Τελευταία ενημέρωση: 2/10/2024
def brute_force(a_list):
"""
Find max subarray from a_list using brute force.
"""
# initialize returns
max_sum, start, end = a_list[0], 0, 0
# brute force
for i in range(len(a_list)):
for j in range(i, len(a_list)):
total = 0
for x in range(i, j + 1):
total += a_list[x]
# replace start and end positions if a larger sum is found
if max_sum < total:
max_sum, start, end = total, i, j
return max_sum, start, end
%%timeit
import random
a_list = [random.randint(-100, 100) for _ in range(1000)]
max_sum, start, end = brute_force(a_list)
print(f"{max_sum=}, {start=}, {end=}")
max_sum=3623, start=389, end=718 max_sum=1826, start=707, end=987 max_sum=1084, start=118, end=280 max_sum=1905, start=718, end=994 max_sum=1996, start=63, end=305 max_sum=2379, start=321, end=891 max_sum=3150, start=22, end=806 max_sum=2932, start=415, end=960 5.23 s ± 65.6 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
def prefix_sums(a_list):
"""
Find max subarray from a_list using prefix sums.
"""
# initialize returns
max_sum, start, end = a_list[0], 0, 0
# prefix table generation
prefix_table = list()
for index, num in enumerate(a_list):
prefix_table.append(num if index == 0 else num + prefix_table[index - 1])
# use prefix table
for i in range(len(a_list)):
for j in range(i, len(a_list)):
total = prefix_table[j] if i == 0 else prefix_table[j] - prefix_table[i - 1]
# replace starting and ending positions if a larger sum is found
if max_sum < total:
max_sum, start, end = total, i, j
return max_sum, start, end
%%timeit
import random
a_list = [random.randint(-100, 100) for _ in range(1000)]
max_sum, start, end = prefix_sums(a_list)
print(f"{max_sum=}, {start=}, {end=}")
max_sum=3446, start=415, end=964 max_sum=2905, start=673, end=943 max_sum=3566, start=119, end=568 max_sum=1206, start=252, end=412 max_sum=2941, start=23, end=414 max_sum=1544, start=282, end=445 max_sum=2676, start=4, end=501 max_sum=2936, start=76, end=640 max_sum=1973, start=91, end=522 max_sum=3105, start=37, end=522 max_sum=2506, start=283, end=762 max_sum=3677, start=10, end=964 max_sum=2374, start=573, end=994 max_sum=5604, start=41, end=948 max_sum=2250, start=561, end=998 max_sum=3347, start=166, end=963 max_sum=1944, start=35, end=422 max_sum=3762, start=12, end=589 max_sum=3074, start=36, end=645 max_sum=2048, start=41, end=617 max_sum=3202, start=79, end=949 max_sum=1149, start=26, end=77 max_sum=2207, start=278, end=438 max_sum=1587, start=741, end=969 max_sum=3600, start=112, end=626 max_sum=1752, start=318, end=988 max_sum=1772, start=721, end=907 max_sum=1995, start=272, end=554 max_sum=1597, start=735, end=978 max_sum=1809, start=28, end=223 max_sum=3462, start=6, end=999 max_sum=1958, start=81, end=347 max_sum=4071, start=36, end=865 max_sum=2720, start=153, end=469 max_sum=1099, start=396, end=647 max_sum=1832, start=183, end=318 max_sum=2810, start=207, end=821 max_sum=948, start=846, end=999 max_sum=3954, start=30, end=987 max_sum=1834, start=52, end=311 max_sum=2677, start=236, end=966 max_sum=5766, start=5, end=955 max_sum=1852, start=245, end=376 max_sum=4198, start=632, end=981 max_sum=2057, start=572, end=826 max_sum=2188, start=600, end=925 max_sum=1789, start=255, end=831 max_sum=1117, start=25, end=129 max_sum=4007, start=322, end=635 max_sum=1664, start=787, end=974 max_sum=3202, start=179, end=944 max_sum=1466, start=16, end=497 max_sum=1373, start=437, end=856 max_sum=2815, start=25, end=551 max_sum=1848, start=3, end=263 max_sum=1054, start=377, end=525 max_sum=2036, start=301, end=510 max_sum=2670, start=40, end=221 max_sum=1359, start=562, end=797 max_sum=1459, start=447, end=554 max_sum=2516, start=400, end=572 max_sum=1752, start=267, end=718 max_sum=4160, start=43, end=707 max_sum=561, start=415, end=495 max_sum=2009, start=87, end=526 max_sum=3805, start=141, end=978 max_sum=3057, start=182, end=990 max_sum=2348, start=22, end=225 max_sum=3746, start=317, end=999 max_sum=1514, start=24, end=187 max_sum=2120, start=67, end=402 max_sum=2784, start=87, end=653 max_sum=2766, start=638, end=953 max_sum=1781, start=314, end=983 max_sum=3129, start=101, end=676 max_sum=3045, start=1, end=539 max_sum=1612, start=679, end=878 max_sum=1970, start=493, end=628 max_sum=1206, start=93, end=230 max_sum=967, start=850, end=973 max_sum=3217, start=86, end=605 34.3 ms ± 236 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)
def kadane(a_list):
"""
Find max subarray from a_list using kadane's algorithm (max suffix sums).
"""
# initialize returns
max_sum, start, end = a_list[0], 0, 0
# generate kadane table
kadane_table = list()
for index, num in enumerate(a_list):
total = num if index == 0 else num + kadane_table[index - 1]
kadane_table.append(0 if total < 0 else total)
# find max and ending position
for index, total in enumerate(kadane_table):
if max_sum < total:
max_sum, end = total, index
# find starting position
for i in range(end, 0, -1):
if kadane_table[i] == 0:
start = i + 1
break
return max_sum, start, end
%%timeit
import random
a_list = [random.randint(-100, 100) for _ in range(1000)]
max_sum, start, end = kadane(a_list)
print(f"{max_sum=}, {start=}, {end=}")
max_sum=2076, start=429, end=790 max_sum=978, start=380, end=430 max_sum=1464, start=69, end=214 max_sum=1738, start=0, end=209 max_sum=2002, start=115, end=575 max_sum=2208, start=569, end=977 max_sum=2925, start=9, end=841 max_sum=1884, start=548, end=963 max_sum=2129, start=406, end=998 max_sum=1210, start=577, end=691 max_sum=1989, start=6, end=942 max_sum=1048, start=2, end=123 max_sum=1842, start=760, end=938 max_sum=1759, start=690, end=804 max_sum=1924, start=642, end=873 max_sum=2773, start=347, end=767 max_sum=3059, start=27, end=949 max_sum=1119, start=672, end=755 max_sum=2586, start=314, end=994 max_sum=2701, start=307, end=998 max_sum=1741, start=677, end=990 max_sum=2605, start=146, end=792 max_sum=1061, start=748, end=792 max_sum=1559, start=462, end=775 max_sum=2797, start=147, end=999 max_sum=2737, start=190, end=915 max_sum=2156, start=529, end=951 max_sum=3621, start=250, end=999 max_sum=3132, start=2, end=668 max_sum=2498, start=352, end=593 max_sum=1864, start=40, end=574 max_sum=1193, start=744, end=997 max_sum=2837, start=6, end=960 max_sum=2066, start=175, end=975 max_sum=1953, start=348, end=521 max_sum=1682, start=0, end=249 max_sum=1674, start=109, end=300 max_sum=3701, start=89, end=987 max_sum=1516, start=663, end=847 max_sum=1316, start=684, end=894 max_sum=2312, start=78, end=970 max_sum=4363, start=390, end=999 max_sum=2682, start=494, end=925 max_sum=1788, start=125, end=337 max_sum=772, start=755, end=878 max_sum=1646, start=307, end=413 max_sum=2977, start=334, end=828 max_sum=1751, start=732, end=896 max_sum=3309, start=126, end=869 max_sum=1635, start=682, end=975 max_sum=1240, start=277, end=428 max_sum=1450, start=38, end=521 max_sum=885, start=32, end=143 max_sum=3517, start=409, end=891 max_sum=1593, start=785, end=999 max_sum=3510, start=203, end=956 max_sum=1081, start=295, end=382 max_sum=2577, start=178, end=503 max_sum=3493, start=113, end=368 max_sum=2692, start=632, end=975 max_sum=1505, start=629, end=919 max_sum=1351, start=303, end=510 max_sum=1746, start=17, end=156 max_sum=1832, start=748, end=931 max_sum=2485, start=418, end=999 max_sum=1649, start=626, end=930 max_sum=2044, start=243, end=452 max_sum=1591, start=817, end=999 max_sum=2353, start=9, end=799 max_sum=3170, start=84, end=968 max_sum=2779, start=73, end=738 max_sum=5266, start=198, end=956 max_sum=2507, start=455, end=756 max_sum=2408, start=86, end=509 max_sum=984, start=462, end=627 max_sum=2162, start=373, end=884 max_sum=5022, start=108, end=986 max_sum=3282, start=125, end=995 max_sum=1915, start=160, end=417 max_sum=4620, start=184, end=913 max_sum=1058, start=437, end=492 max_sum=3253, start=402, end=995 max_sum=2998, start=112, end=979 max_sum=2106, start=11, end=650 max_sum=1578, start=436, end=592 max_sum=2235, start=416, end=954 max_sum=2321, start=192, end=562 max_sum=2545, start=697, end=994 max_sum=1898, start=4, end=336 max_sum=3417, start=10, end=793 max_sum=2860, start=422, end=843 max_sum=1689, start=442, end=833 max_sum=1155, start=563, end=712 max_sum=2554, start=12, end=327 max_sum=3795, start=480, end=802 max_sum=2319, start=728, end=993 max_sum=2724, start=347, end=919 max_sum=2107, start=777, end=984 max_sum=1442, start=792, end=984 max_sum=1684, start=0, end=160 max_sum=2689, start=43, end=998 max_sum=1439, start=31, end=211 max_sum=3445, start=132, end=990 max_sum=2162, start=284, end=840 max_sum=1273, start=96, end=292 max_sum=4872, start=90, end=968 max_sum=2685, start=485, end=813 max_sum=3622, start=27, end=627 max_sum=1498, start=936, end=997 max_sum=3223, start=461, end=820 max_sum=3654, start=139, end=940 max_sum=3492, start=203, end=795 max_sum=2013, start=439, end=963 max_sum=1421, start=464, end=576 max_sum=1323, start=351, end=414 max_sum=3335, start=770, end=983 max_sum=2295, start=0, end=708 max_sum=3483, start=37, end=983 max_sum=1946, start=526, end=742 max_sum=2011, start=15, end=426 max_sum=1935, start=238, end=483 max_sum=2120, start=0, end=907 max_sum=1758, start=514, end=925 max_sum=1322, start=681, end=826 max_sum=1228, start=776, end=926 max_sum=1484, start=745, end=869 max_sum=2361, start=61, end=869 max_sum=2958, start=110, end=854 max_sum=2536, start=8, end=566 max_sum=2129, start=151, end=999 max_sum=1433, start=162, end=317 max_sum=5526, start=74, end=964 max_sum=1208, start=269, end=398 max_sum=2157, start=11, end=324 max_sum=3094, start=72, end=705 max_sum=3057, start=10, end=999 max_sum=2016, start=512, end=799 max_sum=2609, start=478, end=784 max_sum=2003, start=334, end=685 max_sum=2330, start=770, end=984 max_sum=1960, start=553, end=804 max_sum=1966, start=302, end=704 max_sum=1590, start=153, end=384 max_sum=2084, start=567, end=935 max_sum=1199, start=644, end=802 max_sum=2381, start=256, end=501 max_sum=3020, start=53, end=836 max_sum=2689, start=164, end=677 max_sum=5646, start=47, end=842 max_sum=2571, start=360, end=913 max_sum=808, start=799, end=856 max_sum=2473, start=237, end=949 max_sum=2000, start=121, end=400 max_sum=4150, start=296, end=961 max_sum=986, start=0, end=80 max_sum=3104, start=647, end=923 max_sum=2239, start=348, end=989 max_sum=1105, start=419, end=504 max_sum=5420, start=0, end=982 max_sum=1396, start=537, end=706 max_sum=2650, start=324, end=843 max_sum=3212, start=258, end=514 max_sum=3219, start=77, end=902 max_sum=2531, start=20, end=467 max_sum=2355, start=182, end=484 max_sum=1819, start=14, end=150 max_sum=1027, start=563, end=663 max_sum=1614, start=702, end=961 max_sum=3881, start=26, end=738 max_sum=2134, start=542, end=998 max_sum=1143, start=779, end=964 max_sum=3431, start=73, end=999 max_sum=4364, start=618, end=996 max_sum=3059, start=74, end=997 max_sum=2499, start=323, end=956 max_sum=2919, start=90, end=984 max_sum=1987, start=2, end=308 max_sum=2784, start=383, end=855 max_sum=3259, start=91, end=636 max_sum=1274, start=697, end=785 max_sum=4134, start=87, end=699 max_sum=1290, start=160, end=391 max_sum=3412, start=29, end=682 max_sum=2161, start=456, end=864 max_sum=2192, start=450, end=909 max_sum=3224, start=216, end=929 max_sum=1198, start=250, end=444 max_sum=1494, start=0, end=133 max_sum=2614, start=20, end=886 max_sum=2708, start=13, end=393 max_sum=2195, start=440, end=839 max_sum=791, start=355, end=394 max_sum=3184, start=133, end=996 max_sum=1498, start=288, end=450 max_sum=2610, start=581, end=987 max_sum=2891, start=135, end=769 max_sum=2143, start=151, end=972 max_sum=1614, start=508, end=650 max_sum=3078, start=88, end=952 max_sum=1612, start=14, end=959 max_sum=1750, start=736, end=949 max_sum=1973, start=366, end=712 max_sum=1454, start=456, end=786 max_sum=2710, start=3, end=941 max_sum=1920, start=126, end=997 max_sum=1911, start=409, end=673 max_sum=1692, start=625, end=840 max_sum=3289, start=151, end=623 max_sum=5510, start=0, end=946 max_sum=2078, start=510, end=679 max_sum=2053, start=12, end=268 max_sum=1137, start=615, end=787 max_sum=2984, start=0, end=727 max_sum=1174, start=121, end=216 max_sum=1620, start=738, end=931 max_sum=2332, start=303, end=641 max_sum=2209, start=192, end=746 max_sum=1583, start=740, end=826 max_sum=1608, start=532, end=746 max_sum=3311, start=2, end=963 max_sum=2127, start=259, end=456 max_sum=1061, start=202, end=280 max_sum=1191, start=0, end=62 max_sum=1882, start=134, end=479 max_sum=797, start=171, end=310 max_sum=3212, start=497, end=991 max_sum=1747, start=675, end=954 max_sum=5385, start=24, end=913 max_sum=3787, start=15, end=848 max_sum=2641, start=528, end=854 max_sum=2727, start=426, end=952 max_sum=1346, start=79, end=233 max_sum=2290, start=269, end=620 max_sum=5367, start=371, end=975 max_sum=2629, start=189, end=463 max_sum=1675, start=423, end=685 max_sum=1795, start=221, end=382 max_sum=1818, start=288, end=684 max_sum=1407, start=55, end=444 max_sum=1661, start=202, end=423 max_sum=1828, start=258, end=517 max_sum=1657, start=806, end=988 max_sum=1874, start=409, end=882 max_sum=2677, start=253, end=997 max_sum=2644, start=196, end=752 max_sum=2516, start=3, end=234 max_sum=2588, start=6, end=914 max_sum=1404, start=627, end=821 max_sum=2155, start=571, end=921 max_sum=2153, start=695, end=998 max_sum=1310, start=469, end=678 max_sum=2171, start=344, end=826 max_sum=3062, start=6, end=734 max_sum=2328, start=85, end=618 max_sum=3050, start=133, end=612 max_sum=2739, start=34, end=756 max_sum=1262, start=213, end=404 max_sum=3176, start=528, end=998 max_sum=1685, start=186, end=366 max_sum=2144, start=82, end=472 max_sum=775, start=602, end=724 max_sum=2281, start=60, end=474 max_sum=2966, start=59, end=981 max_sum=1337, start=98, end=292 max_sum=3352, start=496, end=844 max_sum=2452, start=373, end=993 max_sum=1497, start=476, end=997 max_sum=1426, start=0, end=508 max_sum=1383, start=40, end=317 max_sum=1961, start=3, end=229 max_sum=1407, start=20, end=224 max_sum=2796, start=380, end=980 max_sum=1429, start=846, end=996 max_sum=1897, start=782, end=992 max_sum=2749, start=472, end=981 max_sum=708, start=266, end=367 max_sum=1462, start=153, end=301 max_sum=2916, start=363, end=864 max_sum=2240, start=221, end=639 max_sum=2051, start=120, end=863 max_sum=2569, start=167, end=337 max_sum=1782, start=123, end=908 max_sum=2541, start=276, end=649 max_sum=2874, start=289, end=993 max_sum=1825, start=715, end=862 max_sum=2604, start=518, end=877 max_sum=2222, start=810, end=994 max_sum=2186, start=22, end=708 max_sum=945, start=855, end=944 max_sum=1848, start=800, end=957 max_sum=2238, start=574, end=870 max_sum=4504, start=19, end=996 max_sum=2046, start=702, end=900 max_sum=1201, start=551, end=622 max_sum=2868, start=46, end=687 max_sum=1964, start=475, end=812 max_sum=1251, start=705, end=865 max_sum=3499, start=196, end=937 max_sum=3402, start=160, end=519 max_sum=1567, start=76, end=261 max_sum=4749, start=82, end=995 max_sum=3489, start=107, end=820 max_sum=1707, start=850, end=987 max_sum=2183, start=174, end=409 max_sum=2049, start=247, end=480 max_sum=2463, start=516, end=769 max_sum=1674, start=401, end=763 max_sum=3040, start=0, end=945 max_sum=1768, start=331, end=998 max_sum=2769, start=154, end=798 max_sum=1480, start=501, end=999 max_sum=3351, start=106, end=541 max_sum=4205, start=18, end=260 max_sum=1329, start=203, end=322 max_sum=2800, start=476, end=875 max_sum=1258, start=501, end=663 max_sum=4362, start=84, end=998 max_sum=1911, start=139, end=291 max_sum=1803, start=589, end=911 max_sum=2899, start=182, end=385 max_sum=3494, start=44, end=630 max_sum=2231, start=581, end=845 max_sum=942, start=383, end=570 max_sum=1798, start=0, end=132 max_sum=2162, start=222, end=578 max_sum=2066, start=357, end=581 max_sum=1920, start=789, end=980 max_sum=2285, start=4, end=787 max_sum=1947, start=239, end=704 max_sum=1074, start=514, end=754 max_sum=2036, start=454, end=863 max_sum=2308, start=81, end=889 max_sum=2467, start=12, end=438 max_sum=2160, start=5, end=657 max_sum=1306, start=531, end=988 max_sum=1148, start=431, 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max_sum=2476, start=0, end=310 max_sum=1431, start=0, end=534 max_sum=1521, start=174, end=543 max_sum=3824, start=421, end=930 max_sum=1803, start=837, end=998 max_sum=3064, start=365, end=896 max_sum=3107, start=264, end=788 max_sum=3328, start=17, end=732 max_sum=1519, start=53, end=379 max_sum=5024, start=31, end=951 max_sum=1707, start=220, end=999 max_sum=2598, start=18, end=451 max_sum=2526, start=401, end=998 max_sum=1854, start=133, end=399 max_sum=2959, start=25, end=793 max_sum=1362, start=358, end=555 max_sum=4188, start=79, end=900 max_sum=2303, start=411, end=998 max_sum=2344, start=389, end=721 max_sum=1633, start=295, end=408 max_sum=1029, start=364, end=515 max_sum=1848, start=747, end=951 max_sum=4317, start=26, end=837 max_sum=1103, start=859, end=906 max_sum=2937, start=68, end=984 max_sum=1382, start=943, end=999 max_sum=3568, start=32, end=718 max_sum=1061, start=732, end=973 max_sum=1234, start=131, end=253 max_sum=3462, start=277, end=954 max_sum=2082, start=579, end=870 max_sum=1201, start=368, end=710 max_sum=809, start=510, end=590 479 µs ± 8.49 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)