Efficiently C# Bit Manipulation and Custom Data Structures
Intuition\nThe problem involves handling alternating groups in a circular array. The first query type requires counting alternating groups of a specific size, w
# Intuition\nThe problem involves handling alternating groups in a circular array. The first query type requires counting alternating groups of a specific size, while the second query type involves updating the color of a tile. Given the circular nature of the problem and the potentially large size of the array, a direct approach would be inefficient. Instead, we can use an advanced data structure like a segment tree or a customized bit manipulation technique to efficiently manage and query the alternating groups.\n\n# Approach\n1.Representation and Transformation: We start by transforming the original colors array into a new array c where the value at each position is XORed based on whether its index is even or odd. This transformation helps in identifying alternating groups more effectively.\n\n2.Data Structures: We use a custom data structure, LST, to efficiently manage and query the positions of transitions between different colors. This allows us to quickly find the start and end of alternating groups.\n\n3.Processing Queries:\n\nFor the first query type ([1, sizei]), we count the number of alternating groups of the specified size by traversing the positions of transitions.\n\nFor the second query type ([2, indexi, colori]), we update the color at the specified index and adjust the internal data structures to reflect this change.\n\n4.Circular Handling: Since the array is circular, special attention is given to handle the wrap-around between the last and first elements.\n\n# Complexity\n- Time complexity:\nThe time complexity for processing each query depends on the operations performed by the LST class. In the worst case, each operation (such as updating or querying) takes $$O(log N)$$.\n\n- Space complexity:\nThe space complexity is dominated by the size of the LST data structures and the auxiliary arrays, leading to a space complexity of $$O(n)$$, where n is the size of the colors array.\n\n# Code\n```\npublic class Solution { // Define the Solution class\n    public List<int> NumberOfAlternatingGroups(int[] colors, int[][] queries) { // Define the NumberOfAlternatingGroups function that takes two parameters colors and queries, and returns a List<int>\n        int n = colors.Length; // Get the length of the colors array and store it in n\n        int[] c = new int[2 * n]; // Create a new array c with length 2 * n\n        for (int i = 0; i < 2 * n; i++) { // Loop from 0 to 2 * n\n            c[i] = colors[i % n] ^ (i % 2 == 0 ? 0 : 1); // Store the XOR result of colors based on whether the index i is even or odd into c\n        }\n\n        LST lst = new LST(2 * n + 3); // Create an instance of the LST class with size 2 * n + 3\n        for (int i = 1; i < 2 * n; i++) { // Loop from 1 to 2 * n\n            if (c[i] != c[i - 1]) { // If the consecutive elements in c are different\n                lst.SetPosition(i); // Set the bit at position i in lst\n            }\n        }\n\n        int[] fs = new int[2 * n + 1]; // Create a new array fs with length 2 * n + 1\n        LST ff = new LST(2 * n + 1); // Create an instance of the LST class with size 2 * n + 1\n        for (int i = 0; i < n; i++) { // Loop from 0 to n\n            if (lst.Get(i)) { // If the bit at position i in lst is set\n                int ne = lst.Next(i + 1); // Get the next set bit in lst starting from i + 1\n                if (ne == -1) ne = 2 * n; // If ne is -1, set ne to 2 * n\n                fs[ne - i]++; // Increment the value in fs at index ne - i\n                ff.SetPosition(ne - i); // Set the bit at position ne - i in ff\n            }\n        }\n\n        List<int> ans = new List<int>(); // Create a List<int> to store the answers\n        foreach (var q in queries) { // Iterate through each query in queries\n            if (q[0] == 1) { // If the first element of the query is 1\n                if (lst.Next(0) == -1) { // If there is no set bit in lst starting from 0\n                    ans.Add(n); // Add n to the answer\n                } else {\n                    int lans = 0; // Initialize the local variable lans to 0\n                    for (int i = ff.Next(q[1]); i != -1; i = ff.Next(i + 1)) { // Iterate through all set bits in ff starting from q[1]\n                        lans += (i - q[1] + 1) * fs[i]; // Calculate and increment the value of lans\n                    }\n                    if (c[2 * n - 1] != c[0]) { // If the last element of c is different from the first element\n                        int f = lst.Next(0); // Get the next set bit in lst starting from 0\n                        if (f >= q[1]) { // If f is greater than or equal to q[1]\n                            lans += (f - q[1] + 1); // Increment the value of lans\n                        }\n                    }\n                    ans.Add(lans); // Add lans to the answer\n                }\n            } else {\n                int ind = q[1], val = q[2]; // Get the second and third elements of the query\n                if (colors[ind] == val) continue; // If the value in colors at index ind is equal to val, skip this iteration\n                colors[ind] ^= 1; // Perform an XOR operation on the value in colors at index ind\n                Go(ind, lst, fs, n, ff, c); // Call the Go function to process the changes\n                Go(ind + n, lst, fs, n, ff, c); // Call the Go function to process the changes\n            }\n        }\n        return ans; // Return the answer\n    }\n\n    private void Go(int ind, LST lst, int[] fs, int n, LST ff, int[] c) { // Define the private Go function that takes multiple parameters\n        if (ind > 0) { // If ind is greater than 0\n            int pre = lst.Prev(ind - 1); // Get the previous set bit in lst starting from ind - 1\n            int nex = lst.Next(pre + 1); // Get the next set bit in lst starting from pre + 1\n            if (nex == -1) nex = 2 * n; // If nex is -1, set nex to 2 * n\n            if (pre != -1 && pre < n && --fs[nex - pre] == 0) { // If pre is not -1, less than n, and decrementing fs at index nex - pre results in 0\n                ff.UnsetPosition(nex - pre); // Unset the bit at position nex - pre in ff\n            }\n        }\n        if (lst.Get(ind)) { // If the bit at position ind in lst is set\n            int pre = ind; // Set pre to ind\n            int nex = lst.Next(ind + 1); // Get the next set bit in lst starting from ind + 1\n            if (nex == -1) nex = 2 * n; // If nex is -1, set nex to 2 * n\n            if (pre != -1 && pre < n && --fs[nex - pre] == 0) { // If pre is not -1, less than n, and decrementing fs at index nex - pre results in 0\n                ff.UnsetPosition(nex - pre); // Unset the bit at position nex - pre in ff\n            }\n        }\n        if (lst.Get(ind + 1)) { // If the bit at position ind + 1 in lst is set\n            int pre = ind + 1; // Set pre to ind + 1\n            int nex = lst.Next(ind + 2); // Get the next set bit in lst starting from ind + 2\n            if (nex == -1) nex = 2 * n; // If nex is -1, set nex to 2 * n\n            if (pre != -1 && pre < n && --fs[nex - pre] == 0) { // If pre is not -1, less than n, and decrementing fs at index nex - pre results in 0\n                ff.UnsetPosition(nex - pre); // Unset the bit at position nex - pre in ff\n            }\n        }\n        lst.UnsetPosition(ind); // Unset the bit at position ind in lst\n        lst.UnsetPosition(ind + 1); // Unset the bit at position ind + 1 in lst\n        c[ind] ^= 1; // Perform an XOR operation on the value in c at position ind\n        if (ind > 0 && c[ind] != c[ind - 1]) { // If ind is greater than 0 and the value in c at position ind is different from the value at ind - 1\n            lst.SetPosition(ind); // Set the bit at position ind in lst\n        }\n        if (ind + 1 < c.Length && c[ind + 1] != c[ind]) { // If ind + 1 is less than the length of c and the value in c at position ind + 1 is different from the value at ind\n            lst.SetPosition(ind + 1); // Set the bit at position ind + 1 in lst\n        }\n\n        if (ind > 0) { // If ind is greater than 0\n            int pre = lst.Prev(ind - 1); // Get the previous set bit in lst starting from ind - 1\n            int nex = lst.Next(pre + 1); // Get the next set bit in lst starting from pre + 1\n            if (nex == -1) nex = 2 * n; // If nex is -1, set nex to 2 * n\n            if (pre != -1 && pre < n && ++fs[nex - pre] == 1) { // If pre is not -1, less than n, and incrementing fs at index nex - pre results in 1\n                ff.SetPosition(nex - pre); // Set the bit at position nex - pre in ff\n            }\n        }\n        if (lst.Get(ind)) { // If the bit at position ind in lst is set\n            int pre = ind; // Set pre to ind\n            int nex = lst.Next(ind + 1); // Get the next set bit in lst starting from ind + 1\n            if (nex == -1) nex = 2 * n; // If nex is -1, set nex to 2 * n\n            if (pre != -1 && pre < n && ++fs[nex - pre] == 1) { // If pre is not -1, less than n, and incrementing fs at index nex - pre results in 1\n                ff.SetPosition(nex - pre); // Set the bit at position nex - pre in ff\n            }\n        }\n        if (lst.Get(ind + 1)) { // If the bit at position ind + 1 in lst is set\n            int pre = ind + 1; // Set pre to ind + 1\n            int nex = lst.Next(ind + 2); // Get the next set bit in lst starting from ind + 2\n            if (nex == -1) nex = 2 * n; // If nex is -1, set nex to 2 * n\n            if (pre != -1 && pre < n && ++fs[nex - pre] == 1) { // If pre is not -1, less than n, and incrementing fs at index nex - pre results in 1\n                ff.SetPosition(nex - pre); // Set the bit at position nex - pre in ff\n            }\n        }\n    }\n\n    public class LST { // Define the LST class\n        public long[][] BitSet { get; private set; } // Define the BitSet property, a long two-dimensional array\n        public int N { get; private set; } // Define the N property, an integer\n\n        public LST(int n) { // Constructor for the LST class, takes one parameter n\n            this.N = n; // Set N to n\n            int d = 1; // Initialize the variable d to 1\n            for (int m = n; m > 1; m >>= 6, d++) ; // Loop to calculate the value of d until m is less than or equal to 1\n\n            BitSet = new long[d][]; // Initialize BitSet as a long two-dimensional array of length d\n            for (int i = 0, m = n >> 6; i < d; i++, m >>= 6) { // Loop to initialize each element of BitSet\n                BitSet[i] = new long[m + 1]; // Initialize BitSet[i] as a long array of length m + 1\n            }\n        }\n\n        public LST SetRange(int r) { // Define the SetRange function, takes one parameter r\n            for (int i = 0; i < BitSet.Length; i++, r = (r + 63) >> 6) { // Loop to set all bits within the range\n                for (int j = 0; j < (r >> 6); j++) { // Loop to set each element in BitSet[i] to -1\n                    BitSet[i][j] = -1L; // Set BitSet[i][j] to -1\n                }\n                if ((r & 63) != 0) BitSet[i][r >> 6] |= (1L << r) - 1; // If the lower 6 bits of r are not 0, set the corresponding bit in BitSet[i][r >> 6]\n            }\n            return this; // Return the current LST instance\n        }\n\n        public LST UnsetRange(int r) { // Define the UnsetRange function, takes one parameter r\n            if (r >= 0) { // If r is greater than or equal to 0\n                for (int i = 0; i < BitSet.Length; i++, r = (r + 63) >> 6) { // Loop to unset all bits within the range\n                    for (int j = 0; j < (r + 63) >> 6; j++) { // Loop to set each element in BitSet[i] to 0\n                        BitSet[i][j] = 0; // Set BitSet[i][j] to 0\n                    }\n                    if ((r & 63) != 0) BitSet[i][r >> 6] &= -(1L << r); // If the lower 6 bits of r are not 0, unset the corresponding bit in BitSet[i][r >> 6]\n                }\n            }\n            return this; // Return the current LST instance\n        }\n\n        public LST SetPosition(int pos) { // Define the SetPosition function, takes one parameter pos\n            if (pos >= 0 && pos < N) { // If pos is greater than or equal to 0 and less than N\n                for (int i = 0; i < BitSet.Length; i++, pos >>= 6) { // Loop to set the bit corresponding to pos\n                    BitSet[i][pos >> 6] |= 1L << pos; // Set the corresponding bit in BitSet[i][pos >> 6]\n                }\n            }\n            return this; // Return the current LST instance\n        }\n\n        public LST UnsetPosition(int pos) { // Define the UnsetPosition function, takes one parameter pos\n            if (pos >= 0 && pos < N) { // If pos is greater than or equal to 0 and less than N\n                for (int i = 0; i < BitSet.Length && (i == 0 || BitSet[i - 1][pos] == 0L); i++, pos >>= 6) { // Loop to unset the bit corresponding to pos\n                    BitSet[i][pos >> 6] &= ~(1L << pos); // Unset the corresponding bit in BitSet[i][pos >> 6]\n                }\n            }\n            return this; // Return the current LST instance\n        }\n\n        public bool Get(int pos) { // Define the Get function, takes one parameter pos\n            return pos >= 0 && pos < N && (BitSet[0][pos >> 6] << ~pos) < 0; // Return whether the bit corresponding to pos is set\n        }\n\n        public LST Toggle(int pos) { // Define the Toggle function, takes one parameter pos\n            return Get(pos) ? UnsetPosition(pos) : SetPosition(pos); // Toggle the bit at pos based on whether it is set\n        }\n\n        public int Prev(int pos) { // Define the Prev function, takes one parameter pos\n            for (int i = 0; i < BitSet.Length && pos >= 0; i++, pos = (pos >> 6) - 1) { // Loop to find the set bit before pos\n                int pre = Prev(BitSet[i][pos >> 6], pos & 63); // Get the set bit before pos & 63 in BitSet[i][pos >> 6]\n                if (pre != -1) { // If pre is not -1\n                    pos = (pos >> 6 << 6) | pre; // Update the value of pos\n                    while (i > 0) pos = (pos << 6) | (63 - LeadingZeros(BitSet[--i][pos])); // Update the value of pos\n                    return pos; // Return pos\n                }\n            }\n            return -1; // If not found, return -1\n        }\n\n        public int Next(int pos) { // Define the Next function, takes one parameter pos\n            for (int i = 0; i < BitSet.Length && (pos >> 6) < BitSet[i].Length; i++, pos = (pos >> 6) + 1) { // Loop to find the set bit after pos\n                int nex = Next(BitSet[i][pos >> 6], pos & 63); // Get the set bit after pos & 63 in BitSet[i][pos >> 6]\n                if (nex != -1) { // If nex is not -1\n                    pos = (pos >> 6 << 6) | nex; // Update the value of pos\n                    while (i > 0) pos = (pos << 6) | TrailingZeros(BitSet[--i][pos]); // Update the value of pos\n                    return pos; // Return pos\n                }\n            }\n            return -1; // If not found, return -1\n        }\n\n        private static int Prev(long set, int n) { // Define the static private Prev function, takes a long set and an int n\n            long h = set << ~n; // Calculate the value of h\n            if (h == 0L) return -1; // If h is 0, return -1\n            return -LeadingZeros(h) + n; // Otherwise, return -LeadingZeros(h) + n\n        }\n\n        private static int Next(long set, int n) { // Define the static private Next function, takes a long set and an int n\n            long h = set >> n; // Calculate the value of h\n            if (h == 0L) return -1; // If h is 0, return -1\n            return TrailingZeros(h) + n; // Otherwise, return TrailingZeros(h) + n\n        }\n\n        private static int LeadingZeros(long x) { // Define the static private LeadingZeros function, takes a long x\n            if (x == 0) return 64; // If x is 0, return 64\n            int count = 0; // Initialize count to 0\n            for (int i = 62; i >= 0; i -= 2) { // Loop to check every two bits\n                if ((x & (3L << i)) == 0) { // If the two bits of x are 0\n                    count += 2; // Increment count\n                } else {\n                    if ((x & (2L << i)) == 0) { // If the higher bit is 0\n                        count++; // Increment count\n                    }\n                    break; // Break the loop\n                }\n            }\n            return count; // Return count\n        }\n\n        private static int TrailingZeros(long x) { // Define the static private TrailingZeros function, takes a long x\n            if (x == 0) return 64; // If x is 0, return 64\n            int count = 0; // Initialize count to 0\n            for (int i = 0; i < 64; i += 2) { // Loop to check every two bits\n                if ((x & (3L << i)) == 0) { // If the two bits of x are 0\n                    count += 2; // Increment count\n                } else {\n                    if ((x & (1L << i)) == 0) { // If the lower bit is 0\n                        count++; // Increment count\n                    }\n                    break; // Break the loop\n                }\n            }\n            return count; // Return count\n        }\n\n        public override string ToString() { // Override the ToString function\n            List<int> list = new List<int>(); // Create a List<int> to store the positions of set bits\n            for (int pos = Next(0); pos != -1; pos = Next(pos + 1)) { // Loop to get all the positions of set bits\n                list.Add(pos); // Add pos to the list\n            }\n            return string.Join(", ", list); // Return the string representation of the list\n        }\n    }\n}\n```
['C#']