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STLv1 不支持范围概念，需要传入首尾迭代器\n    \u002F\u002F 初始值是 0，二元函数是 (a, b) -> a + b\n    auto sum = std::accumulate(a.begin(), a.end(), 0, std::plus{});\n\n    std::cout \u003C\u003C sum \u003C\u003C std::endl;\n}\n","cpp","",[931,1005,1006,1026,1038,1055,1062,1079,1129,1134,1140,1146,1206,1211,1238],{"__ignoreMap":1003},[1007,1008,1011,1015,1019,1023],"span",{"class":1009,"line":1010},"line",1,[1007,1012,1014],{"class":1013},"s7zQu","#include",[1007,1016,1018],{"class":1017},"sMK4o"," \u003C",[1007,1020,1022],{"class":1021},"sfazB","iostream",[1007,1024,1025],{"class":1017},">\n",[1007,1027,1029,1031,1033,1036],{"class":1009,"line":1028},2,[1007,1030,1014],{"class":1013},[1007,1032,1018],{"class":1017},[1007,1034,1035],{"class":1021},"vector",[1007,1037,1025],{"class":1017},[1007,1039,1041,1043,1045,1048,1051],{"class":1009,"line":1040},3,[1007,1042,1014],{"class":1013},[1007,1044,1018],{"class":1017},[1007,1046,1047],{"class":1021},"numeric",[1007,1049,1050],{"class":1017},">",[1007,1052,1054],{"class":1053},"sHwdD"," \u002F\u002F 注意：这玩意儿不定义在 \u003Calgorithm>\n",[1007,1056,1058],{"class":1009,"line":1057},4,[1007,1059,1061],{"emptyLinePlaceholder":1060},true,"\n",[1007,1063,1065,1069,1073,1076],{"class":1009,"line":1064},5,[1007,1066,1068],{"class":1067},"spNyl","int",[1007,1070,1072],{"class":1071},"s2Zo4"," main",[1007,1074,1075],{"class":1017},"()",[1007,1077,1078],{"class":1017}," {\n",[1007,1080,1082,1086,1089,1093,1096,1100,1103,1106,1108,1111,1113,1116,1118,1121,1123,1126],{"class":1009,"line":1081},6,[1007,1083,1085],{"class":1084},"sBMFI","    std",[1007,1087,1088],{"class":1017},"::",[1007,1090,1092],{"class":1091},"sTEyZ","vector a",[1007,1094,1095],{"class":1017},"{",[1007,1097,1099],{"class":1098},"sbssI","1",[1007,1101,1102],{"class":1017},",",[1007,1104,1105],{"class":1098}," 2",[1007,1107,1102],{"class":1017},[1007,1109,1110],{"class":1098}," 3",[1007,1112,1102],{"class":1017},[1007,1114,1115],{"class":1098}," 4",[1007,1117,1102],{"class":1017},[1007,1119,1120],{"class":1098}," 5",[1007,1122,1102],{"class":1017},[1007,1124,1125],{"class":1098}," 6",[1007,1127,1128],{"class":1017},"};\n",[1007,1130,1132],{"class":1009,"line":1131},7,[1007,1133,1061],{"emptyLinePlaceholder":1060},[1007,1135,1137],{"class":1009,"line":1136},8,[1007,1138,1139],{"class":1053},"    \u002F\u002F 由于 STLv1 不支持范围概念，需要传入首尾迭代器\n",[1007,1141,1143],{"class":1009,"line":1142},9,[1007,1144,1145],{"class":1053},"    \u002F\u002F 初始值是 0，二元函数是 (a, b) -> a + b\n",[1007,1147,1149,1152,1155,1158,1161,1163,1166,1169,1172,1175,1178,1181,1184,1186,1189,1191,1194,1196,1198,1200,1203],{"class":1009,"line":1148},10,[1007,1150,1151],{"class":1067},"    auto",[1007,1153,1154],{"class":1091}," sum ",[1007,1156,1157],{"class":1017},"=",[1007,1159,1160],{"class":1084}," std",[1007,1162,1088],{"class":1017},[1007,1164,1165],{"class":1071},"accumulate",[1007,1167,1168],{"class":1017},"(",[1007,1170,1171],{"class":1091},"a",[1007,1173,1174],{"class":1017},".",[1007,1176,1177],{"class":1071},"begin",[1007,1179,1180],{"class":1017},"(),",[1007,1182,1183],{"class":1091}," a",[1007,1185,1174],{"class":1017},[1007,1187,1188],{"class":1071},"end",[1007,1190,1180],{"class":1017},[1007,1192,1193],{"class":1098}," 0",[1007,1195,1102],{"class":1017},[1007,1197,1160],{"class":1084},[1007,1199,1088],{"class":1017},[1007,1201,1202],{"class":1091},"plus{}",[1007,1204,1205],{"class":1017},");\n",[1007,1207,1209],{"class":1009,"line":1208},11,[1007,1210,1061],{"emptyLinePlaceholder":1060},[1007,1212,1214,1216,1218,1221,1224,1226,1228,1230,1232,1235],{"class":1009,"line":1213},12,[1007,1215,1085],{"class":1084},[1007,1217,1088],{"class":1017},[1007,1219,1220],{"class":1091},"cout ",[1007,1222,1223],{"class":1017},"\u003C\u003C",[1007,1225,1154],{"class":1091},[1007,1227,1223],{"class":1017},[1007,1229,1160],{"class":1084},[1007,1231,1088],{"class":1017},[1007,1233,1234],{"class":1091},"endl",[1007,1236,1237],{"class":1017},";\n",[1007,1239,1241],{"class":1009,"line":1240},13,[1007,1242,1243],{"class":1017},"}\n",[1245,1246,1247],"blockquote",{},[918,1248,1249,1250,1252,1253,1255,1256,1258],{},"之所以命名为 ",[931,1251,987],{},"，是因为二元函数参数带有默认实参，默认值就是 ",[931,1254,995],{},"。此时，使用 ",[931,1257,987],{}," 就是将一个范围内的元素求和，即累积（Accumulate）的含义。",[918,1260,1261,1262,1264,1265,1268,1269,1272,1273,1275,1276,1278],{},"类似地，如果将初始值设为 ",[931,1263,1099],{},"，二元函数设为乘法 ",[931,1266,1267],{},"std::multiply{}","，则效果就是求范围上所有元素的积。此外，如果传入的元素是 ",[931,1270,1271],{},"std::string"," 类型的，那么以 ",[931,1274,995],{}," 函数 ",[931,1277,987],{}," 整个范围就是在拼接字符串。你可以自己试一试，想一想原理。",[954,1280,1281],{"id":1281},"并行归约",[918,1283,1284],{},"其实折叠只是一种特殊的归约。更通用的从二元函数导出的归约具有如下定义：",[918,1286,1287],{},"$${x_1,\\cdots,x_n}\\xrightarrow{\\operatorname{reduce}(f,\\cdot)}\\begin{cases}\nx_1,&n=1\\\nf\\big(\\operatorname{reduce}(f,{x_,\\cdots,x_}),\\operatorname{reduce}(f,{x_{i_{k+1}},\\cdots,x_})\\big),&n>1\n\\end{cases}$$",[918,1289,1290],{},"其中 ${i}$ 是 ${1,\\cdots,n}$ 的任意排列，$1\\leqslant k\\leqslant n$ 是任意正整数。",[918,1292,1293,1294,1297],{},"数学定义看上去总是令人脑袋大。简单来说，这是一个递归的定义：对于具有两个元素或更大的范围，则任意地拆成两半分别递归计算其归约结果。计算完成后，将两个结果用二元函数 ",[931,1295,1296],{},"f"," 运算，得到当前归约的结果。",[918,1299,1300],{},"它不同于折叠的最大区别在于“任意”二字。在对 $n$ 个元素的范围折叠时，总是先计算好前 $n - 1$ 个元素的折叠结果，然后再与 $x_n$ 运算。而这里则是任意划分运算顺序；它可以先将前 $k$ 个元素归约，然后再处理剩余的；甚至可以先对 $x_1,x_3,x_5,\\cdots$ 归约，然后再对 $x_2,x_4,x_6\\cdots$ 归约。具体如何运算取决于编译器和硬件如何处理。仍然是之前求和的例子，某次运算的过程可能如下图所示：",[918,1302,1303],{},[975,1304],{"alt":1305,"src":1306},"reduce","https:\u002F\u002Fs1.ax1x.com\u002F2022\u002F07\u002F02\u002Fj1esZ8.png",[918,1308,1309,1310,1314],{},"而到了不同的计算机、或者不同的编译器就会编译出不一样的运算顺序。这种任意性带来潜在的危险：只有当你的二元运算",[1311,1312,1313],"strong",{},"满足交换律和结合律","时，才能保证结果是惟一的；否则，结果是不确值。但它也带来了好处：可以让计算机做并行优化。",[918,1316,1317,1318,1321,1322,1321,1325,1328,1329,1331,1332,1331,1335,1338],{},"并行优化是指，现代的计算机一般有两个以上的计算核心（核）。多个计算核心可以同时工作以加快整体运算速度；在之前的图示中，",[931,1319,1320],{},"0 + 1"," ",[931,1323,1324],{},"2 + 3",[931,1326,1327],{},"4 + 5"," 就可以同时计算。普通的折叠必须一步步挨排算，先算 ",[931,1330,1320],{},"，然后 ",[931,1333,1334],{},"1 + 2",[931,1336,1337],{},"3 + 3","……总共需要花费六次计算时间，并行优化后则只用三次计算时间就可以完成。",[918,1340,1341,1342,1345,1346,1348],{},"STLv1 中，",[931,1343,1344],{},"std::reduce"," 实现可并行优化的归约。使用上，前两个参数一般是首尾迭代器；第三个参数是初始值；第四个参数是二元函数。初始值是为了保证传入的范围不是空的（因为刚才的数学定义没有规定空范围的归约结果），默认实参是零值（具体而言，默认初始化的值）；二元函数同样默认为 ",[931,1347,995],{},"。",[1245,1350,1351],{},[918,1352,1353,1355],{},[931,1354,987],{}," 没有初始值的默认实参。",[998,1357,1359],{"className":1000,"code":1358,"language":1002,"meta":1003,"style":1003},"#include \u003Ciostream>\n#include \u003Cvector>\n#include \u003Cnumeric> \u002F\u002F 注意：这玩意儿也不定义在 \u003Calgorithm>\n\nint main() {\n    std::vector a{1, 2, 3, 4, 5, 6};\n\n    \u002F\u002F 初始值是 0，二元函数是 (a, b) -> a + b\n    auto sum = std::reduce(a.begin(), a.end(), 0, std::plus{});\n    \u002F\u002F 或者使用默认实参\n    std::reduce(a.begin(), a.end());\n\n    std::cout \u003C\u003C sum \u003C\u003C std::endl;\n}\n",[931,1360,1361,1371,1381,1394,1398,1408,1442,1446,1450,1494,1499,1526,1530,1552],{"__ignoreMap":1003},[1007,1362,1363,1365,1367,1369],{"class":1009,"line":1010},[1007,1364,1014],{"class":1013},[1007,1366,1018],{"class":1017},[1007,1368,1022],{"class":1021},[1007,1370,1025],{"class":1017},[1007,1372,1373,1375,1377,1379],{"class":1009,"line":1028},[1007,1374,1014],{"class":1013},[1007,1376,1018],{"class":1017},[1007,1378,1035],{"class":1021},[1007,1380,1025],{"class":1017},[1007,1382,1383,1385,1387,1389,1391],{"class":1009,"line":1040},[1007,1384,1014],{"class":1013},[1007,1386,1018],{"class":1017},[1007,1388,1047],{"class":1021},[1007,1390,1050],{"class":1017},[1007,1392,1393],{"class":1053}," \u002F\u002F 注意：这玩意儿也不定义在 \u003Calgorithm>\n",[1007,1395,1396],{"class":1009,"line":1057},[1007,1397,1061],{"emptyLinePlaceholder":1060},[1007,1399,1400,1402,1404,1406],{"class":1009,"line":1064},[1007,1401,1068],{"class":1067},[1007,1403,1072],{"class":1071},[1007,1405,1075],{"class":1017},[1007,1407,1078],{"class":1017},[1007,1409,1410,1412,1414,1416,1418,1420,1422,1424,1426,1428,1430,1432,1434,1436,1438,1440],{"class":1009,"line":1081},[1007,1411,1085],{"class":1084},[1007,1413,1088],{"class":1017},[1007,1415,1092],{"class":1091},[1007,1417,1095],{"class":1017},[1007,1419,1099],{"class":1098},[1007,1421,1102],{"class":1017},[1007,1423,1105],{"class":1098},[1007,1425,1102],{"class":1017},[1007,1427,1110],{"class":1098},[1007,1429,1102],{"class":1017},[1007,1431,1115],{"class":1098},[1007,1433,1102],{"class":1017},[1007,1435,1120],{"class":1098},[1007,1437,1102],{"class":1017},[1007,1439,1125],{"class":1098},[1007,1441,1128],{"class":1017},[1007,1443,1444],{"class":1009,"line":1131},[1007,1445,1061],{"emptyLinePlaceholder":1060},[1007,1447,1448],{"class":1009,"line":1136},[1007,1449,1145],{"class":1053},[1007,1451,1452,1454,1456,1458,1460,1462,1464,1466,1468,1470,1472,1474,1476,1478,1480,1482,1484,1486,1488,1490,1492],{"class":1009,"line":1142},[1007,1453,1151],{"class":1067},[1007,1455,1154],{"class":1091},[1007,1457,1157],{"class":1017},[1007,1459,1160],{"class":1084},[1007,1461,1088],{"class":1017},[1007,1463,1305],{"class":1071},[1007,1465,1168],{"class":1017},[1007,1467,1171],{"class":1091},[1007,1469,1174],{"class":1017},[1007,1471,1177],{"class":1071},[1007,1473,1180],{"class":1017},[1007,1475,1183],{"class":1091},[1007,1477,1174],{"class":1017},[1007,1479,1188],{"class":1071},[1007,1481,1180],{"class":1017},[1007,1483,1193],{"class":1098},[1007,1485,1102],{"class":1017},[1007,1487,1160],{"class":1084},[1007,1489,1088],{"class":1017},[1007,1491,1202],{"class":1091},[1007,1493,1205],{"class":1017},[1007,1495,1496],{"class":1009,"line":1148},[1007,1497,1498],{"class":1053},"    \u002F\u002F 或者使用默认实参\n",[1007,1500,1501,1503,1505,1507,1509,1511,1513,1515,1517,1519,1521,1523],{"class":1009,"line":1208},[1007,1502,1085],{"class":1084},[1007,1504,1088],{"class":1017},[1007,1506,1305],{"class":1071},[1007,1508,1168],{"class":1017},[1007,1510,1171],{"class":1091},[1007,1512,1174],{"class":1017},[1007,1514,1177],{"class":1071},[1007,1516,1180],{"class":1017},[1007,1518,1183],{"class":1091},[1007,1520,1174],{"class":1017},[1007,1522,1188],{"class":1071},[1007,1524,1525],{"class":1017},"());\n",[1007,1527,1528],{"class":1009,"line":1213},[1007,1529,1061],{"emptyLinePlaceholder":1060},[1007,1531,1532,1534,1536,1538,1540,1542,1544,1546,1548,1550],{"class":1009,"line":1240},[1007,1533,1085],{"class":1084},[1007,1535,1088],{"class":1017},[1007,1537,1220],{"class":1091},[1007,1539,1223],{"class":1017},[1007,1541,1154],{"class":1091},[1007,1543,1223],{"class":1017},[1007,1545,1160],{"class":1084},[1007,1547,1088],{"class":1017},[1007,1549,1234],{"class":1091},[1007,1551,1237],{"class":1017},[1007,1553,1555],{"class":1009,"line":1554},14,[1007,1556,1243],{"class":1017},[918,1558,1559,1560,1563],{},"此外，为了让编译器启用并行优化，还需要在参数列表中额外地传入一个提示 ",[931,1561,1562],{},"std::execution::par","：",[998,1565,1567],{"className":1000,"code":1566,"language":1002,"meta":1003,"style":1003},"#include \u003Ciostream>\n#include \u003Cvector>\n#include \u003Cnumeric> \u002F\u002F 注意：这玩意儿也不定义在 \u003Calgorithm>\n#include \u003Cexecution>\n\nint main() {\n    std::vector a{1, 2, 3, 4, 5, 6};\n\n    auto sum = std::reduce(std::execution::par, a.begin(), a.end());\n\n    std::cout \u003C\u003C sum \u003C\u003C std::endl;\n}\n",[931,1568,1569,1579,1589,1601,1612,1616,1626,1660,1664,1710,1714,1736],{"__ignoreMap":1003},[1007,1570,1571,1573,1575,1577],{"class":1009,"line":1010},[1007,1572,1014],{"class":1013},[1007,1574,1018],{"class":1017},[1007,1576,1022],{"class":1021},[1007,1578,1025],{"class":1017},[1007,1580,1581,1583,1585,1587],{"class":1009,"line":1028},[1007,1582,1014],{"class":1013},[1007,1584,1018],{"class":1017},[1007,1586,1035],{"class":1021},[1007,1588,1025],{"class":1017},[1007,1590,1591,1593,1595,1597,1599],{"class":1009,"line":1040},[1007,1592,1014],{"class":1013},[1007,1594,1018],{"class":1017},[1007,1596,1047],{"class":1021},[1007,1598,1050],{"class":1017},[1007,1600,1393],{"class":1053},[1007,1602,1603,1605,1607,1610],{"class":1009,"line":1057},[1007,1604,1014],{"class":1013},[1007,1606,1018],{"class":1017},[1007,1608,1609],{"class":1021},"execution",[1007,1611,1025],{"class":1017},[1007,1613,1614],{"class":1009,"line":1064},[1007,1615,1061],{"emptyLinePlaceholder":1060},[1007,1617,1618,1620,1622,1624],{"class":1009,"line":1081},[1007,1619,1068],{"class":1067},[1007,1621,1072],{"class":1071},[1007,1623,1075],{"class":1017},[1007,1625,1078],{"class":1017},[1007,1627,1628,1630,1632,1634,1636,1638,1640,1642,1644,1646,1648,1650,1652,1654,1656,1658],{"class":1009,"line":1131},[1007,1629,1085],{"class":1084},[1007,1631,1088],{"class":1017},[1007,1633,1092],{"class":1091},[1007,1635,1095],{"class":1017},[1007,1637,1099],{"class":1098},[1007,1639,1102],{"class":1017},[1007,1641,1105],{"class":1098},[1007,1643,1102],{"class":1017},[1007,1645,1110],{"class":1098},[1007,1647,1102],{"class":1017},[1007,1649,1115],{"class":1098},[1007,1651,1102],{"class":1017},[1007,1653,1120],{"class":1098},[1007,1655,1102],{"class":1017},[1007,1657,1125],{"class":1098},[1007,1659,1128],{"class":1017},[1007,1661,1662],{"class":1009,"line":1136},[1007,1663,1061],{"emptyLinePlaceholder":1060},[1007,1665,1666,1668,1670,1672,1674,1676,1678,1680,1683,1685,1687,1689,1692,1694,1696,1698,1700,1702,1704,1706,1708],{"class":1009,"line":1142},[1007,1667,1151],{"class":1067},[1007,1669,1154],{"class":1091},[1007,1671,1157],{"class":1017},[1007,1673,1160],{"class":1084},[1007,1675,1088],{"class":1017},[1007,1677,1305],{"class":1071},[1007,1679,1168],{"class":1017},[1007,1681,1682],{"class":1084},"std",[1007,1684,1088],{"class":1017},[1007,1686,1609],{"class":1084},[1007,1688,1088],{"class":1017},[1007,1690,1691],{"class":1091},"par",[1007,1693,1102],{"class":1017},[1007,1695,1183],{"class":1091},[1007,1697,1174],{"class":1017},[1007,1699,1177],{"class":1071},[1007,1701,1180],{"class":1017},[1007,1703,1183],{"class":1091},[1007,1705,1174],{"class":1017},[1007,1707,1188],{"class":1071},[1007,1709,1525],{"class":1017},[1007,1711,1712],{"class":1009,"line":1148},[1007,1713,1061],{"emptyLinePlaceholder":1060},[1007,1715,1716,1718,1720,1722,1724,1726,1728,1730,1732,1734],{"class":1009,"line":1208},[1007,1717,1085],{"class":1084},[1007,1719,1088],{"class":1017},[1007,1721,1220],{"class":1091},[1007,1723,1223],{"class":1017},[1007,1725,1154],{"class":1091},[1007,1727,1223],{"class":1017},[1007,1729,1160],{"class":1084},[1007,1731,1088],{"class":1017},[1007,1733,1234],{"class":1091},[1007,1735,1237],{"class":1017},[1007,1737,1738],{"class":1009,"line":1213},[1007,1739,1243],{"class":1017},[918,1741,1742,1743,1746,1747,1750],{},"这个东西的学名叫“执行策略”（Execution 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