TopCoder Statistics - Problem Statement

Problem Statement

This problem is about points in a two-dimensional plane. We will deal with L-shapes: triples of points that resemble the letter L.

Three (not necessarily distinct) points form an L-shape if we can label them A, B, and C in such a way that points A and B have the same x-coordinate, and points B and C have the same y-coordinate. For example:

The points (0, 3), (1, 9), (1, 3) form an L-shape. If we label them as C = (0, 3), A = (1, 9) and B = (1, 3), we have A.x = B.x and B.y = C.y.
The points (5, 5), (5, 5), and (5, 5) also form an L-shape.
The points (1, 9), (1, 9), and (7, 7) do not form an L-shape.

Let a step be an operation in which we change any one coordinate of any one point by 1. The L-score of a triple of points is defined as the smallest number of such steps needed to transform that triple of points into an L-shape.

You are given a set of N points in the plane: for each valid i, the coordinates of one point are (i, y[i]). No two points have same x-coordinate, and no two points have same y-coordinate. Compute and return the sum of L-scores over all N*(N-1)*(N-2)/6 triples of points.

Note that you don't actually move any of the N given points. The L-score calculations for different triples of points are completely independent.

Definition

Class:: LShape
Method:: sumL
Parameters:: int[]
Returns:: long
Method signature:: long sumL(int[] y)
(be sure your method is public)

Notes

Note the unusual time limit.

Constraints

N (the number of elements in y) will be between 3 and 3000, inclusive.
Each element in y will be between 0 and 2999, inclusive.
Elements in y will be distinct.

Examples

{4, 9, 3}

Returns: 2

We are given three points: (0, 4), (1, 9), and (2, 3). There is only one triple of points, and its L-score is 2. One way of changing this triple into an L-shape in two steps looks as follows: Move the first point from (0, 4) to (0, 3). Move the third point from (2, 3) to (1, 3). After these two steps we have the L-shape (0, 3), (1, 9), (1, 3). Another optimal strategy is to move the first point from (0, 4) to (1, 4) and then to (1, 3), yielding the L-shape (1, 3), (1, 9), (2, 3).
{0, 1, 2, 3}

Returns: 10

The given four points are (0, 0), (1, 1), (2, 2) and (3, 3). There are N*(N-1)*(N-2)/6 = 4*3*2/6 = 4 triples and their scores are 2, 3, 3, 2, so the total answer is 10. For example, a triple (0, 0), (2, 2), (3, 3) has score 3 and it's optimal to move the point (2, 2) to (0, 3) in 3 moves.
{0, 2, 1, 3}

Returns: 8

Each of four triples has score 2, so the answer is 8.
{6, 2, 3, 0, 1, 4, 7}

Returns: 107
{1400, 200, 1499, 0, 1, 500, 600, 700, 777, 1498}

Returns: 30708
{2,0,1}

Returns: 2
{9,2,4}

Returns: 3
{3,1,2,0}

Returns: 8
{862,2272,1874,44}

Returns: 2436
{28,13,0,29,21}

Returns: 78
{785,42,1375,1104,1060,1099}

Returns: 3486
{1,5,4,2,0,6,3}

Returns: 94
{549,1945,2654,2065,2397,1276,1751,1995}

Returns: 18696
{11,10,12,0,6,7,13,1,4}

Returns: 388
{28,20,0,16,17,7,21,10,2,11}

Returns: 852
{1945,851,2738,2889,2351,668,2614,288,1839,2443,262,67,17,2301}

Returns: 137203
{15,20,35,14,33,13,11,0,30,18,3,6,19,17,9,7,34,28,21}

Returns: 8568
{361,2135,1207,2817,277,568,2286,1658,357,460,1575,1815,2401,2428,411,2225,1705,2193,1680,2759,1874,2215,1420,1350,2996,1165,1985,1481,2424,382}

Returns: 1480372
{80,79,12,75,41,51,91,61,31,74,84,25,89,53,77,57,88,95,3,49,11,70,66,37,15,13,93,92,16,44,65,20,14,27,6,2,46,72,96,4,50,0,8,85,94,36,34,86,68,17,63}

Returns: 483314
{16,2907,166,624,1539,2114,1017,813,959,2173,268,2704,1255,110,1419,1462,1952,2326,2691,1692,995,1069,1854,2391,2821,1235,1789,828,2424,365,1880,2074,1463,916,1509,1968,771,691,483,1649,2320,2158,115,102,2231,1713,2218,899,1491,2056,2291,389,1640,1516,928,2272,2304,397,2256,2993,2016,145,713,1655,555,1973,141,2799,703,1513,1866,743,885,2222,1776,2297,2786,1512,1832,1138,1322,438,2969,1666,1676,1867,697,411,192,1052,1777,1934,1769,1482,826,830,2935,479,1845,2987}

Returns: 62204921
{290,1582,1321,2676,211,1022,498,967,1277,1712,1172,412,1449,2713,2808,589,319,84,2898,2292,2886,9,1224,1421,508,2708,956,12,2497,681,2234,2961,1545,537,1162,2921,306,1997,2028,2844,1584,2308,194,335,261,2265,1344,810,2445,1579,2300,356,373,670,1798,466,2372,2052,1486,733,1413,2235,1230,1532,1518,355,1534,2215,202,1051,1403,719,456,1667,2902,817,1467,2730,858,1105,576,178,1705,1879,1928,1129,2385,2434,1077,1336,835,1095,2454,2637,305,449,1742,2740,2527,839,716,531,2274,1050,878,188,812,1630,2327,2838,1694,2512,2862,2983,200,2371,2851,1103,1505,1328,2647,1343,152,2563,1844,2795,2241,2982,2757,2051,1724,2419,1886,1476,1586,962,2305,1950,2616,2645,2275,1192,1515,2717,1184,1152,2285,2084,1204,345,30,1302,2134,2067,1871,2871,778,1326,777,1882,436,874,2348,1160,1631,2463,148,315,1923,1684,377,2008,246,928,1734,2480,2803,2513,1642,582,1234,1747,2294,2203,1861,2358,900,1352,1801,1646,2696,535,400,558,1391,1790,2395,1926,709,204,2022,976,1665,2211,1102,593,195,867,945,2145,1334,2225,1131,1896,1782,2138,1819,2368,830,1173,1267,1015,2193,2570,1908,2180,2464,1962,1722,2332,2405,2195,371,1813}

Returns: 872963663
{1591,2363,1459,1685,1506,2612,2051,1577,729,1224,20,506,2069,1602,1951,523,2864,1908,402,2337,1413,2231,1779,1088,2434,2293,143,1225,101,1068,1994,347,1541,1766,1887,842,154,2800,2664,2315,1586,1989,687,2503,913,495,1512,1353,1167,133,1749,2310,2141,38,2679,994,734,805,1667,2352,1769,2671,895,1324,81,2117,418,2092,2707,2036,1636,2994,681,510,1423,756,1658,761,1263,863,1765,2811,85,2475,2110,2041,1894,1677,42,1614,1008,545,2739,72,1005,68,1233,409,1544,2176,2441,2580,1759,434,2174,2620,163,699,1981,1745,2082,2113,236,1840,2723,2888,975,2505,357,2462,1387,875,477,1277,26,1185,315,2940,1431,2794,2859,1226,1993,719,2087,605,1053,375,1316,2469,843,1905,619,747,1251,1122,2350,2642,281,1231,2123,1853,851,2635,2563,302,1923,1443,921,2819,1703,2697,343,591,775,1059,1500,2847,624,2921,276,2344,2945,52,2655,2438,1869,420,883,180,824,2155,2711,332,2909,1764,2770,1044,1709,1184,2645,644,580,1230,1132,95,2380,260,2768,2418,695,1361,1248,1968,2603,1622,2019,1927,2245,1354,1210,1635,220,746,19,445,646,2018,56,621,2584,594,1939,2079,1922,465,2978,2152,366,1397,2990,2200,1877,782,2408,177,1407,1628,2755,2883,1711,772,2949,2057,2151,1297,2546,2198,1470,1642,1474,826,790,1434,2404,2413,2677,1072,186,2497,1736,2136,1585,1550,2942,2365,1693,2427,2332,2084,1680,2706,2965,417,1463,1657,973,1962,1406,2290,1102,1744,541,2440,2132,2340,561,384,1598,1737,972,486,1727,2203,2026,1860,2832,1179,1954,388,1279,941,2722,2297,213,1930,1421,2566,2201,324,1545,2017,692,2639,1573,996,193,660,700,2234,1732,2232,869,2382,267,798,1338,244,1562,1332,1511,234,380,370,2898,1501,2790,2805,2747,953,1203,2775,1637,405,152,2196,1079,2632,789,617,2510,2323,1763,1756,1134,2672,2309,904,12,2447,2887,1391,2210,2009,2506,780,1751,1173,728,1808,2384,2409,1273,1619,2149,2347,2022,2808,848,2802,2187,720,647,2837,117,1668,241,107,849,818,583,1888,2483,1404,2271,730,2420,161,118,1298,2175,877,2972,1467,1176,1302,862,1498,529,1714,2029,1250,1395,275,1186,727,2614,2417,938,914,1382,513,604,1790,1563,2993,2222,1403,2486,894,1961,2241,688,2970,2185,2325,2574,1180,2287,673,1899,2789,1104,837,1659,2261,569,190,2705,2653,2324,2330,1806,2463,424,2772,1773,2320,2392,2217,1672,1799,48,1916,2393,1804,1938,448,5,2651,1949,2425,509,2650,432,679,1058,1089,2634,2134,1979,2013,1792,1201,774,2316,2341,261,652,1119,2047,2138,1472,1521,2708,1366,2150,2473,680,577}

Returns: 9660873278
{1221,1319,2115,1784,520,2136,1052,788,927,2029,639,1814,1409,638,361,731,855,1770,946,413,214,637,1001,301,1153,1632,1986,2118,601,2170,1210,1349,1896,1132,1012,2007,728,1621,304,1889,1198,396,2096,1837,548,2120,69,1191,695,2042,1201,267,381,854,1647,470,1615,1549,921,1366,2217,814,401,1819,1172,1551,404,1624,1924,1798,852,2013,2150,2094,1808,1878,1597,726,14,230,1898,358,2198,2238,1721,2091,1673,1408,2206,1411,472,1060,781,1053,724,1237,581,101,226,372,1442,83,1176,1742,276,1296,147,1775,218,1935,1163,2104,1623,67,270,1121,2095,1187,1746,1848,1840,1575,874,1120,2044,1125,1277,896,582,729,265,2020,1695,181,1829,1244,947,1918,1301,1488,2105,1656,1963,20,1853,1894,1486,1753,799,1744,1110,1239,681,771,2015,2283,860,1705,220,308,1284,255,1476,2012,128,1801,1983,599,1809,1007,913,373,608,5,1913,1357,1948,2112,611,736,1368,610,1485,609,727,1250,1400,1361,1500,1600,671,2063,240,703,1559,1214,933,1791,1014,948,934,1541,2088,436,1028,161,104,168,2274,969,44,1663,2008,935,1723,1696,687,2099,1359,366,26,701,88,764,455,443,481,2207,1658,291,1579,136,974,1467,1392,452,958,1558,1996,1144,334,2107,1080,261,167,938,353,1175,1207,142,1493,1047,250,1287,519,395,319,1455,529,93,1883,325,543,1630,249,129,999,196,1117,1328,2010,1038,407,700,293,279,259,632,2162,878,204,1802,1533,405,1755,176,2023,290,1129,1403,269,1472,906,1565,474,2130,1990,414,71,1222,1645,766,1027,1134,1109,522,2061,1716,307,164,1855,1818,2116,85,332,1431,926,1090,1631,779,1625,941,789,491,656,1257,347,2281,1218,1620,1516,1882,963,415,210,2237,833,1599,70,1420,531,1302,1944,1276,1937,1341,1143,826,800,76,587,1981,1861,1435,1283,1505,517,1576,1955,213,676,1489,1326,1229,387,1183,1072,1475,497,1367,359,1279,1806,1249,1358,92,1469,1810,780,224,732,2079,1628,1224,1397,1920,816,1545,1727,1252,956,53,1648,1139,2081,419,668,1364,469,616,784,649,758,1953,1641,145,86,880,570,1003,897,1417,2047,821,428,1614,2167,1661,110,239,672,2224,155,859,2180,998,1644,650,73,1209,1601,124,266,1057,1108,1704,1304,1833,281,87,1329,1234,1928,557,1965,1649,542,918,2066,2024,563,47,942,810,1790,496,2203,1976,1089,337,1796,1333,663,438,1609,1313,665,1968,2075,1353,705,886,513,1232,1035,1506,2216,1118,1880,1308,1969,1264,723,615,1714,394,757,577,1438,1230,383,435,2189,1414,467,1781,1767,1192,1255,1895,1305,412,2134,1030,2215,1685,1227,959,1288,80,2228,391,2132,1554,986,1907,1292,747,621,2214,1973,501,785,1852,360,274,1360,533,690,1879,1887,813,25,1352,1792,1282,2110,117,798,322,2089,1550,525,805,838,1295,493,144,1689,1184,643,1724,765,427,187,2208,790,1740,850,1612,81,492,546,1037,922,112,953,1534,549,1687,741,1604,930,661,911,444,162,993,429,845,2246,1569,1238,488,1104,2129,2082,1863,1136,212,742,1067,475,399,692,1128,822,514,1917,836,889,662,905,796,2249,1269,335,1483,1240,1839,1487,1987,1849,2027,688,964,883,345,590,2048,629,1771,235,655,2159,1381,1691,2039,647,2230,1919,1088,2240,992,2,105,424,2272,641,1034,1841,1421,2239,867,1375,985,131,2069,158,2193,804,2103,2269,1165,979,560,1754,15,1881,227,512,1504,408,1540,356,1961,1502,439,1095,1491,635,1535,754,1271,630,990,871,2122,135,1365,950,1446,1407,997,398,776,1334,984,313,449,1859,2045,1555,1646,1484,1054,1205,425,2019,1180,51,1950,2017,2235,29,565,1154,1024,1494,477,2016,1194,720,1577,280,847,2220,2218,1888,1199,793,526,1995,1092,1457,1870,1019,1557,55,1571,640,1762,1078,537,1433,1653,9,713,2100,2197,1219,1522,338,2090,2126,1141,1836,711,2028,1465,223,1312,371,97,126,82,1773,1564,738,451,440,1650,1524,211,46,1051,2222,2128,774,1441,1044,2196,2101,564,2001,1026,1374,566,1275,1116,2163,1606,148,1954,909,2058,524,2187,273,1815,315,1710,667,416,2195,1429,806,1940,193,1677,594,1004,2030,1525,2227,1688,1532,1167,2226,592,1768,532,1916,10,2265,2229,653,1280,660,1371,1496,1884,1899,1091,2250,1391,1607,349,1470,2188,919,1127,137,1914,1835,2205,134,2113,468,1131,1761,1258,1303,722,344,2186,669,981,1638,664,936,1949,99,1971,1951,1566,406,1297,1006,484,716,1082,823,1196,1309,1927,2199,527,1378,423,1854,63,206,1138,1694,1521,673,2086,121,817,578,869,770,916,1466,1036,1543,613,1622,339,842,320,2083,138,907,1588,1162,2169,192,1123,1752,541,1905,686,299,500,2131,7,1016,42,1748,704,737,1776,1530,1967,1477,2279,831,1185,2142,246,377,1709,185,910,623,767,1145,329,538,1635,521,1388,544,768,2177,1759,65,393,106,1711,1115,951,1263,960,228,382,1009,1464,1085,125,2041,2185,457,2157,1778,324,2076,237,348,902,1398,2127,2003,364,1041,658,848,186,199,2183,1921,1561,675,955,1758,744,2255,378,1511,719,1640,1795,983,292,402,2263,1593,929,1100,1616,2143,756,23,2106,1432,389,12,2270,2051,1473,115,2252,2135,1766,256,735,775,420,680,2223,1444,631,1877,588,2035,1501,445,1639,1803,1048,1980,1821,1507,177,1657,819,1730,1135,2055,839,1741,706,1846,489,1731,1443,2148,1389,1426,2026,1729,1804,2209,1834,1822,1479,1925,2251,628,36,1717,1311,1188,271,857,2098,971,900,1964,108,2166,40,403,965,1637,450,1747,1346,1298,1029,1573,264,1979,1482,2160,2200,64,1324,828,3,478,1265,2184,2072,996,367,1659,113,949,551,1860,236,1399,323,1156,1226,1251,2050,620,2244,257,244,483,648,1418,1636,830,829,1936,21,1174,561,1061,1574,130,6,1402,1634,152,1932,614,1531,1749,1706,2151,418,410,1285,811,1099,1763,1140,384,2046,751,156,740,195,1512,559,2123,580,818,2262,222,1412,893,1743,2033,2204,1605,8,1866,200,123,2271,2121,1515,2053,957,1171,2139,1059,1193,1286,1362,286,2114,120,2175,480,2074,198,19,534,351,864,56,1537,574,1947,2267,146,1461,989,939,1437,1217,2070,1197,473,490,1966,721,1675,1262,1033,604,576,1934,1582,1539,1901,1520,446,1526,1666,890,511,1307,495,1289,1674,247,1405,1273,2144,1377,1166}

Returns: 155399717644
{197,595,2077,2110,2405,2111,53,2874,198,2770,2461,1589,158,206,906,604,1476,1784,2911,775,4,1928,2042,2057,535,1042,2698,1674,2731,1501,1183,2641,109,195,2658,2654,1791,1845,2151,2112,1173,2396,1163,981,870,2429,611,2819,1634,2230,2869,2788,262,636,2350,1947,1788,2608,1595,2436,114,374,2036,2184,2066,2421,1152,2228,1156,695,660,2447,1904,2137,935,1378,365,142,1837,929,2695,2647,2054,482,2848,2227,73,815,2330,1321,0,1702,2211,1129,1799,215,2919,1106,1559,2220,428,381,389,366,740,955,483,2765,2672,2095,1803,1250,2520,1575,182,2910,2427,475,2309,2609,269,462,2601,2947,2311,863,744,590,2454,1420,2375,86,59,192,173,878,2559,388,1526,2873,865,1666,2886,1808,2294,2267,2591,1436,1752,667,1522,1717,275,712,2012,1935,1437,370,2760,1409,1794,2031,1755,516,210,2665,1165,1658,717,2399,774,1582,2361,2337,1018,1153,363,2605,2984,2256,2316,1656,2670,18,1866,817,630,1498,1962,125,688,8,2617,2588,2810,664,2498,2046,1258,556,2782,1109,912,1185,443,1157,2808,140,406,2655,599,1038,721,1508,2389,1360,1386,2907,156,2972,1855,1827,1713,1804,486,2414,457,1635,2579,1426,2045,514,1644,778,2360,2512,1489,202,2250,273,1288,247,1672,214,2529,620,1167,672,691,1337,399,2562,412,1663,1623,2206,291,2974,1231,165,1234,2080,511,2023,1089,1775,144,933,1535,713,824,1218,1533,781,869,1551,2828,578,52,2353,403,850,1188,1801,2273,2174,1189,1470,7,2682,2295,1041,2006,1714,1302,2085,152,317,383,2928,2312,1391,1227,450,2729,2542,34,1402,2062,1338,1746,2667,2478,407,1727,841,984,1012,2768,1653,2357,1934,1430,889,2170,1124,1044,2688,832,665,2417,1270,1453,2208,2742,1300,2192,1874,1407,119,909,495,1729,2221,1381,1955,582,1141,2965,2164,256,110,5,2168,1857,757,2825,978,2341,2989,1675,1119,1146,139,2264,442,567,2916,1695,2385,1384,800,484,750,2015,2611,1123,2899,1240,1483,887,1976,429,2277,2875,997,1754,94,2552,541,2727,2138,504,510,2532,1015,47,2986,1834,1001,1046,2108,761,155,2897,2022,1991,2282,2568,1920,2924,218,833,2515,2558,2867,1236,2691,542,2800,566,2666,1851,1688,2381,334,1161,2437,1511,2598,2416,854,1035,798,908,2556,1554,2140,223,1886,2741,2853,2593,2160,1558,1965,606,2737,1058,844,446,1610,337,2508,2968,1966,2269,2521,1390,1083,320,1484,244,2098,942,1072,130,2933,693,2289,2387,2474,2577,1181,666,2119,1396,2044,1940,673,430,1190,2726,2317,2442,2299,2596,1203,1361,213,1751,1340,835,974,1764,651,820,1620,813,526,914,2234,2223,2005,277,251,423,2047,451,2567,2284,1369,2153,1776,899,826,1114,2490,1900,1070,1858,571,1703,1330,608,720,1467,1811,1862,1443,855,2513,1175,2669,1869,127,1512,2450,1979,485,1322,2649,414,537,1269,1645,2578,964,2866,1474,2651,1757,1186,1574,2068,174,2479,1341,2944,905,2872,233,2724,634,650,2612,2448,2238,2856,99,2535,2232,799,338,2566,164,2958,2941,1912,533,112,748,2514,212,1100,926,161,2456,1200,216,706,2978,2850,2285,936,1241,1908,1145,1291,513,2446,631,448,498,2349,240,2597,2262,2797,1985,2011,714,987,700,1131,1638,2420,1882,2214,2996,2821,2412,2278,2481,417,2977,1376,103,970,1427,1697,2710,2876,2152,2495,621,1918,790,1942,2801,2195,1871,2730,1876,1252,2038,2322,1405,225,990,2586,1669,1285,1879,201,1333,60,893,2631,208,2689,1684,2557,2124,1692,805,818,591,2205,2502,1395,581,453,1401,2718,683,871,143,658,299,2408,980,2822,219,2781,724,1743,1032,1004,2704,839,2717,349,1383,356,675,2945,1844,1134,2072,1591,2133,2145,2093,2985,891,2891,2477,2313,2602,554,2247,782,674,2970,1593,1741,2171,1599,372,1367,1256,288,960,2019,550,2325,1774,1676,2721,1971,6,1438,847,2165,343,2397,678,2523,1206,452,368,84,1481,2871,1233,393,1249,2268,857,1579,2912,341,36,786,881,21,391,1922,2014,1416,2900,488,2713,466,2440,884,1411,159,1075,2522,2756,1690,1138,189,1194,1792,725,2016,2279,1150,1766,1651,2026,340,1975,464,45,1490,361,1253,2271,601,2993,22,2,369,2638,548,1961,2320,763,2348,193,930,1069,1957,90,2132,41,2218,1967,1767,1033,92,2187,2438,331,722,586,296,2185,148,1488,2365,2680,1084,2143,2837,2683,540,1277,2486,1868,38,44,2245,1605,1388,1652,1683,147,410,1704,1007,633,1631,1896,1419,1640,1531,455,301,596,2902,1768,2676,281,325,1497,671,420,2482,529,2321,2957,618,438,2163,2975,2517,977,2835,1354,2534,1352,354,610,2100,2394,1191,1102,2527,1701,1329,655,2784,239,859,530,1442,497,597,698,2048,1573,2842,2335,1353,2287,1514,2500,2799,2407,133,227,2157,2990,2003,2892,2870,1424,2554,1485,108,1472,56,120,1397,2203,2509,2592,1748,360,2240,885,505,2142,1549,898,1960,3,2049,2518,1515,1732,136,2531,1148,2131,1541,1553,1475,1877,1759,1734,1274,2511,924,1158,2473,315,2759,2750,1406,985,2525,2864,1178,1863,1399,2926,203,1733,246,359,1006,1030,292,2662,1563,1062,1909,2940,2101,2051,2359,1245,1479,81,2785,1532,1389,1504,506,1494,2419,2286,2460,809,1071,1617,2383,681,993,927,2209,1088,2696,1404,2148,1847,738,2908,371,1607,335,543,2755,992,228,1687,1370,2485,2021,1584,2528,1943,577,1373,2548,2475,477,2356,1951,2182,2679,1809,221,1603,625,2492,1893,2636,966,2172,524,2251,2963,2159,1848,63,1155,1486,1014,2956,2435,1142,1796,1614,2879,760,441,2795,2074,2274,80,1523,1228,546,895,2839,951,1066,1339,1829,792,23,2125,551,2629,71,1937,2896,1219,1063,536,772,853,2831,1760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Returns: 369449442757
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Returns: 816477777014
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Returns: 1843354165709
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Returns: 2263487175759
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Returns: 3157850238634
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Returns: 3286732467673
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Returns: 3540462104631
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Returns: 3610668471048
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Returns: 3850957245562
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Returns: 4005942010014
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Returns: 4006947321988
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Returns: 4010845784077
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Returns: 4017529912269
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Returns: 4020857982314
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Returns: 6745499250500
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Returns: 6745499250500
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Returns: 6745499250500

Statistics

Problem Statement for "LShape"

Problem Statement

Definition

Notes

Constraints

Examples