【笔记】TAOCP Vol4

组合个数

/// <summary>
/// <paramref name="n"/>个元素集合中所有<paramref name="m"/>个元素组合的个数。
/// <paramref name="m"/> &gt;= 0; <paramref name="n"/> &gt;= 0
/// </summary>
public static int BinomialCoefficient(int n, int m)
{if (n < 0)throw new ArgumentOutOfRangeException("n should be nonegative");if (m < 0)throw new ArgumentOutOfRangeException("m should be nonegative");if (n < m) return 0;if (n == m) return 1;if (m == 0) return 1;return BinomialCoefficient(n - 1, m - 1) + BinomialCoefficient(n - 1, m);
}

Lexicographic(字典顺序)

给定两个整数n, m(1 < m < n)，升序返回int[]集合，数组里的数字代表索引。譬如n = 6, m = 3时，返回

210,310,320,321,410,420,421,430,431,432,510,520,521,530,531,532,540,541,542,543

对于任意n个元素的集合，根据索引选取对应元素就可以得到该集合的所有m个元素组合。

/// <summary>
/// <paramref name="n"/>个元素集合中所有<paramref name="m"/>个元素组合。
/// </summary>
public static IEnumerable<int[]> Lexicographic(int n, int m)
{if (n < 2)throw new ArgumentOutOfRangeException("n shouble be greater than 1");if (m < 1 || m > n)throw new ArgumentOutOfRangeException("m shouble be positive and not greater than " + n);var pos = new int[m + 3];pos[m + 1] = n;for (int i = 1; i <= m; ++i)pos[i] = i - 1;var result = new List<int[]>();while (true){var combination = pos.Skip(1).Take(m).Reverse().ToArray();result.Add(combination);var index = 1;while (pos[index] + 1 == pos[index + 1]){pos[index] = index - 1;++index;}if (index > m) break;++pos[index];}return result;
}/// <summary>
/// <paramref name="n"/>个元素集合中所有<paramref name="m"/>个元素组合。
/// </summary>
public static IEnumerable<int[]> FastLexicographic(int n, int m)
{if (n < 2)throw new ArgumentOutOfRangeException("n shouble be greater than 1");if (m < 1 || m > n)throw new ArgumentOutOfRangeException("m shouble be positive and not greater than " + n);var pos = new int[m + 3];pos[m + 1] = n;for (int i = 1; i <= m; ++i)pos[i] = i - 1;var result = new List<int[]>();var index = m;while (true){var combination = pos.Skip(1).Take(m).Reverse().ToArray();result.Add(combination);int tmp;if (index > 0)tmp = index;else{if (pos[1] + 1 < pos[2]){++pos[1];continue;}elseindex = 2;while (true){pos[index - 1] = index - 2;tmp = pos[index] + 1;if (tmp != pos[index + 1]) break;++index;}if (index > m) break;}pos[index] = tmp;--index;}return result;
}/// <summary>
/// <paramref name="n"/>个元素集合中所有<paramref name="m"/>个元素组合。
/// </summary>
public static IEnumerable<int[]> DecreasingLexicographic(int n, int m)
{if (n < 2)throw new ArgumentOutOfRangeException("n shouble be greater than 1");if (m < 1 || m > n)throw new ArgumentOutOfRangeException("m shouble be positive and not greater than " + n);var pos = new int[m + 1];for (int i = 1; i < pos.Length; ++i)pos[i] = i + n - m - 1;var result = new List<int[]>();var index = 1;while (true){var combination = pos.Skip(1).Take(m).Reverse().ToArray();result.Add(combination);if (index > m) break;--pos[index];if (pos[index] < index){++index;continue;}while (index > 1){pos[index - 1] = pos[index] - 1;--index;}}return result;
}

Revolving-Door

n = 6, m = 3时返回

210,320,321,310,430,431,432,420,421,410,540,541,542,543,530,531,532,520,521,510

/// <summary>
/// <paramref name="n"/>个元素集合中所有<paramref name="m"/>个元素组合。
/// </summary>
public static IEnumerable<int[]> RevolvingDoor(int n, int m)
{if (n < 2)throw new ArgumentOutOfRangeException("n shouble be greater than 1");if (m < 1 || m > n)throw new ArgumentOutOfRangeException("m shouble be positive and not greater than " + n);var pos = new int[m + 3];pos[m + 1] = n;for (int i = 1; i <= m; ++i)pos[i] = i - 1;var result = new List<int[]>();var flag = true;while (flag){var combination = pos.Skip(1).Take(m).Reverse().ToArray();result.Add(combination);var condition = false;int index;if (IsEven(m)){if (pos[1] > 0){--pos[1];continue;}else{index = 2;condition = true;}}else{if (pos[1] + 1 < pos[2]){++pos[1];continue;}elseindex = 2;}while (true){if (!condition){if (pos[index] >= index){pos[index] = pos[index - 1];pos[index - 1] = index - 2;break;}else++index;}elsecondition = false;if (pos[index] + 1 < pos[index + 1]){pos[index - 1] = pos[index];++pos[index];break;}else if (++index > m){flag = false;break;}}}return result;
}

Chase Sequence(这几个方法中最好的组合方法)

/// <summary>
/// <paramref name="n"/>个元素集合中所有<paramref name="m"/>个元素组合。
/// </summary>
public static IEnumerable<int[]> ChaseSequence(int n, int m)
{if (n < 2)throw new ArgumentOutOfRangeException("n shouble be greater than 1");if (m < 1 || m > n)throw new ArgumentOutOfRangeException("m shouble be positive and not greater than " + n);var pos = new int[n];for (var i = n - m; i < pos.Length; ++i)pos[i] = 1;var aux = new int[n + 1];Array.Fill(aux, 1);var result = new List<int[]>();var tmp = n - m;while (true){var combination = new int[m];for (int i = 0, j = 0; i < pos.Length; ++i)if (pos[i] == 1)combination[j++] = i;result.Add(combination);var index = tmp;if (aux[index] == 0)while (aux[index] != 1)aux[index++] = 1;if (index == n) break;aux[index] = 0;if (pos[index] == 0){if (!IsEven(index) && pos[index - 1] == 0){pos[index] = 1;pos[index - 2] = 0;if (tmp == index - 2)tmp = index;else if (tmp == index - 1)tmp = index - 2;}else{pos[index] = 1;pos[index - 1] = 0;if (tmp == index && index > 1)tmp = index - 1;else if (tmp == index - 1)tmp = index;}}else{if (IsEven(index) && pos[index - 2] == 0){pos[index - 2] = 1;pos[index] = 0;if (tmp == index)tmp = Math.Max(index - 2, 1);else if (tmp == index - 2)tmp = index - 1;}else{pos[index - 1] = 1;pos[index] = 0;if (tmp == index && index > 1){tmp = index - 1;}else if (tmp == index - 1){tmp = index;}}}}return result;
}public static bool IsEven(int n) => (n & 1) == 0;

Fixed Length Binary Sequence(n个0和m个1的所有组合)

譬如numberOfZero = 2, numberOfOne = 3返回

11100,01110,10110,11010,01101,10101,01011,00111,10011,11001

/// <summary>
/// 返回长度为<paramref name="numberOfZero"/>+<paramref name="numberOfOne"/>的二进制字符串,
/// 其中包含<paramref name="numberOfZero"/>个0和<paramref name="numberOfOne"/>个1
/// </summary>
public static IEnumerable<string> FixedLengthBinarySequence(int numberOfZero, int numberOfOne)
{if (numberOfZero < 1)throw new ArgumentOutOfRangeException("numberOfZero shouble be positive");if (numberOfOne < 1)throw new ArgumentOutOfRangeException("numberOfOne shouble be positive");var Length = numberOfZero + numberOfOne;var aux = new int[Length + 1];for (var i = 0; i < numberOfOne; ++i)aux[i] = 1;var index1 = numberOfOne - 1;var index2 = numberOfOne - 1;var result = new List<string>();while (true){var temp = aux.Take(Length).ToArray();result.Add(string.Join("", temp));aux[index1++] = 0;if (aux[index1] == 1){aux[index2++] = 1;continue;}if (index1 == Length) break;aux[index1] = 1;if (index2 > 0){aux[index2] = 1;aux[0] = 0;index1 = 1;index2 = 0;}}return result;
}

Fenichel(指定长度及最大和值的所有组合)

譬如sum = 2, len = 3返回

200,110,020,101,011,002,100,010,001,000

/// <summary>
/// 返回长度为<paramref name="length"/>的数字字符串，其所有数字字符的和不超过<paramref name="sum"/>
/// </summary>
public static IEnumerable<string> Fenichel(int sum, int length)
{if (sum < 1)throw new ArgumentOutOfRangeException("sum shouble be positive");if (length < 1)throw new ArgumentOutOfRangeException("length shouble be positive");var pos = new int[length + 1];pos[0] = sum;var index = 0;var result = new List<string>();while (true){var temp = pos.Take(length).ToArray();result.Add(string.Join("", temp));if (pos[0] == 0){if (index == length) break;pos[0] = pos[index] - 1;pos[index++] = 0;}else{--pos[0];index = 1;}++pos[index];}return result;
}

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【笔记】TAOCP Vol4 - Combination