Given a rooted tree (assume root is 1) of N nodes and Q queries, each of the form (Val, Node). For each query, the task is to find the number of nodes with values smaller than Val in sub-tree of Node, including itself.
Note that by definition, nodes in this tree are unique.
Examples:
Input: N = 7, Q = 3 Val = 4, Node = 4 Val = 7, Node = 6 Val = 5, Node = 1 Given tree:
Output: 2 1 4 Explanation: For given queries: Q1 -> Val = 4, Node = 4 The required nodes are 2 and 3 Hence the output is 2 Q2 -> Val = 7, Node = 6 The required node is 6 only Hence the output is 1 Q3 -> Val = 5, Node = 1 The required nodes are 1, 2, 3 and 4 Hence the output is 4
Naive approach: A simple approach to solve this problem would be to run DFS from a given node for each query and count the number of nodes smaller than the given value. The parent of a given node must be excluded from the DFS.
Time complexity: O(N*Q), where Q is the number of queries and N is the number of nodes in the tree.
Efficient Approach: We can reduce the problem of finding the number of elements in a sub-tree to finding them in contiguous segments of an array. To generate such a representation we run a DFS from the root node and push the node into an array when we enter into it the first time and while exiting for the last time. This representation of a tree is known as Euler Tour of the tree.
For example,
The Euler Tour of the above tree will be:
1 4 2 2 3 3 5 5 4 6 7 7 6 1
This representation of tree has the property that the sub-tree of every node X is contained within the first and last occurrence of X in the array. Each node appears exactly twice. So counting the number of nodes smaller than Val between the first and last occurrence of Node will give us twice the answer of that query.
Using this representation, the queries can be processed offline in O(log(N)) per query using a binary indexed tree.
Pre-processing:
- We store the index of 1st and last occurrence of each node in the Tour in two arrays, start and end. Let start[X] and end[X] represent these indices for node X. This can be done in O(N)
- In the Euler Tour we store the position of the element along with node as a pair (indexInTree, indexInTour), and then sort according to indexInTree. Let this array be sortedTour
- Similarly, we maintain an array of Queries of the form (Val, Node) and sort according to Val. Let this array be sortedQuery
- Initialize a Binary Indexed Tree of size 2N with all entries as 0. Let this be bit
- Then proceed as follows. Maintain a pointer each in sortedTour and sortedQuery
- For each query in sortedQuery from the beginning select the nodes from sortedTour having indexInTree < Val, and increment their indexInTour in bit. Then the answer of that query would be half of the sum from start[Node] to end[Node]
- For the next query in sortedQuery we select any previously un-selected nodes from sortedTour having indexInTree < Val, and increment their indexInTour in bit and answer the query as done before.
- Repeating this process for each query we can answer them in O(Qlog(N)).
Below is the C++ implementation of the above approach:
CPP
// C++ program Queries to find the Number of// Nodes having Smaller Values from a Given// Value in the Subtree of a Node#include <bits/stdc++.h>using namespace std; // Takes a tree and generates a Euler tour for that// tree in 'tour' parameter This is a modified form// of DFS in which we push the node while entering for// the first time and when exiting from itvoid eulerTour(vector<vector<int> >& tree, vector<int>& vst, int root, vector<int>& tour){ // Push node in tour while entering tour.push_back(root); // DFS vst[root] = 1; for (auto& x : tree[root]) { if (!vst[x]) { eulerTour(tree, vst, x, tour); } } // Push node in tour while exiting tour.push_back(root);} // Creates the start and end array as described in// pre-processing section. Traverse the tour from// left to right and update start for 1st occurrence// and end for 2nd occurrence of each nodevoid createStartEnd(vector<int>& tour, vector<int>& start, vector<int>& end){ for (int i = 1; i < tour.size(); ++i) { if (start[tour[i]] == -1) { start[tour[i]] = i; } else { end[tour[i]] = i; } }} // Returns a sorted array of pair containing node and// tourIndex as described in pre-processing sectionvector<pair<int, int> >createSortedTour(vector<int>& tour){ vector<pair<int, int> > arr; for (int i = 1; i < tour.size(); ++i) { arr.push_back(make_pair(tour[i], i)); } sort(arr.begin(), arr.end()); return arr;} // Binary Indexed Tree Implementation// This function will add 1 from the positionvoid increment(vector<int>& bit, int pos){ for (; pos < bit.size(); pos += pos & -pos) { bit[pos]++; }} // It will give the range sumint query(vector<int>& bit, int start, int end){ --start; int s1 = 0, s2 = 0; for (; start > 0; start -= start & -start) { s1 += bit[start]; } for (; end > 0; end -= end & -end) { s2 += bit[end]; } return s2 - s1;} // Function to calculate the ans for each querymap<pair<int, int>, int> cal(int N, int Q, vector<vector<int>> tree, vector<pair<int, int>> queries){ // Preprocessing // Creating the vector to store the tours and queries vector<int> tour, vst(N + 1, 0), start(N + 1, -1), end(N + 1, -1), bit(2 * N + 4, 0); // For one based indexing in tour. // Simplifies code for Binary Indexed Tree. // We will ignore the 1st element in tour tour.push_back(-1); // Create Euler Tour eulerTour(tree, vst, 1, tour); // Create Start and End Array createStartEnd(tour, start, end); // Create sortedTour and sortedQuery auto sortedTour = createSortedTour(tour); auto sortedQuery = queries; sort(sortedQuery.begin(), sortedQuery.end()); // For storing answers to query map<pair<int, int>, int> queryAns; // We maintain pointers each for sortedTour and // sortedQuery.For each element X in sortedTour // we first process any queries with val smaller // than X's node and update queryptr to first // unprocessed query.Then update the position // of X in BIT. int tourptr = 0, queryptr = 0; while (queryptr < sortedQuery.size()) { // Queries lies in the range then while (queryptr < sortedQuery.size() && sortedQuery[queryptr].first <= sortedTour[tourptr].first){ auto node = sortedQuery[queryptr].second; // Making the query on BIT and dividing by 2. queryAns[sortedQuery[queryptr]] = query(bit, start[node], end[node]) / 2; ++queryptr; } if (tourptr < sortedTour.size()) { increment(bit, sortedTour[tourptr].second); ++tourptr; } } return queryAns;} // Driver Codeint main(){ int N = 7, Q = 3; // Tree edges vector<vector<int> > tree = { {}, { 4, 6 }, { 4 }, { 4 }, { 1, 2, 3, 5 }, { 4 }, { 1, 7 }, { 6 } }; // Queries vector vector<pair<int, int> > queries = { make_pair(4, 1), make_pair(7, 6), make_pair(5, 1) }; // Calling the function map<pair<int, int>, int> queryAns = cal(N, Q, tree, queries); // Print answer in order of input. for (auto& x : queries) { cout << queryAns[x] << '\n'; } return 0;} |
Python3
# Python3 implementation for the above approach# Import necessary librariesfrom typing import List, Tuple, Dictfrom collections import defaultdict# Recursive function to perform euler tour on the treedef eulerTour(tree: List[List[int]], vst: List[int], root: int, tour: List[int]) -> None: # Append the root to the tour tour.append(root) # Mark the root as visited vst[root] = 1 # Traverse through all the children of the root for x in tree[root]: # If a child is not visited, # perform euler tour on that child if not vst[x]: eulerTour(tree, vst, x, tour) # Append the root again to # complete the cycle of euler tour tour.append(root)# Create start and end index for each node in the euler tourdef createStartEnd(tour: List[int], start: List[int], end: List[int]) -> None: # Traverse through the euler tour and # store the start and end index for each node for i in range(1, len(tour)): if start[tour[i]] == -1: start[tour[i]] = i else: end[tour[i]] = i # Create a sorted euler tourdef createSortedTour(tour: List[int]) -> List[Tuple[int, int]]: # Create a tuple with the node and its index in the euler tour arr = [(tour[i], i) for i in range(1, len(tour))] # Sort the array of tuples by the node values arr.sort() # Return the sorted array of tuples return arr # Increment the value of a position in the binary indexed treedef increment(bit: List[int], pos: int) -> None: # Traverse through the binary indexed tree and increment # the value at the given position and its ancestors while pos < len(bit): bit[pos] += 1 pos += pos & -posdef query(bit: List[int], start: int, end: int) -> int: # Subtract 1 from start position as indexing starts from 0 start -= 1 s1, s2 = 0, 0 # Calculate sum of BIT from start to root of the tree while start > 0: s1 += bit[start] start -= start & -start # Calculate sum of BIT from end to root of the tree while end > 0: s2 += bit[end] end -= end & -end # Return the difference between s2 and s1 return s2 - s1def cal(N: int, Q: int, tree: List[List[int]], queries: List[Tuple[int, int]]) -> Dict[Tuple[int, int], int]: # Initialize variables tour, vst, start, end, bit = [-1], [0]*(N+1), [-1]*(N+1), [-1]*(N+1), [0]*(2*N+4) # Perform Euler tour to create the tour list eulerTour(tree, vst, 1, tour) # Create start and end arrays createStartEnd(tour, start, end) # Sort the tour list based on node values sortedTour = createSortedTour(tour) # Sort the queries list sortedQuery = sorted(queries) # Initialize a defaultdict to store the query results queryAns = defaultdict(int) # Initialize pointers tourptr, queryptr = 0, 0 # Iterate through sortedQuery list while queryptr < len(sortedQuery): # Iterate through queries whose node values are # less than or equal to the current tour node while queryptr < len(sortedQuery) and sortedQuery[queryptr][0] <= sortedTour[tourptr][0]: # Get the node value from the current query node = sortedQuery[queryptr][1] # Calculate and store the query result in queryAns dictionary queryAns[sortedQuery[queryptr]] = query(bit, start[node], end[node]) // 2 # Increment the query pointer queryptr += 1 # Increment BIT at the position # corresponding to the current tour node if tourptr < len(sortedTour): increment(bit, sortedTour[tourptr][1]) # Increment the tour pointer tourptr += 1 # Return the queryAns dictionary return queryAns# Driver Codeif __name__ == "__main__": import math N = 7 Q = 3 # Tree edges tree = { 1: [4, 6], 2: [4], 3: [4], 4: [1, 2, 3, 5], 5: [4], 6: [1, 7], 7: [6] } # Queries vector queries = [(4, 1), (7, 6), (5, 1)] # Calling the function queryAns = cal(N, Q, tree, queries) # Print answer in order of input. for query in queries: print(queryAns[query])# This code is contributed by amit_mangal_ |
Output:
3 1 4
Auxiliary Space: O(Q+N*log(N))
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