Query to check if a range is made up of consecutive elements
Given an array of n non-consecutive integers and Q queries, the task is to check whether for the given range l and r, the elements are consecutive or not.
Examples:
Input: arr = { 2, 4, 3, 7, 6, 1}, Q = { (1, 3), (3, 5), (5, 6) }
Output: Yes, No, No
Explanation: Array elements from (1, 3) = {2, 4, 3}, (3, 5) = {3, 7, 6}, (5, 6) = {6, 1}
So the answer is Yes, No, No
Input: arr = {1, 2, 3, 4, 5}, Q = { (1, 2), (4, 5), (1, 5) }
Output:Yes, Yes, Yes
Naive Approach:
The sub array between range l and r can be sorted and checked if it contains consequetive elements or not.
Time Complexity: O(q*n*log n)
where n is the size of the array and q is the number of queries.
Efficient approach:
- A segment tree can be used to get the minimum and maximum in a range.
- If the minimum element of a sub array is x and the maximum element is y then we can there is no element less than x and more than y, so we can say that x<=arr[i]<=y.
- As there are no repeated elements, so it can be concluded that if the difference of maximum and minimum element + 1 is equal to the length of the range (r-l+1) then the range consists of consecutive elements.
Below is the implementation of the above approach.
CPP
// C++ implementation of the above approach. #include <bits/stdc++.h> using namespace std; // Segment tree int tree_min[1000001], tree_max[1000001]; // Functions to return // minimum and maximum int min(int a, int b) { return a < b ? a : b; } int max(int a, int b) { return a > b ? a : b; } // Segment tree for minimum element void build_min(int array[], int node, int left, int right) { // If left is equal to right if (left == right) tree_min[node] = array[left]; else { // Divide the segment into equal halves build_min(array, 2 * node, left, (left + right) / 2); build_min(array, 2 * node + 1, (left + right) / 2 + 1, right); // Update the element as minimum // of the divided two subarrays tree_min[node] = min(tree_min[2 * node], tree_min[2 * node + 1]); } } // Query to find a minimum // in a given ranges int query_min(int node, int c_l, int c_r, int l, int r) { // Out of range if (c_l > r || c_r < l) return 1e9; // Within the range completely if (c_l >= l && c_r <= r) return tree_min[node]; else // Divide the range into two halves // Query for each half // and return the minimum of two return min(query_min(2 * node, c_l, (c_l + c_r) / 2, l, r), query_min(2 * node + 1, (c_l + c_r) / 2 + 1, c_r, l, r)); } // Segment tree for maximum element void build_max(int array[], int node, int left, int right) { // If left is equal to right if (left == right) tree_max[node] = array[left]; else { // Divide the segment into equal halves build_max(array, 2 * node, left, (left + right) / 2); build_max(array, 2 * node + 1, (left + right) / 2 + 1, right); // Update the element as maximum // of the divided two subarrays tree_max[node] = max(tree_max[2 * node], tree_max[2 * node + 1]); } } // Query to find maximum // in a given ranges int query_max(int node, int c_l, int c_r, int l, int r) { // Out of range if (c_l > r || c_r < l) return -1; // Within the range completely if (c_l >= l && c_r <= r) return tree_max[node]; else // Divide the range into two halves // and query for each half // and return the maximum of two return max(query_max(2 * node, c_l, (c_l + c_r) / 2, l, r), query_max(2 * node + 1, (c_l + c_r) / 2 + 1, c_r, l, r)); } // Build the tree void init(int array[], int n) { build_min(array, 1, 0, n - 1); build_max(array, 1, 0, n - 1); } // Check if the given range is Consecutive bool isConsecutive(int x, int y, int n) { return ((query_max(1, 0, n - 1, x, y) - query_min(1, 0, n - 1, x, y)) == (y - x)); } // Driver code int main() { int arr[] = { 2, 4, 3, 7, 6, 1 }; int query[][2] = { { 1, 3 }, { 3, 5 }, { 5, 6 } }; int n = sizeof(arr) / sizeof(int), q = 3; init(arr, n); for (int i = 0; i < q; i++) { int l, r; l = query[i][0]; r = query[i][1]; cout << (isConsecutive(l - 1, r - 1, n) ? "Yes" : "No") << "\n"; } return 0; } |
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Java
// Java implementation of the above approach. class GFG { // Segment tree static int[] tree_min = new int[1000001]; static int[] tree_max = new int[1000001]; // Functions to return // minimum and maximum static int min(int a, int b) { return a < b ? a : b; } static int max(int a, int b) { return a > b ? a : b; } // Segment tree for minimum element static void build_min(int array[], int node, int left, int right) { // If left is equal to right if (left == right) tree_min[node] = array[left]; else { // Divide the segment into equal halves build_min(array, 2 * node, left, (left + right) / 2); build_min(array, 2 * node + 1, (left + right) / 2 + 1, right); // Update the element as minimum // of the divided two subarrays tree_min[node] = Math.min(tree_min[2 * node], tree_min[2 * node + 1]); } } // Query to find a minimum // in a given ranges static int query_min(int node, int c_l, int c_r, int l, int r) { // Out of range if (c_l > r || c_r < l) return (int) 1e9; // Within the range completely if (c_l >= l && c_r <= r) return tree_min[node]; else // Divide the range into two halves // Query for each half // and return the minimum of two return min(query_min(2 * node, c_l, (c_l + c_r) / 2, l, r), query_min(2 * node + 1, (c_l + c_r) / 2 + 1, c_r, l, r)); } // Segment tree for maximum element static void build_max(int array[], int node, int left, int right) { // If left is equal to right if (left == right) tree_max[node] = array[left]; else { // Divide the segment into equal halves build_max(array, 2 * node, left, (left + right) / 2); build_max(array, 2 * node + 1, (left + right) / 2 + 1, right); // Update the element as maximum // of the divided two subarrays tree_max[node] = Math.max(tree_max[2 * node], tree_max[2 * node + 1]); } } // Query to find maximum // in a given ranges static int query_max(int node, int c_l, int c_r, int l, int r) { // Out of range if (c_l > r || c_r < l) return -1; // Within the range completely if (c_l >= l && c_r <= r) return tree_max[node]; else // Divide the range into two halves // and query for each half // and return the maximum of two return Math.max(query_max(2 * node, c_l, (c_l + c_r) / 2, l, r), query_max(2 * node + 1, (c_l + c_r) / 2 + 1, c_r, l, r)); } // Build the tree static void init(int array[], int n) { build_min(array, 1, 0, n - 1); build_max(array, 1, 0, n - 1); } // Check if the given range is Consecutive static boolean isConsecutive(int x, int y, int n) { return ((query_max(1, 0, n - 1, x, y) - query_min(1, 0, n - 1, x, y)) == (y - x)); } // Driver code public static void main(String[] args) { int arr[] = { 2, 4, 3, 7, 6, 1 }; int query[][] = { { 1, 3 }, { 3, 5 }, { 5, 6 } }; int n = arr.length, q = 3; init(arr, n); for (int i = 0; i < q; i++) { int l, r; l = query[i][0]; r = query[i][1]; System.out.print((isConsecutive(l - 1, r - 1, n) ? "Yes" : "No") + "\n"); } } } // This code is contributed by PrinciRaj1992 |
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Python
# Python3 implementation of the above approach. # Segment tree tree_min = [0] * 1000001 tree_max = [0] * 1000001 # Segment tree for minimum element def build_min(array, node,left, right): # If left is equal to right if (left == right): tree_min[node] = array[left] else : # Divide the segment into equal halves build_min(array, 2 * node, left,(left + right) // 2) build_min(array, 2 * node + 1,(left + right) // 2 + 1, right) # Update the element as minimum # of the divided two subarrays tree_min[node] = min(tree_min[2 * node], tree_min[2 * node + 1]) # Query to find a minimum # in a given ranges def query_min(node, c_l, c_r, l, r): # Out of range if (c_l > r or c_r < l): return 10**9 # Within the range completely if (c_l >= l and c_r <= r): return tree_min[node] else: # Divide the range into two halves # Query for each half # and return the minimum of two return min(query_min(2 * node, c_l,(c_l + c_r) // 2, l, r), query_min(2 * node + 1, (c_l + c_r) // 2 + 1,c_r, l, r)) # Segment tree for maximum element def build_max(array, node, left, right): # If left is equal to right if (left == right): tree_max[node] = array[left] else : # Divide the segment into equal halves build_max(array, 2 * node, left,(left + right) // 2) build_max(array, 2 * node + 1,(left + right) // 2 + 1, right) # Update the element as maximum # of the divided two subarrays tree_max[node] = max(tree_max[2 * node],tree_max[2 * node + 1]) # Query to find maximum # in a given ranges def query_max(node, c_l, c_r, l, r): # Out of range if (c_l > r or c_r < l): return -1 # Within the range completely if (c_l >= l and c_r <= r): return tree_max[node] else: # Divide the range into two halves # and query for each half # and return the maximum of two return max(query_max(2 * node, c_l,(c_l + c_r) // 2, l, r),query_max(2 * node + 1,(c_l + c_r) // 2 + 1,c_r, l, r)) # Build the tree def init(array, n): build_min(array, 1, 0, n - 1) build_max(array, 1, 0, n - 1) # Check if the given range is Consecutive def isConsecutive(x, y, n): return ((query_max(1, 0, n - 1, x, y) - query_min(1, 0, n - 1, x, y)) == (y - x)) # Driver code arr = [2, 4, 3, 7, 6, 1] query = [ [1, 3] , [3, 5 ], [5, 6] ] n = len(arr) q = 3init(arr, n) for i in range(q): l = query[i][0] r = query[i][1] if (isConsecutive(l - 1, r - 1, n) ): print("Yes") else: print("No") # This code is contributed by mohit kumar 29 |
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C#
// C# implementation of the above approach. using System; class GFG { // Segment tree static int[] tree_min = new int[1000001]; static int[] tree_max = new int[1000001]; // Functions to return // minimum and maximum static int min(int a, int b) { return a < b ? a : b; } static int max(int a, int b) { return a > b ? a : b; } // Segment tree for minimum element static void build_min(int []array, int node, int left, int right) { // If left is equal to right if (left == right) tree_min[node] = array[left]; else { // Divide the segment into equal halves build_min(array, 2 * node, left, (left + right) / 2); build_min(array, 2 * node + 1, (left + right) / 2 + 1, right); // Update the element as minimum // of the divided two subarrays tree_min[node] = Math.Min(tree_min[2 * node], tree_min[2 * node + 1]); } } // Query to find a minimum // in a given ranges static int query_min(int node, int c_l, int c_r, int l, int r) { // Out of range if (c_l > r || c_r < l) return (int) 1e9; // Within the range completely if (c_l >= l && c_r <= r) return tree_min[node]; else // Divide the range into two halves // Query for each half // and return the minimum of two return min(query_min(2 * node, c_l, (c_l + c_r) / 2, l, r), query_min(2 * node + 1, (c_l + c_r) / 2 + 1, c_r, l, r)); } // Segment tree for maximum element static void build_max(int []array, int node, int left, int right) { // If left is equal to right if (left == right) tree_max[node] = array[left]; else { // Divide the segment into equal halves build_max(array, 2 * node, left, (left + right) / 2); build_max(array, 2 * node + 1, (left + right) / 2 + 1, right); // Update the element as maximum // of the divided two subarrays tree_max[node] = Math.Max(tree_max[2 * node], tree_max[2 * node + 1]); } } // Query to find maximum // in a given ranges static int query_max(int node, int c_l, int c_r, int l, int r) { // Out of range if (c_l > r || c_r < l) return -1; // Within the range completely if (c_l >= l && c_r <= r) return tree_max[node]; else // Divide the range into two halves // and query for each half // and return the maximum of two return Math.Max(query_max(2 * node, c_l, (c_l + c_r) / 2, l, r), query_max(2 * node + 1, (c_l + c_r) / 2 + 1, c_r, l, r)); } // Build the tree static void init(int []array, int n) { build_min(array, 1, 0, n - 1); build_max(array, 1, 0, n - 1); } // Check if the given range is Consecutive static bool isConsecutive(int x, int y, int n) { return ((query_max(1, 0, n - 1, x, y) - query_min(1, 0, n - 1, x, y)) == (y - x)); } // Driver code public static void Main(String[] args) { int []arr = {2, 4, 3, 7, 6, 1}; int [,]query = {{ 1, 3 }, { 3, 5 }, { 5, 6 }}; int n = arr.Length, q = 3; init(arr, n); for (int i = 0; i < q; i++) { int l, r; l = query[i,0]; r = query[i,1]; Console.Write((isConsecutive(l - 1, r - 1, n) ? "Yes" : "No") + "\n"); } } } // This code is contributed by Rajput-Ji |
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Output:
Yes No No
Time Complexity: O(Q*log(n))
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