Block Placement Queries

Block Placement Queries - Problem

There exists an infinite number line, with its origin at 0 and extending towards the positive x-axis. You are given a 2D array queries, which contains two types of queries:

For a query of type 1, queries[i] = [1, x]. Build an obstacle at distance x from the origin. It is guaranteed that there is no obstacle at distance x when the query is asked.

For a query of type 2, queries[i] = [2, x, sz]. Check if it is possible to place a block of size sz anywhere in the range [0, x] on the line, such that the block entirely lies in the range [0, x]. A block cannot be placed if it intersects with any obstacle, but it may touch it. Note that you do not actually place the block. Queries are separate.

Return a boolean array results, where results[i] is true if you can place the block specified in the i-th query of type 2, and false otherwise.

Input & Output

Example 1 — Basic Operations

$ Input: queries = [[1,2],[2,3,3],[2,5,2]]

› Output: [false,true]

💡 Note: Add obstacle at 2. Check if size-3 block fits in [0,3]: gap 0→2 is size 2, gap 2→3 is size 1, max gap < 3, so false. Check if size-2 block fits in [0,5]: gap 2→5 is size 3 ≥ 2, so true.

Example 2 — Multiple Obstacles

$ Input: queries = [[1,1],[1,4],[2,6,2],[2,6,3]]

› Output: [true,false]

💡 Note: Add obstacles at 1 and 4. Check size-2 block in [0,6]: gaps are 0→1(1), 1→4(2), 4→6(2), max gap=2≥2, so true. Check size-3 block: max gap=2<3, so false.

Example 3 — No Obstacles

$ Input: queries = [[2,5,3],[1,2],[2,5,6]]

› Output: [true,false]

💡 Note: Check size-3 block in [0,5] with no obstacles: entire range has size 6≥3, so true. Add obstacle at 2. Check size-6 block: max gap is now 3<6, so false.

Constraints

1 ≤ queries.length ≤ 10⁴
queries[i].length == 2 or 3
1 ≤ x ≤ 5 × 10⁴
1 ≤ sz ≤ 5 × 10⁴

Visualization

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Asked in

G Google 35 a Amazon 28 M Microsoft 22

The key insight is to track gaps between obstacles and find the maximum available space. The optimal approach uses TreeMap/sorted list with binary search for O(q log n) time complexity. Time: O(q log n), Space: O(n).

Common Approaches

✓ Binary Search with TreeMap

⏱️ Time: O(q × log n) Space: O(n)

Maintain obstacles in a TreeMap for O(log n) insertions and range queries. For block placement, use binary search to find obstacles within range and calculate gaps efficiently.

Brute Force with Set

⏱️ Time: O(q × x) Space: O(n)

For each type-2 query, iterate through all possible starting positions and check if the block can fit without intersecting any obstacles. Use a set to store obstacle positions for O(1) lookup.

Sorted List with Gap Tracking

⏱️ Time: O(q × n) Space: O(n)

Keep obstacles in a sorted list and for each block placement query, find the largest gap between consecutive obstacles. If the largest gap is ≥ block size, placement is possible.

Binary Search with TreeMap — Algorithm Steps

Use TreeMap to store obstacles with O(log n) operations
Binary search to find obstacles in query range
Calculate gaps and find maximum gap efficiently

Visualization

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Step-by-Step Walkthrough

TreeMap Storage

Maintain obstacles in sorted TreeMap for fast operations

Binary Search Range

Use binary search to find obstacles ≤ x

Calculate Max Gap

Find largest gap in O(k) time where k = obstacles in range

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <stdbool.h>

#define MAX_OBSTACLES 10000
#define MAX_QUERIES 10000

int compare_ints(const void *a, const void *b) {
    return (*(int*)a - *(int*)b);
}

int binary_search_right(int arr[], int n, int target) {
    int left = 0, right = n;
    while (left < right) {
        int mid = (left + right) / 2;
        if (arr[mid] <= target) {
            left = mid + 1;
        } else {
            right = mid;
        }
    }
    return left;
}

int solution(int queries[][3], int query_sizes[], int num_queries, bool* results) {
    int obstacles[MAX_OBSTACLES];
    int obstacle_count = 0;
    int result_count = 0;
    
    for (int i = 0; i < num_queries; i++) {
        if (queries[i][0] == 1) {  // Add obstacle
            obstacles[obstacle_count++] = queries[i][1];
            qsort(obstacles, obstacle_count, sizeof(int), compare_ints);
        } else {  // Check block placement
            int x = queries[i][1], sz = queries[i][2];
            
            // Find obstacles in range [0, x] using binary search
            int right_idx = binary_search_right(obstacles, obstacle_count, x);
            
            // Find maximum gap
            int max_gap = 0;
            
            if (right_idx == 0) {
                max_gap = x + 1;
            } else {
                // Gap from 0 to first obstacle
                if (obstacles[0] > max_gap) {
                    max_gap = obstacles[0];
                }
                
                // Gaps between consecutive obstacles
                for (int j = 1; j < right_idx; j++) {
                    int gap = obstacles[j] - obstacles[j-1] - 1;
                    if (gap > max_gap) {
                        max_gap = gap;
                    }
                }
                
                // Gap from last obstacle to x
                int gap = x - obstacles[right_idx-1];
                if (gap > max_gap) {
                    max_gap = gap;
                }
            }
            
            results[result_count++] = (max_gap >= sz);
        }
    }
    
    return result_count;
}

int main() {
    int queries[MAX_QUERIES][3];
    int query_sizes[MAX_QUERIES];
    bool results[MAX_QUERIES];
    
    // Mock data for demonstration
    queries[0][0] = 1; queries[0][1] = 2;
    queries[1][0] = 2; queries[1][1] = 3; queries[1][2] = 3;
    queries[2][0] = 2; queries[2][1] = 5; queries[2][2] = 2;
    query_sizes[0] = 2; query_sizes[1] = 3; query_sizes[2] = 3;
    
    int result_count = solution(queries, query_sizes, 3, results);
    
    printf("[");
    for (int i = 0; i < result_count; i++) {
        printf(results[i] ? "true" : "false");
        if (i < result_count - 1) printf(",");
    }
    printf("]\n");
    
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(q × log n)

Each query performs O(log n) operations on TreeMap, and gap calculation is O(k) where k is obstacles in range

✓ Linear Growth

Space Complexity

O(n)

TreeMap stores up to n obstacle positions

⚡ Linearithmic Space

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Binary Search with TreeMap — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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