_state1(pos)
        return
    else:
        print 'we should never get here!'
        return

def do_state1(pos):
    global max_sum
    global vector
    global high
    max_sum += vector[pos]
    high = pos+1                # [low, high)
    pos += 1
    if not check_range(pos):
        do_exit()
        return
    if vector[pos] > 0:
        do_state1(pos)
        return
    else:
        do_state2(pos)
        return

def do_state2(pos):
    global minus_sum
    global vector
    global cur_low
    minus_sum += vector[pos]
    pos += 1
    if not check_range(pos):
        do_exit()
        return
    if vector[pos] > 0:
        cur_low = pos
        do_state3(pos)
        return
    else:
        do_state2(pos)
        return

def do_state3(pos):
    global vector
    global cur_sum
    cur_sum += vector[pos]
    pos += 1
    if not check_range(pos):
        determine_and_reset(pos)
        do_exit()
        return
    if vector[pos] > 0:
        do_state3(pos)
        return
    else:
        determine_and_reset(pos)
        do_state2(pos)
        return

def determine_and_reset(pos):
    global minus_sum
    global max_sum
    global cur_sum
    global cur_low
    global vector
    global low
    global high
    sum_total = max_sum + minus_sum + cur_sum
    if sum_total > max_sum and sum_total > cur_sum:
        high = pos
        low = low
        max_sum = sum_total
    elif cur_sum >= sum_total and cur_sum > max_sum:
        high = pos
        low = cur_low
        max_sum = cur_sum
    elif max_sum >= sum_total and max_sum >= cur_sum:
        high = high
        low = low
    else:
        print 'we should never get here!'
    cur_sum = 0
    cur_low = 0
    minus_sum = 0

    
def test1():
    global vector
    vector = [1, 2, 3, 4]
    print vector
    do_start(0)

def test2():
    global vector
    vector = [-11, -2, -3, -4]
    print vector
    do_start(0)

def test3():
    global vector
    vector = [-1, -2, -3, 4]
    print vector
    do_start(0)

def test4():
    global vector
    vector = [1, -2, -3, -4]
    print vector
    do_start(0)

def test5():
    global vector
    vector = [-1, 2, 3, -4]
    print vector
    do_start(0)

def test6():
    global vector
    vector = [31, -41, 59, 26, -53, 58, 97, -93, -23, 84]
    print vector
    do_start(0)


def main():
    test1()
    test2()
    test3()
    test4()
    test5()
    test6()
    return 0

if __name__ == '__main__':
    main()