Extended Mo's Algorithm with ≈ O(1) time complexity
Last Updated : 28 Feb, 2024
Given an array of n elements and q range queries (range sum in this article) with no updates, task is to answer these queries with efficient time and space complexity. The time complexity of a range query after applying square root decomposition comes out to be O(?n). This square-root factor can be decreased to a constant linear factor by applying square root decomposition on the block of the array which was decomposed earlier.
Prerequisite: Mo's Algorithm | Prefix Array
Approach :
As we apply square root decomposition to the given array, querying a range-sum comes in O(?n) time.
Here, calculate the sum of blocks which are in between the blocks under consideration(corner blocks), which takes O(?n) iterations.
Initial Array :
Initial ArrayDecomposition of array into blocks :
Decomposition at level-1
And the calculation time for the sum on the starting block and ending block both takes O(?n) iterations.
Which will leaves us per query time complexity of :
= O(?n) + O(?n) + O(?n)
= 3 * O(?n)
~O(?n)
Here, we can reduce the runtime complexity of our query algorithm cleverly by calculating blockwise prefix sum and using it to calculate the sum accumulated in the blocks which lie between the blocks under consideration. Consider the code below :
interblock_sum[x1][x2] = prefixData[x2 - 1] - prefixData[x1];
Time taken for calculation of above table is :
= O(?n) * O(?n)
~O(n)
NOTE : We haven't taken the sum of blocks x1 & x2 under consideration as they might be carrying partial data.
Prefix Array :
Prefix ArraySuppose we want to query for the sum for range from 4 to 11, we consider the sum between block 0 and block 2 (excluding the data contained in block 0 and block 1), which can be calculated using the sum in the green coloured blocks represented in the above image.
Sum between block 0 and block 2 = 42 - 12 = 30
For calculation of rest of the sum present in the yellow blocks, consider the prefix array at the decomposition level-2 and repeat the process again.
Here, observe that we have reduced our time complexity per query significantly, though our runtime remains similar to our last approach :
Our new time complexity can be calculated as :
= O(?n) + O(1) + O(?n)
= 2 * O(?n)
~O(?n)
Square-root Decomposition at Level-2 :
Decomposition at level-2Further, we apply square root decomposition again on every decomposed block retained from the previous decomposition. Now at this level, we have approximately ??n sub-blocks in each block which were decomposed at last level. So, we need to run a range query on these blocks only two times, one time for starting block and one time for ending block.
Precalculation Time taken for level 2 decomposition :
No of blocks at level 1 ~ ?n
No of blocks at level 2 ~ ??n
Level-2 Decomposition Runtime of a level-1 decomposed block :
= O(?n)
Overall runtime of level-2 decomposition over all blocks :
= O(?n) * O(?n)
~O(n)
Now, we can query our level-2 decomposed blocks in O(??n) time.
So, we have reduced our overall time complexity from O(?n) to O(??n)
Time complexity taken in querying edge blocks :
= O(??n) + O(1) + O(??n)
= 2 * O(??n)
~O(??n)
Total Time complexity can be calculated as :
= O(??n)+O(1)+O(??n)
= 2 * O(??n)
~O(??n)
Square-root Decomposition at Level-3 :
Decomposition at level-3Using this method we can decompose our array again and again recursively d times to reduce our time complexity to a factor of constant linearity.
O(d * n1/(2^d)) ~ O(k), as d increases this factor converges to a constant linear term
The code presented below is a representation of triple square root decomposition where d = 3:
O(q * d * n1/(2 ^ 3)) ~ O(q * k) ~ O(q)
[where q represents number of range queries]
C++ // CPP code for offline queries in // approx constant time. #include<bits/stdc++.h> using namespace std; int n1; // Structure to store decomposed data typedef struct { vector<int> data; vector<vector<int>> rdata; int blocks; int blk_sz; }sqrtD; vector<vector<sqrtD>> Sq3; vector<sqrtD> Sq2; sqrtD Sq1; // Square root Decomposition of // a given array sqrtD decompose(vector<int> arr) { sqrtD sq; int n = arr.size(); int blk_idx = -1; sq.blk_sz = sqrt(n); sq.data.resize((n/sq.blk_sz) + 1, 0); // Calculation of data in blocks for (int i = 0; i < n; i++) { if (i % sq.blk_sz == 0) { blk_idx++; } sq.data[blk_idx] += arr[i]; } int blocks = blk_idx + 1; sq.blocks = blocks; // Calculation of prefix data int prefixData[blocks]; prefixData[0] = sq.data[0]; for(int i = 1; i < blocks; i++) { prefixData[i] = prefixData[i - 1] + sq.data[i]; } sq.rdata.resize(blocks + 1, vector<int>(blocks + 1)); // Calculation of data between blocks for(int i = 0 ;i < blocks; i++) { for(int j = i + 1; j < blocks; j++) { sq.rdata[i][j] = sq.rdata[j][i] = prefixData[j - 1] - prefixData[i]; } } return sq; } // Square root Decomposition at level3 vector<vector<sqrtD>> tripleDecompose(sqrtD sq1, sqrtD sq2,vector<int> &arr) { vector<vector<sqrtD>> sq(sq1.blocks, vector<sqrtD>(sq1.blocks)); int blk_idx1 = -1; for(int i = 0; i < sq1.blocks; i++) { int blk_ldx1 = blk_idx1 + 1; blk_idx1 = (i + 1) * sq1.blk_sz - 1; blk_idx1 = min(blk_idx1,n1 - 1); int blk_idx2 = blk_ldx1 - 1; for(int j = 0; j < sq2.blocks; ++j) { int blk_ldx2 = blk_idx2 + 1; blk_idx2 = blk_ldx1 + (j + 1) * sq2.blk_sz - 1; blk_idx2 = min(blk_idx2, blk_idx1); vector<int> ::iterator it1 = arr.begin() + blk_ldx2; vector<int> ::iterator it2 = arr.begin() + blk_idx2 + 1; vector<int> vec(it1, it2); sq[i][j] = decompose(vec); } } return sq; } // Square root Decomposition at level2 vector<sqrtD> doubleDecompose(sqrtD sq1, vector<int> &arr) { vector<sqrtD> sq(sq1.blocks); int blk_idx = -1; for(int i = 0; i < sq1.blocks; i++) { int blk_ldx = blk_idx + 1; blk_idx = (i + 1) * sq1.blk_sz - 1; blk_idx = min(blk_idx, n1 - 1); vector<int> ::iterator it1 = arr.begin() + blk_ldx; vector<int> ::iterator it2 = arr.begin() + blk_idx + 1; vector<int> vec(it1, it2); sq[i] = decompose(vec); } return sq; } // Square root Decomposition at level1 void singleDecompose(vector<int> &arr) { sqrtD sq1 = decompose(arr); vector<sqrtD> sq2(sq1.blocks); sq2 = doubleDecompose(sq1, arr); vector<vector<sqrtD>> sq3(sq1.blocks, vector<sqrtD>(sq2[0].blocks)); sq3 = tripleDecompose(sq1, sq2[0],arr); // ASSIGNMENT TO GLOBAL VARIABLES Sq1 = sq1; Sq2.resize(sq1.blocks); Sq2 = sq2; Sq3.resize(sq1.blocks, vector<sqrtD>(sq2[0].blocks)); Sq3 = sq3; } // Function for query at level 3 int queryLevel3(int start,int end, int main_blk, int sub_main_blk, vector<int> &arr) { int blk_sz= Sq3[0][0].blk_sz; // Element Indexing at level2 decomposition int nstart = start - main_blk * Sq1.blk_sz - sub_main_blk * Sq2[0].blk_sz; int nend = end - main_blk * Sq1.blk_sz - sub_main_blk * Sq2[0].blk_sz; // Block indexing at level3 decomposition int st_blk = nstart / blk_sz; int en_blk = nend / blk_sz; int answer = Sq3[main_blk][sub_main_blk].rdata[st_blk][en_blk]; // If start and end point don't lie in same block if(st_blk != en_blk) { int left = 0, en_idx = main_blk * Sq1.blk_sz + sub_main_blk * Sq2[0].blk_sz + (st_blk + 1) * blk_sz -1; for(int i = start; i <= en_idx; i++) { left += arr[i]; } int right = 0, st_idx = main_blk * Sq1.blk_sz + sub_main_blk * Sq2[0].blk_sz + (en_blk) * blk_sz; for(int i = st_idx; i <= end; i++) { right += arr[i]; } answer += left; answer += right; } else { for(int i = start; i <= end; i++) { answer += arr[i]; } } return answer; } // Function for splitting query to level two int queryLevel2(int start, int end, int main_blk, vector<int> &arr) { int blk_sz = Sq2[0].blk_sz; // Element Indexing at level1 decomposition int nstart = start - (main_blk * Sq1.blk_sz); int nend = end - (main_blk * Sq1.blk_sz); // Block indexing at level2 decomposition int st_blk = nstart / blk_sz; int en_blk = nend / blk_sz; // Interblock data level2 decomposition int answer = Sq2[main_blk].rdata[st_blk][en_blk]; if(st_blk == en_blk) { answer += queryLevel3(start, end, main_blk, st_blk, arr); } else { answer += queryLevel3(start, (main_blk * Sq1.blk_sz) + ((st_blk + 1) * blk_sz) - 1, main_blk, st_blk, arr); answer += queryLevel3((main_blk * Sq1.blk_sz) + (en_blk * blk_sz), end, main_blk, en_blk, arr); } return answer; } // Function to return answer according to query int Query(int start,int end,vector<int>& arr) { int blk_sz = Sq1.blk_sz; int st_blk = start / blk_sz; int en_blk = end / blk_sz; // Interblock data level1 decomposition int answer = Sq1.rdata[st_blk][en_blk]; if(st_blk == en_blk) { answer += queryLevel2(start, end, st_blk, arr); } else { answer += queryLevel2(start, (st_blk + 1) * blk_sz - 1, st_blk, arr); answer += queryLevel2(en_blk * blk_sz, end, en_blk, arr); } // returning final answer return answer; } // Driver code int main() { n1 = 16; vector<int> arr = {7, 2, 3, 0, 5, 10, 3, 12, 18, 1, 2, 3, 4, 5, 6, 7}; singleDecompose(arr); int q = 5; pair<int, int> query[q] = {{6, 10}, {7, 12}, {4, 13}, {4, 11}, {12, 16}}; for(int i = 0; i < q; i++) { int a = query[i].first, b = query[i].second; printf("%d\n", Query(a - 1, b - 1, arr)); } return 0; } import java.util.Arrays; class sqrtD { int[] data; int[][] rdata; int blocks; int blk_sz; // Constructor for sqrtD sqrtD(int n) { blk_sz = (int) Math.sqrt(n); data = new int[(n / blk_sz) + 1]; Arrays.fill(data, 0); } } public class OfflineQueries { static int n1; static sqrtD Sq1; static sqrtD[] Sq2; static sqrtD[][] Sq3; // Function to decompose an array using square root decomposition static sqrtD decompose(int[] arr) { int n = arr.length; sqrtD sq = new sqrtD(n); int blk_idx = -1; // Calculation of data in blocks for (int i = 0; i < n; i++) { if (i % sq.blk_sz == 0) { blk_idx++; } sq.data[blk_idx] += arr[i]; } int blocks = blk_idx + 1; sq.blocks = blocks; // Calculation of prefix data int[] prefixData = new int[blocks]; prefixData[0] = sq.data[0]; for (int i = 1; i < blocks; i++) { prefixData[i] = prefixData[i - 1] + sq.data[i]; } sq.rdata = new int[blocks + 1][blocks + 1]; // Calculation of data between blocks for (int i = 0; i < blocks; i++) { for (int j = i + 1; j < blocks; j++) { sq.rdata[i][j] = sq.rdata[j][i] = prefixData[j - 1] - prefixData[i]; } } return sq; } // Function to perform triple decomposition static sqrtD[][] tripleDecompose(sqrtD sq1, sqrtD sq2, int[] arr) { sqrtD[][] sq = new sqrtD[sq1.blocks][sq2.blocks]; int blk_idx1 = -1; for (int i = 0; i < sq1.blocks; i++) { int blk_ldx1 = blk_idx1 + 1; blk_idx1 = (i + 1) * sq1.blk_sz - 1; blk_idx1 = Math.min(blk_idx1, n1 - 1); int blk_idx2 = blk_ldx1 - 1; for (int j = 0; j < sq2.blocks; ++j) { int blk_ldx2 = blk_idx2 + 1; blk_idx2 = blk_ldx1 + (j + 1) * sq2.blk_sz - 1; blk_idx2 = Math.min(blk_idx2, blk_idx1); int[] vec = Arrays.copyOfRange(arr, blk_ldx2, blk_idx2 + 1); sq[i][j] = decompose(vec); } } return sq; } // Function to perform double decomposition static sqrtD[] doubleDecompose(sqrtD sq1, int[] arr) { sqrtD[] sq = new sqrtD[sq1.blocks]; int blk_idx = -1; for (int i = 0; i < sq1.blocks; i++) { int blk_ldx = blk_idx + 1; blk_idx = (i + 1) * sq1.blk_sz - 1; blk_idx = Math.min(blk_idx, n1 - 1); int[] vec = Arrays.copyOfRange(arr, blk_ldx, blk_idx + 1); sq[i] = decompose(vec); } return sq; } // Function to perform single decomposition static void singleDecompose(int[] arr) { Sq1 = decompose(arr); Sq2 = doubleDecompose(Sq1, arr); Sq3 = tripleDecompose(Sq1, Sq2[0], arr); } // Function to query at level 3 static int queryLevel3(int start, int end, int main_blk, int sub_main_blk, int[] arr) { int blk_sz = Sq3[0][0].blk_sz; // Element Indexing at level2 decomposition int nstart = start - main_blk * Sq1.blk_sz - sub_main_blk * Sq2[0].blk_sz; int nend = end - main_blk * Sq1.blk_sz - sub_main_blk * Sq2[0].blk_sz; // Block indexing at level3 decomposition int st_blk = nstart / blk_sz; int en_blk = nend / blk_sz; int answer = Sq3[main_blk][sub_main_blk].rdata[st_blk][en_blk]; // If start and end point don't lie in the same block if (st_blk != en_blk) { int left = 0, en_idx = main_blk * Sq1.blk_sz + sub_main_blk * Sq2[0].blk_sz + (st_blk + 1) * blk_sz - 1; for (int i = start; i <= en_idx; i++) { left += arr[i]; } int right = 0, st_idx = main_blk * Sq1.blk_sz + sub_main_blk * Sq2[0].blk_sz + (en_blk) * blk_sz; for (int i = st_idx; i <= end; i++) { right += arr[i]; } answer += left; answer += right; } else { for (int i = start; i <= end; i++) { answer += arr[i]; } } return answer; } // Function to query at level 2 static int queryLevel2(int start, int end, int main_blk, int[] arr) { int blk_sz = Sq2[0].blk_sz; // Element Indexing at level1 decomposition int nstart = start - (main_blk * Sq1.blk_sz); int nend = end - (main_blk * Sq1.blk_sz); // Block indexing at level2 decomposition int st_blk = nstart / blk_sz; int en_blk = nend / blk_sz; // Interblock data level2 decomposition int answer = Sq2[main_blk].rdata[st_blk][en_blk]; if (st_blk == en_blk) { answer += queryLevel3(start, end, main_blk, st_blk, arr); } else { answer += queryLevel3(start, (main_blk * Sq1.blk_sz) + ((st_blk + 1) * blk_sz) - 1, main_blk, st_blk, arr); answer += queryLevel3((main_blk * Sq1.blk_sz) + (en_blk * blk_sz), end, main_blk, en_blk, arr); } return answer; } // Function to return the answer according to the query static int Query(int start, int end, int[] arr) { int blk_sz = Sq1.blk_sz; int st_blk = start / blk_sz; int en_blk = end / blk_sz; // Interblock data level1 decomposition int answer = Sq1.rdata[st_blk][en_blk]; if (st_blk == en_blk) { answer += queryLevel2(start, end, st_blk, arr); } else { answer += queryLevel2(start, (st_blk + 1) * blk_sz - 1, st_blk, arr); answer += queryLevel2(en_blk * blk_sz, end, en_blk, arr); } // Returning the final answer return answer; } // Driver code public static void main(String[] args) { n1 = 16; int[] arr = {7, 2, 3, 0, 5, 10, 3, 12, 18, 1, 2, 3, 4, 5, 6, 7}; singleDecompose(arr); int q = 5; int[][] query = {{6, 10}, {7, 12}, {4, 13}, {4, 11}, {12, 16}}; for (int i = 0; i < q; i++) { int a = query[i][0], b = query[i][1]; System.out.println(Query(a - 1, b - 1, arr)); } } } import math class sqrtD: def __init__(self): self.data = [] self.rdata = [] self.blocks = 0 self.blk_sz = 0 # Function to decompose an array into blocks and calculate block-wise and cumulative sums def decompose(arr): sq = sqrtD() n = len(arr) blk_idx = -1 sq.blk_sz = int(math.sqrt(n)) sq.data = [0] * ((n // sq.blk_sz) + 1) # Calculation of block-wise sums for i in range(n): if i % sq.blk_sz == 0: blk_idx += 1 sq.data[blk_idx] += arr[i] blocks = blk_idx + 1 sq.blocks = blocks # Calculation of cumulative sums between blocks prefix_data = [0] * blocks prefix_data[0] = sq.data[0] for i in range(1, blocks): prefix_data[i] = prefix_data[i - 1] + sq.data[i] sq.rdata = [[0] * (blocks + 1) for _ in range(blocks + 1)] for i in range(blocks): for j in range(i + 1, blocks): sq.rdata[i][j] = sq.rdata[j][i] = prefix_data[j - 1] - prefix_data[i] return sq # Function to decompose two arrays at level 3 def triple_decompose(sq1, sq2, arr): sq = [[None] * sq1.blocks for _ in range(sq1.blocks)] blk_idx1 = -1 for i in range(sq1.blocks): blk_ldx1 = blk_idx1 + 1 blk_idx1 = min((i + 1) * sq1.blk_sz - 1, n1 - 1) blk_idx2 = blk_ldx1 - 1 for j in range(sq2.blocks): blk_ldx2 = blk_idx2 + 1 blk_idx2 = min(blk_ldx1 + (j + 1) * sq2.blk_sz - 1, blk_idx1) vec = arr[blk_ldx2:blk_idx2 + 1] sq[i][j] = decompose(vec) return sq # Function to decompose one array at level 2 def double_decompose(sq1, arr): sq = [None] * sq1.blocks blk_idx = -1 for i in range(sq1.blocks): blk_ldx = blk_idx + 1 blk_idx = min((i + 1) * sq1.blk_sz - 1, n1 - 1) vec = arr[blk_ldx:blk_idx + 1] sq[i] = decompose(vec) return sq # Function to decompose one array at level 1 def single_decompose(arr): global Sq1, Sq2, Sq3 sq1 = decompose(arr) sq2 = double_decompose(sq1, arr) sq3 = triple_decompose(sq1, sq2[0], arr) Sq1 = sq1 Sq2 = sq2 Sq3 = sq3 # Function for querying at level 3 def query_level3(start, end, main_blk, sub_main_blk, arr): blk_sz = Sq3[0][0].blk_sz nstart = start - main_blk * Sq1.blk_sz - sub_main_blk * Sq2[0].blk_sz nend = end - main_blk * Sq1.blk_sz - sub_main_blk * Sq2[0].blk_sz st_blk = nstart // blk_sz en_blk = nend // blk_sz answer = Sq3[main_blk][sub_main_blk].rdata[st_blk][en_blk] if st_blk != en_blk: left = 0 en_idx = main_blk * Sq1.blk_sz + sub_main_blk * Sq2[0].blk_sz + (st_blk + 1) * blk_sz - 1 for i in range(start, en_idx + 1): left += arr[i] right = 0 st_idx = main_blk * Sq1.blk_sz + sub_main_blk * Sq2[0].blk_sz + (en_blk) * blk_sz for i in range(st_idx, end + 1): right += arr[i] answer += left answer += right else: for i in range(start, end + 1): answer += arr[i] return answer # Function for querying at level 2 def query_level2(start, end, main_blk, arr): blk_sz = Sq2[0].blk_sz nstart = start - main_blk * Sq1.blk_sz nend = end - main_blk * Sq1.blk_sz st_blk = nstart // blk_sz en_blk = nend // blk_sz answer = Sq2[main_blk].rdata[st_blk][en_blk] if st_blk == en_blk: answer += query_level3(start, end, main_blk, st_blk, arr) else: answer += query_level3(start, (main_blk * Sq1.blk_sz) + ((st_blk + 1) * blk_sz) - 1, main_blk, st_blk, arr) answer += query_level3((main_blk * Sq1.blk_sz) + (en_blk * blk_sz), end, main_blk, en_blk, arr) return answer # Function to handle queries def Query(start, end, arr): blk_sz = Sq1.blk_sz st_blk = start // blk_sz en_blk = end // blk_sz answer = Sq1.rdata[st_blk][en_blk] if st_blk == en_blk: answer += query_level2(start, end, st_blk, arr) else: answer += query_level2(start, (st_blk + 1) * blk_sz - 1, st_blk, arr) answer += query_level2(en_blk * blk_sz, end, en_blk, arr) return answer # Driver code n1 = 16 arr = [7, 2, 3, 0, 5, 10, 3, 12, 18, 1, 2, 3, 4, 5, 6, 7] single_decompose(arr) q = 5 queries = [(6, 10), (7, 12), (4, 13), (4, 11), (12, 16)] for query in queries: a, b = query print(Query(a - 1, b - 1, arr))
using System; using System.Collections.Generic; public class MainClass { // Global variable to store the size of blocks public static int n1; // Class to represent a block in the decomposed data // structure public class SqrtD { // Data stored in the block public List<int> data; // Prefix sum data between blocks public List<List<int> > rdata; // Number of blocks public int blocks; // Size of each block public int blk_sz; // Constructor to initialize lists public SqrtD() { data = new List<int>(); rdata = new List<List<int> >(); } } // Global variables to store decomposed data structure public static List<List<SqrtD> > Sq3; public static List<SqrtD> Sq2; public static SqrtD Sq1; // Function to decompose the input array into blocks public static SqrtD Decompose(List<int> arr) { SqrtD sq = new SqrtD(); int n = arr.Count; int blk_idx = -1; sq.blk_sz = (int)Math.Sqrt(n); sq.data = new List<int>((n / sq.blk_sz) + 1); for (int i = 0; i < sq.data.Capacity; i++) { sq.data.Add(0); } for (int i = 0; i < n; i++) { if (i % sq.blk_sz == 0) { blk_idx++; } sq.data[blk_idx] += arr[i]; } int blocks = blk_idx + 1; sq.blocks = blocks; int[] prefixData = new int[blocks]; prefixData[0] = sq.data[0]; for (int i = 1; i < blocks; i++) { prefixData[i] = prefixData[i - 1] + sq.data[i]; } sq.rdata = new List<List<int> >(); for (int i = 0; i <= blocks; i++) { sq.rdata.Add( new List<int>(new int[blocks + 1])); } for (int i = 0; i < blocks; i++) { for (int j = i + 1; j < blocks; j++) { sq.rdata[i][j] = sq.rdata[j][i] = prefixData[j - 1] - prefixData[i]; } } return sq; } // Function to decompose the data structure into triple // levels public static List<List<SqrtD> > TripleDecompose(SqrtD sq1, SqrtD sq2, List<int> arr) { List<List<SqrtD> > sq = new List<List<SqrtD> >(sq1.blocks); for (int i = 0; i < sq1.blocks; i++) { sq.Add(new List<SqrtD>(new SqrtD[sq1.blocks])); } int blk_idx1 = -1; for (int i = 0; i < sq1.blocks; i++) { int blk_ldx1 = blk_idx1 + 1; blk_idx1 = (i + 1) * sq1.blk_sz - 1; blk_idx1 = Math.Min(blk_idx1, n1 - 1); int blk_idx2 = blk_ldx1 - 1; for (int j = 0; j < sq2.blocks; ++j) { int blk_ldx2 = blk_idx2 + 1; blk_idx2 = blk_ldx1 + (j + 1) * sq2.blk_sz - 1; blk_idx2 = Math.Min(blk_idx2, blk_idx1); List<int> vec = new List<int>(arr.GetRange( blk_ldx2, blk_idx2 - blk_ldx2 + 1)); sq[i][j] = Decompose(vec); } } return sq; } // Function to decompose the data structure into double // levels public static List<SqrtD> DoubleDecompose(SqrtD sq1, List<int> arr) { List<SqrtD> sq = new List<SqrtD>(sq1.blocks); int blk_idx = -1; for (int i = 0; i < sq1.blocks; i++) { int blk_ldx = blk_idx + 1; blk_idx = (i + 1) * sq1.blk_sz - 1; blk_idx = Math.Min(blk_idx, n1 - 1); List<int> vec = new List<int>(arr.GetRange( blk_ldx, blk_idx - blk_ldx + 1)); sq.Add(Decompose(vec)); } return sq; } // Function to decompose the data structure into single // level public static void SingleDecompose(List<int> arr) { Sq1 = Decompose(arr); Sq2 = DoubleDecompose(Sq1, arr); Sq3 = TripleDecompose(Sq1, Sq2[0], arr); } // Function to handle queries at level 3 of // decomposition public static int QueryLevel3(int start, int end, int main_blk, int sub_main_blk, List<int> arr) { int blk_sz = Sq3[0][0].blk_sz; int nstart = start - main_blk * Sq1.blk_sz - sub_main_blk * Sq2[0].blk_sz; int nend = end - main_blk * Sq1.blk_sz - sub_main_blk * Sq2[0].blk_sz; int st_blk = nstart / blk_sz; int en_blk = nend / blk_sz; int answer = Sq3[main_blk][sub_main_blk] .rdata[st_blk][en_blk]; if (st_blk != en_blk) { int left = 0, en_idx = main_blk * Sq1.blk_sz + sub_main_blk * Sq2[0].blk_sz + (st_blk + 1) * blk_sz - 1; for (int i = start; i <= en_idx; i++) { left += arr[i]; } int right = 0, st_idx = main_blk * Sq1.blk_sz + sub_main_blk * Sq2[0].blk_sz + (en_blk)*blk_sz; for (int i = st_idx; i <= end; i++) { right += arr[i]; } answer += left; answer += right; } else { for (int i = start; i <= end; i++) { answer += arr[i]; } } return answer; } // Function to handle queries at level 2 of // decomposition public static int QueryLevel2(int start, int end, int main_blk, List<int> arr) { int blk_sz = Sq2[0].blk_sz; int nstart = start - (main_blk * Sq1.blk_sz); int nend = end - (main_blk * Sq1.blk_sz); int st_blk = nstart / blk_sz; int en_blk = nend / blk_sz; int answer = Sq2[main_blk].rdata[st_blk][en_blk]; if (st_blk == en_blk) { answer += QueryLevel3(start, end, main_blk, st_blk, arr); } else { answer += QueryLevel3( start, (main_blk * Sq1.blk_sz) + ((st_blk + 1) * blk_sz) - 1, main_blk, st_blk, arr); answer += QueryLevel3( (main_blk * Sq1.blk_sz) + (en_blk * blk_sz), end, main_blk, en_blk, arr); } return answer; } // Function to handle queries at level 1 of // decomposition public static int Query(int start, int end, List<int> arr) { int blk_sz = Sq1.blk_sz; int st_blk = start / blk_sz; int en_blk = end / blk_sz; int answer = Sq1.rdata[st_blk][en_blk]; if (st_blk == en_blk) { answer += QueryLevel2(start, end, st_blk, arr); } else { answer += QueryLevel2(start, (st_blk + 1) * blk_sz - 1, st_blk, arr); answer += QueryLevel2(en_blk * blk_sz, end, en_blk, arr); } return answer; } public static void Main(string[] args) { n1 = 16; List<int> arr = new List<int>{ 7, 2, 3, 0, 5, 10, 3, 12, 18, 1, 2, 3, 4, 5, 6, 7 }; SingleDecompose(arr); int q = 5; Tuple<int, int>[] query = { Tuple.Create(6, 10), Tuple.Create(7, 12), Tuple.Create(4, 13), Tuple.Create(4, 11), Tuple.Create(12, 16) }; for (int i = 0; i < q; i++) { int a = query[i].Item1, b = query[i].Item2; Console.WriteLine(Query(a - 1, b - 1, arr)); } } } // Structure to store decomposed data class sqrtD { constructor() { this.data = []; this.rdata = []; this.blocks = 0; this.blk_sz = 0; } } let Sq1 = new sqrtD(); let Sq2 = []; let Sq3 = []; // Square root Decomposition of a given array function decompose(arr) { const sq = new sqrtD(); const n = arr.length; let blk_idx = -1; sq.blk_sz = Math.floor(Math.sqrt(n)); sq.data = Array(Math.ceil(n / sq.blk_sz)).fill(0); // Calculation of data in blocks for (let i = 0; i < n; i++) { if (i % sq.blk_sz === 0) { blk_idx++; } sq.data[blk_idx] += arr[i]; } const blocks = blk_idx + 1; sq.blocks = blocks; // Calculation of prefix data const prefixData = new Array(blocks).fill(0); prefixData[0] = sq.data[0]; for (let i = 1; i < blocks; i++) { prefixData[i] = prefixData[i - 1] + sq.data[i]; } sq.rdata = new Array(blocks + 1).fill().map(() => new Array(blocks + 1).fill(0)); // Calculation of data between blocks for (let i = 0; i < blocks; i++) { for (let j = i + 1; j < blocks; j++) { sq.rdata[i][j] = sq.rdata[j][i] = prefixData[j - 1] - prefixData[i]; } } return sq; } // Square root Decomposition at level3 function tripleDecompose(sq1, sq2, arr) { const sq = new Array(sq1.blocks).fill().map(() => new Array(sq2.blocks)); let blk_idx1 = -1; for (let i = 0; i < sq1.blocks; i++) { let blk_ldx1 = blk_idx1 + 1; blk_idx1 = (i + 1) * sq1.blk_sz - 1; blk_idx1 = Math.min(blk_idx1, arr.length - 1); let blk_idx2 = blk_ldx1 - 1; for (let j = 0; j < sq2.blocks; ++j) { let blk_ldx2 = blk_idx2 + 1; blk_idx2 = blk_ldx1 + (j + 1) * sq2.blk_sz - 1; blk_idx2 = Math.min(blk_idx2, blk_idx1); const vec = arr.slice(blk_ldx2, blk_idx2 + 1); sq[i][j] = decompose(vec); } } return sq; } // Square root Decomposition at level2 function doubleDecompose(sq1, arr) { const sq = []; let blk_idx = -1; for (let i = 0; i < sq1.blocks; i++) { let blk_ldx = blk_idx + 1; blk_idx = (i + 1) * sq1.blk_sz - 1; blk_idx = Math.min(blk_idx, arr.length - 1); const vec = arr.slice(blk_ldx, blk_idx + 1); sq.push(decompose(vec)); } return sq; } // Square root Decomposition at level1 function singleDecompose(arr) { Sq1 = decompose(arr); Sq2 = doubleDecompose(Sq1, arr); Sq3 = tripleDecompose(Sq1, Sq2[0], arr); } // Function for query at level 3 function queryLevel3(start, end, main_blk, sub_main_blk, arr) { const blk_sz = Sq3[0][0].blk_sz; // Element Indexing at level2 decomposition const nstart = start - main_blk * Sq1.blk_sz - sub_main_blk * Sq2[0].blk_sz; const nend = end - main_blk * Sq1.blk_sz - sub_main_blk * Sq2[0].blk_sz; // Block indexing at level3 decomposition const st_blk = Math.floor(nstart / blk_sz); const en_blk = Math.floor(nend / blk_sz); let answer = Sq3[main_blk][sub_main_blk].rdata[st_blk][en_blk]; // If start and end point don't lie in same block if (st_blk !== en_blk) { let left = 0; let en_idx = main_blk * Sq1.blk_sz + sub_main_blk * Sq2[0].blk_sz + (st_blk + 1) * blk_sz - 1; for (let i = start; i <= en_idx; i++) { left += arr[i]; } let right = 0; let st_idx = main_blk * Sq1.blk_sz + sub_main_blk * Sq2[0].blk_sz + en_blk * blk_sz; for (let i = st_idx; i <= end; i++) { right += arr[i]; } answer += left; answer += right; } else { for (let i = start; i <= end; i++) { answer += arr[i]; } } return answer; } // Function for splitting query to level two function queryLevel2(start, end, main_blk, arr) { const blk_sz = Sq2[0].blk_sz; // Element Indexing at level1 decomposition const nstart = start - (main_blk * Sq1.blk_sz); const nend = end - (main_blk * Sq1.blk_sz); // Block indexing at level2 decomposition const st_blk = Math.floor(nstart / blk_sz); const en_blk = Math.floor(nend / blk_sz); // Interblock data level2 decomposition let answer = Sq2[main_blk].rdata[st_blk][en_blk]; if (st_blk === en_blk) { answer += queryLevel3(start, end, main_blk, st_blk, arr); } else { answer += queryLevel3(start, main_blk * Sq1.blk_sz + (st_blk + 1) * blk_sz - 1, main_blk, st_blk, arr); answer += queryLevel3(main_blk * Sq1.blk_sz + en_blk * blk_sz, end, main_blk, en_blk, arr); } return answer; } // Function to return answer according to query function Query(start, end, arr) { const blk_sz = Sq1.blk_sz; const st_blk = Math.floor(start / blk_sz); const en_blk = Math.floor(end / blk_sz); // Interblock data level1 decomposition let answer = Sq1.rdata[st_blk][en_blk]; if (st_blk === en_blk) { answer += queryLevel2(start, end, st_blk, arr); } else { answer += queryLevel2(start, (st_blk + 1) * blk_sz - 1, st_blk, arr); answer += queryLevel2(en_blk * blk_sz, end, en_blk, arr); } // returning final answer return answer; } // Driver code function main() { const n1 = 16; const arr = [7, 2, 3, 0, 5, 10, 3, 12, 18, 1, 2, 3, 4, 5, 6, 7]; singleDecompose(arr); const q = 5; const query = [[6, 10], [7, 12], [4, 13], [4, 11], [12, 16]]; for (let i = 0; i < q; i++) { const [a, b] = query[i]; console.log(Query(a - 1, b - 1, arr)); } } main(); Time Complexity : O(q * d * n1/(2^3)) ? O(q * k) ? O(q)
Auxiliary Space : O(k * n) ? O(n)
Note : This article is to only explain the method of decomposing the square root to further decomposition.
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