Statistics

Problem Statement for "CheeseRolling"

Problem Statement

N people (where N is a power of 2) are taking part in a single-elimination tournament in cheese rolling. The diagram below illustrates the structure of the tournament bracket.



The people entering the tournament are numbered from 0 to N-1. For each potential cheese rolling match you know who would win the match. You are given this information encoded as a String[] wins with N elements, each containing N characters. For each valid i and j, wins[i][j] is 'Y' if person i beats person j. Otherwise, wins[i][j] is 'N'. The relation is not necessarily transitive: it may be the case that person i beats person j, person j beats person k, and person k beats person i.


There are N! (N factorial) ways to assign the people to positions in the bracket. Different assignments may produce a different winner of the tournament. Return a long[] with N elements. For each valid i, element i of the return value should be the exact number of assignments for which person i wins the tournament.

Definition

Class:
CheeseRolling
Method:
waysToWin
Parameters:
String[]
Returns:
long[]
Method signature:
long[] waysToWin(String[] wins)
(be sure your method is public)

Constraints

  • N will be between 2 and 16, inclusive.
  • N will be a power of 2.
  • wins will contain exactly N elements.
  • Each element of wins will have a length of exactly N.
  • Each element of wins will be composed of the characters 'Y' and 'N'.
  • For each i from 0 to N-1, wins[i][i] = 'N'.
  • For all distinct integers i and j from 0 to N-1, exactly one of wins[i][j] and wins[j][i] will be 'Y'.

Examples

  1. {"NN", "YN"}

    Returns: {0, 2 }

    There are 2 ways to assign the players: Player 0 goes to position 0 and player 1 goes to position 1. Player 1 goes to position 0 and player 0 goes to position 1. In both assignments, player 1 will win the match against player 0 because wins[1][0] = 'Y'.

  2. {"NYNY", "NNYN", "YNNY", "NYNN"}

    Returns: {8, 0, 16, 0 }

  3. {"NYNYNYNY", "NNYNYNYY", "YNNNNNNN", "NYYNNYNY", "YNYYNYYY", "NYYNNNNN", "YNYYNYNN", "NNYNNYYN"}

    Returns: {4096, 8960, 0, 2048, 23808, 0, 1408, 0 }

  4. {"NYNNNNYYNYYNNYNN", "NNNNNNNNNYYNYYNY", "YYNYYNNNNYYYYYYN", "YYNNYYYNYNNYYYNY", "YYNNNYYNYYNNNNYY", "YYYNNNNYYNNYYNYN", "NYYNNYNYNYNYYYYN", "NYYYYNNNYYNYNYYY", "YYYNNNYNNYYYYNNN", "NNNYNYNNNNNNYYNY", "NNNYYYYYNYNYYYNN", "YYNNYNNNNYNNYNNY", "YNNNYNNYNNNNNYNN", "NNNNYYNNYNNYNNYY", "YYNYNNNNYYYYYNNN", "YNYNNYYNYNYNYNYN"}

    Returns: {331616878592, 37267079168, 2426798866432, 2606831599616, 994941665280, 1162501849088, 1888166674432, 4632734203904, 832881524736, 84707409920, 3007127748608, 55490052096, 17818550272, 254672666624, 629921447936, 1959311671296 }

  5. {"NYYYYYYYYYYYYYYY", "NNYYYYYYYYYYYYYY", "NNNYYYYYYYYYYYYY", "NNNNYYYYYYYYYYYY", "NNNNNYYYYYYYYYYY", "NNNNNNYYYYYYYYYY", "NNNNNNNYYYYYYYYY", "NNNNNNNNYYYYYYYY", "NNNNNNNNNYYYYYYY", "NNNNNNNNNNYYYYYY", "NNNNNNNNNNNYYYYY", "NNNNNNNNNNNNYYYY", "NNNNNNNNNNNNNYYY", "NNNNNNNNNNNNNNYY", "NNNNNNNNNNNNNNNY", "NNNNNNNNNNNNNNNN"}

    Returns: {20922789888000, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 }

    Player 0 wins no matter how the positions are assigned, so the answer is 16! = 20922789888000.

  6. {"NYNY","NNNY","YYNN","NNYN"}

    Returns: {8, 0, 16, 0 }

  7. {"NNYN","YNNN","NYNN","YYYN"}

    Returns: {0, 0, 0, 24 }

  8. {"NNNY","YNNY","YYNN","NNYN"}

    Returns: {0, 8, 16, 0 }

  9. {"NNNN","YNNN","YYNY","YYNN"}

    Returns: {0, 0, 24, 0 }

  10. {"NNNNNYNN","YNYNYNNY","YNNYNNNN","YYNNNNYY","YNYYNYYY","NYYYNNNN","YYYNNYNN","YNYNNYYN"}

    Returns: {0, 6912, 0, 4480, 24576, 1408, 2048, 896 }

  11. {"NNNYYYYY","YNNNNNYN","YYNNYNNN","NYYNYYNY","NYNNNNYY","NYYNYNYN","NNYYNNNY","NYYNNYNN"}

    Returns: {22656, 0, 1408, 10240, 384, 2432, 2176, 1024 }

  12. {"NYNYNYNY","NNYNYNYY","YNNNNNNN","NYYNNYNY","YNYYNYYY","NYYNNNNN","YNYYNYNN","NNYNNYYN"}

    Returns: {4096, 8960, 0, 2048, 23808, 0, 1408, 0 }

  13. {"NYNYYNYY","NNYNYYYY","YNNYYNNY","NYNNYNYY","NNNNNYYY","YNYYNNNN","NNYNNYNY","NNNNNYNN"}

    Returns: {14720, 9088, 7552, 3328, 256, 4864, 512, 0 }

  14. {"NNNNNNYNYYNNNYYN","YNYNYYNYYNYNYNNY","YNNNYNNYYNYYNNYY","YYYNNYYYYYNYNNYY","YNNYNNYYNYYYYYYY","YNYNYNYNYYYYYNNN","NYYNNNNNYYNYYNYN","YNNNNYYNNYNYYNNY","NNNNYNNYNNYNNYNY","NYYNNNNNYNYNYNNY","YNNYNNYYNNNNNYYN","YYNNNNNNYYYNYYYN","YNYYNNNNYNYNNNNN","NYYYNYYYNYNNYNNN","NYNNNYNYYYNNYYNN","YNNNNYYNNNYYYYYN"}

    Returns: {30040588288, 2170413056000, 1032712028160, 5360170696704, 6088356921344, 1478110412800, 287463833600, 214525149184, 181157625856, 144355753984, 281618874368, 469078540288, 65955627008, 2059355881472, 435793723392, 623681175552 }

  15. {"NYNNNYYNYYNYYYYN","NNNYYYNYYYYNYNYN","YYNYNYYNNYYYNYYN","YNNNNYYYYNYNYNYN","YNYYNNYYNYYNNNNN","NNNNYNNYYYNYYYNN","NYNNNYNNNNYNNNNY","YNYNNNYNYNNYYYYY","NNYNYNYNNNNYYNNY","NNNYNNYYYNYNNNNN","YNNNNYNYYNNNNYYY","NYNYYNYNNYYNNYYN","NNYNYNYNNYYYNNYY","NYNYYNYNYYNNYNYN","NNNNYYYNYYNNNNNN","YYYYYYNNNYNYNYYN"}

    Returns: {1077396144128, 1754904461312, 2746923646976, 1000729870336, 820680032256, 572898705408, 41721626624, 3640348213248, 508378578944, 66093121536, 1050268565504, 553727066112, 1336769675264, 572694822912, 15265202176, 5163990155264 }

  16. {"NNYYNYNYYYNYNYYN","YNNNYYNNYNYYNNNN","NYNNNNYNYNYYYNNN","NYYNYYNYNYYNYNNN","YNYNNNNNNNNYYYNY","NNYNYNNYYNNNNYNN","YYNYYYNYNYNNYYYY","NYYNYNNNNNYNNNYY","NNNYYNYYNNNYNNYN","NYYNYYNYYNYNNYYY","YNNNYYYNYNNNYYNY","NNNYNYYYNYYNNNYY","YYNNNYNYYYNYNNYN","NYYYNNNYYNNYYNYN","NYYYYYNNNNYNNNNN","YYYYNYNNYNNNYYYN"}

    Returns: {1667478618112, 233966403584, 530295488512, 824265048064, 443519074304, 23755522048, 6939609366528, 137287663616, 387530915840, 1956016455680, 2303021285376, 2571640471552, 889922322432, 632551800832, 105898541056, 1276030910464 }

  17. {"NYNNNNYYNYYNNYNN","NNNNNNNNNYYNYYNY","YYNYYNNNNYYYYYYN","YYNNYYYNYNNYYYNY","YYNNNYYNYYNNNNYY","YYYNNNNYYNNYYNYN","NYYNNYNYNYNYYYYN","NYYYYNNNYYNYNYYY","YYYNNNYNNYYYYNNN","NNNYNYNNNNNNYYNY","NNNYYYYYNYNYYYNN","YYNNYNNNNYNNYNNY","YNNNYNNYNNNNNYNN","NNNNYYNNYNNYNNYY","YYNYNNNNYYYYYNNN","YNYNNYYNYNYNYNYN"}

    Returns: {331616878592, 37267079168, 2426798866432, 2606831599616, 994941665280, 1162501849088, 1888166674432, 4632734203904, 832881524736, 84707409920, 3007127748608, 55490052096, 17818550272, 254672666624, 629921447936, 1959311671296 }

  18. {"NYNYNYNNYNYYYNNY","NNNNYYYYNYNNYNYY","YYNYYYNNNYNNYYYY","NYNNNNYYNNNYYYNN","YNNYNNNNNYYNNYYN","NNNYYNYNYYYNNYYN","YNYNYNNNYNNNNNYN","YNYNYYYNYYYNYYYY","NYYYYNNNNYNYYNNN","YNNYNNYNNNYYYNYN","NYYYNNYNYNNYNNNN","NYYNYYYYNNNNYNNN","NNNNYYYNNNYNNYNY","YYNNNNYNYYYYNNNN","YNNYNNNNYNYYYYNY","NNNYYYYNYYYYNYNN"}

    Returns: {885134983168, 1890747416576, 2865795432448, 401523703808, 109519568896, 325261099008, 89119260672, 9766860488704, 565816328192, 261356584960, 260551639040, 1506438381568, 73463922688, 368376709120, 728593498112, 824230871040 }

  19. {"NYYNNNNNYNYNYNNN","NNYYYNNYYNYNYNNY","NNNNNYNYYYNYNNYY","YNYNYYYNYNYYYYNY","YNYNNNYNYYNYYNNY","YYNNYNYNYNNYNNYN","YYYNNNNNYYYYNNYY","YNNYYYYNNYNNNNYY","NNNNNNNYNNNNYYYY","YYNYNYNNYNNNNNNY","NNYNYYNYYYNNNNNN","YYNNNNNYYYYNNNYN","NNYNNYYYNYYYNYYN","YYYNYYYYNYYYNNYY","YYNYYNNNNYYNNNNY","YNNNNYNNNNYYYNNN"}

    Returns: {49631068160, 1655057580032, 133099814912, 7316190691328, 415300354048, 258408251392, 443880603648, 1016443240448, 217264652288, 201121333248, 83421855744, 170946330624, 2934036103168, 5560120737792, 440022990848, 27844280320 }

  20. {"NNNNYNNYYNYYNNYY","YNYYNYNNNYNNYNNY","YNNYYYNNYNNNYNNN","YNNNNYYYYNNNYNNY","NYNYNNNNYNYYYNNY","YNNNYNNNNNNYYYNY","YYYNYYNNYNNNYYYY","NYYNYYYNNNNNNYNN","NYNNNYNYNYNNNYNY","YNYYYYYYNNYYNNYN","NYYYNYYYYNNYNNNY","NYYYNNYYYNNNNNYN","YNNNNNNYYYYYNNYN","YYYYYNNNNYYYYNYN","NYYYYYNYYNYNNNNY","NNYNNNNYNYNYYYNN"}

    Returns: {460724862976, 441999949824, 97390657536, 463713337344, 401143889920, 294866780160, 3886726414336, 413830643712, 371627622400, 4278679240704, 1388528435200, 527135735808, 932294852608, 5396181647360, 1061146296320, 506799521792 }

  21. {"NYNY","NNNN","YYNY","NYNN"}

    Returns: {0, 0, 24, 0 }

  22. {"NYYYYNYY","NNNNNNYN","NYNYYNYY","NYNNYNYY","NYNNNNYN","YYYYYNYY","NNNNNNNN","NYNNYNYN"}

    Returns: {0, 0, 0, 0, 0, 40320, 0, 0 }

  23. {"NNNNNNNNNNNNNNNN","YNNNNNYYYNYNYNNY","YYNNNNYYYNYYYNNY","YYYNYYYYYNYYYNNY","YYYNNYYYYNYYYNNY","YYYNNNYYYNYYYNNY","YNNNNNNNNNYNNNNN","YNNNNNYNYNYNNNNN","YNNNNNYNNNYNNNNN","YYYYYYYYYNYYYNNY","YNNNNNNNNNNNNNNN","YYNNNNYYYNYNYNNY","YNNNNNYYYNYNNNNN","YYYYYYYYYYYYYNNY","YYYYYYYYYYYYYYNY","YNNNNNYYYNYNYNNN"}

    Returns: {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20922789888000, 0 }

  24. {"NN","YN"}

    Returns: {0, 2 }

  25. {"NNNNYNNN","YNYNNYYN","YNNNNYYN","YYYNNYNN","NYYYNNNN","YNNNYNYN","YNNYYNNN","YYYYYYYN"}

    Returns: {0, 0, 0, 0, 0, 0, 0, 40320 }

  26. {"NNNYNYYNNYNYYYNN","YNNYYNYNNNNYNYNN","YYNNNYNYNNNYYYNN","NNYNYYYYNYNYYNNN","YNYNNYNYNYNYYYNN","NYNNNNYNNYNYNNNN","NNYNYNNYNNYYYNYN","YYNNNYNNYNNNYYYN","YYYYYYYNNYYYYNNN","NYYNNNYYNNNYNNYN","YYYYYYNYNYNNYYNN","NNNNNNNYNNYNYYNN","NYNNNYNNNYNNNYYN","NNNYNYYNYYNNNNNN","YYYYYYNNYNYYNYNN","YYYYYYYYYYYYYYYN"}

    Returns: {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20922789888000 }

  27. {"NYYYYYYYYYYYYYYY", "NNYYYYYYYYYYYYYY", "NNNYYYYYYYYYYYYY", "NNNNYYYYYYYYYYYY", "NNNNNYYYYYYYYYYY", "NNNNNNYYYYYYYYYY", "NNNNNNNYYYYYYYYY", "NNNNNNNNYYYYYYYY", "NNNNNNNNNYYYYYYY", "NNNNNNNNNNYYYYYY", "NNNNNNNNNNNYYYYY", "NNNNNNNNNNNNYYYY", "NNNNNNNNNNNNNYYY", "NNNNNNNNNNNNNNYY", "NNNNNNNNNNNNNNNY", "NNNNNNNNNNNNNNNN" }

    Returns: {20922789888000, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 }

  28. {"NYNY", "NNYN", "YNNY", "NYNN" }

    Returns: {8, 0, 16, 0 }

  29. {"NYNNNNYYNYYNNYNN", "NNNNNNNNNYYNYYNY", "YYNYYNNNNYYYYYYN", "YYNNYYYNYNNYYYNY", "YYNNNYYNYYNNNNYY", "YYYNNNNYYNNYYNYN", "NYYNNYNYNYNYYYYN", "NYYYYNNNYYNYNYYY", "YYYNNNYNNYYYYNNN", "NNNYNYNNNNNNYYNY", "NNNYYYYYNYNYYYNN", "YYNNYNNNNYNNYNNY", "YNNNYNNYNNNNNYNN", "NNNNYYNNYNNYNNYY", "YYNYNNNNYYYYYNNN", "YNYNNYYNYNYNYNYN" }

    Returns: {331616878592, 37267079168, 2426798866432, 2606831599616, 994941665280, 1162501849088, 1888166674432, 4632734203904, 832881524736, 84707409920, 3007127748608, 55490052096, 17818550272, 254672666624, 629921447936, 1959311671296 }

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