Find Building Where Alice and Bob Can Meet

2940. Find Building Where Alice and Bob Can Meet

Difficulty: Hard

Topics: Array, Binary Search, Stack, Binary Indexed Tree, Segment Tree, Heap (Priority Queue), Monotonic Stack

You are given a 0-indexed array heights of positive integers, where heights[i] represents the height of the i^th building.

If a person is in building i, they can move to any other building j if and only if i < j and heights[i] < heights[j].

You are also given another array queries where queries[i] = [a_i, b_i]. On the i^th query, Alice is in building a_i while Bob is in building b_i.

Return an array ans where ans[i] is the index of the leftmost building where Alice and Bob can meet on the i^th query. If Alice and Bob cannot move to a common building on query i, set ans[i] to -1.

Example 1:

• Input: heights = [6,4,8,5,2,7], queries = [[0,1],[0,3],[2,4],[3,4],[2,2]]
• Output: [2,5,-1,5,2]
• Explanation: In the first query, Alice and Bob can move to building 2 since heights[0] < heights[2] and heights[1] < heights[2].
  • In the second query, Alice and Bob can move to building 5 since heights[0] < heights[5] and heights[3] < heights[5].
  • In the third query, Alice cannot meet Bob since Alice cannot move to any other building.
  • In the fourth query, Alice and Bob can move to building 5 since heights[3] < heights[5] and heights[4] < heights[5].
  • In the fifth query, Alice and Bob are already in the same building.
  • For ans[i] != -1, It can be shown that ans[i] is the leftmost building where Alice and Bob can meet.
  • For ans[i] == -1, It can be shown that there is no building where Alice and Bob can meet.

Example 2:

• Input: heights = [5,3,8,2,6,1,4,6], queries = [[0,7],[3,5],[5,2],[3,0],[1,6]]
• Output: [7,6,-1,4,6]
• Explanation: In the first query, Alice can directly move to Bob's building since heights[0] < heights[7].
  • In the second query, Alice and Bob can move to building 6 since heights[3] < heights[6] and heights[5] < heights[6].
  • In the third query, Alice cannot meet Bob since Bob cannot move to any other building.
  • In the fourth query, Alice and Bob can move to building 4 since heights[3] < heights[4] and heights[0] < heights[4].
  • In the fifth query, Alice can directly move to Bob's building since heights[1] < heights[6].
  • For ans[i] != -1, It can be shown that ans[i] is the leftmost building where Alice and Bob can meet.
  • For ans[i] == -1, It can be shown that there is no building where Alice and Bob can meet.

Constraints:

• 1 <= heights.length <= 5 * 10⁴
• 1 <= heights[i] <= 10⁹
• 1 <= queries.length <= 5 * 10⁴
• queries[i] = [a_i, b_i]
• 0 <= a_i, b_i <= heights.length - 1

Hint:

1. For each query [x, y], if x > y, swap x and y. Now, we can assume that x <= y.
2. For each query [x, y], if x == y or heights[x] < heights[y], then the answer is y since x ≤ y.
3. Otherwise, we need to find the smallest index t such that y < t and heights[x] < heights[t]. Note that heights[y] <= heights[x], so heights[x] < heights[t] is a sufficient condition.
4. To find index t for each query, sort the queries in descending order of y. Iterate over the queries while maintaining a monotonic stack which we can binary search over to find index t.

Solution:

The problem requires determining the leftmost building where Alice and Bob can meet given their starting buildings and movement rules. Each query involves finding a meeting point based on building heights. This is challenging due to the constraints on movement and the need for efficient computation.

Key Points

1. Alice and Bob can move to another building if its height is strictly greater than the current building.
2. For each query, find the leftmost valid meeting point, or return -1 if no such building exists.
3. The constraints demand a solution better than a naive O(n²) approach.

Approach

1. Observations:

   • If a == b, Alice and Bob are already at the same building.
   • If heights[a] < heights[b], Bob's building is the meeting point.
   • Otherwise, find the smallest building index t > b where:
     • heights[a] < heights[t]
     • heights[b] <= heights[t] (as b is already less than a in height comparison).

2. Optimization Using Monotonic Stack:

   • A monotonic stack helps efficiently track the valid buildings Alice and Bob can move to. Buildings are added to the stack in a way that ensures heights are in decreasing order, enabling fast binary searches.

3. Query Sorting:

   • Sort the queries in descending order of b to process buildings with larger indices first. This ensures that we build the stack efficiently as we move from higher to lower indices.

4. Binary Search on Stack:

   • For each query, use binary search on the monotonic stack to find the smallest index t that satisfies the conditions.

Plan

1. Sort queries based on the larger of the two indices (b) in descending order.
2. Traverse the array backward while maintaining a monotonic stack of valid indices.
3. For each query, check trivial cases (a == b or heights[a] < heights[b]).
4. For non-trivial cases, use the stack to find the leftmost valid building via binary search.
5. Return the results in the original query order.

Solution Steps

1. Preprocess Queries:

   • Ensure a <= b in each query for consistency.
   • Sort queries by b in descending order.

2. Iterate Through Queries:

   • Maintain a monotonic stack as we traverse the array.
   • For each query:
     • If a == b, the answer is b.
     • If heights[a] < heights[b], the answer is b.
     • Otherwise, use the stack to find the smallest valid index t > b.

   • Binary Search on Stack:

     • Use binary search to quickly find the smallest index t on the stack that satisfies heights[t] > heights[a].

   • Restore Original Order:

     • Map results back to the original query indices.

   • Return Results.

   Let's implement this solution in PHP: 2940. Find Building Where Alice and Bob Can Meet
   

   <?php
   /**
    * @param Integer[] $heights
    * @param Integer[][] $queries
    * @return Integer[]
    */
   function leftmostBuildingQueries($heights, $queries) {
       ...
       ...
       ...
       /**
        * go to ./solution.php
        */
   }
   
   /**
    * @param $queries
    * @return array
    */
   private function getIndexedQueries($queries) {
       ...
       ...
       ...
       /**
        * go to ./solution.php
        */
   }
   
   /**
    * @param $stack
    * @param $a
    * @param $heights
    * @return mixed|null
    */
   private function findUpperBound($stack, $a, $heights) {
       ...
       ...
       ...
       /**
        * go to ./solution.php
        */
   }
   
   class IndexedQuery {
       public $queryIndex;
       public $a; // Alice's index
       public $b; // Bob's index
   
       /**
        * @param $queryIndex
        * @param $a
        * @param $b
        */
       public function __construct($queryIndex, $a, $b) {
           $this->queryIndex = $queryIndex;
           $this->a = $a;
           $this->b = $b;
       }
   }
   
   // Test the function
   $heights = [6, 4, 8, 5, 2, 7];
   $queries = [[0, 1], [0, 3], [2, 4], [3, 4], [2, 2]];
   print_r(leftmostBuildingQueries($heights, $queries));
   
   $heights = [5, 3, 8, 2, 6, 1, 4, 6];
   $queries = [[0, 7], [3, 5], [5, 2], [3, 0], [1, 6]];
   print_r(leftmostBuildingQueries($heights, $queries));
   ?>
   

   Explanation:

   1. Sorting Queries: The queries are sorted by b in descending order to process larger indices first, which allows us to update our monotonic stack as we process.
   2. Monotonic Stack: The stack is used to keep track of building indices where Alice and Bob can meet. We only keep buildings that have a height larger than any previously seen buildings in the stack.
   3. Binary Search: When answering each query, we use binary search to efficiently find the smallest index t where the conditions are met.

   Example Walkthrough

   Input:

   • heights = [6,4,8,5,2,7]
   • queries = [[0,1],[0,3],[2,4],[3,4],[2,2]]

   Process:

   1. Sort Queries:

      • Indexed queries: [(2,4), (3,4), (0,3), (0,1), (2,2)]

   2. Build Monotonic Stack:

      • Start at the highest index and add indices to the stack:
        • At index 5: Stack = [5]
        • At index 4: Stack = [5, 4]
        • ...

   3. Query Processing:

      • For query [0,1], heights[0] < heights[1]: Result = 2.
      • ...

   Output:

   [2, 5, -1, 5, 2]

   Time Complexity

   1. Query Sorting: O(Q log Q) where Q is the number of queries.
   2. Monotonic Stack Construction: O(N) where N is the length of heights.
   3. Binary Search for Each Query: O(Q log N).

   Overall: O(N Q log (Q N)).

   Output for Example

   Input:
   

   $heights = [6, 4, 8, 5, 2, 7];
   $queries = [[0, 1], [0, 3], [2, 4], [3, 4], [2, 2]];
   

   Output:
   

   print_r(findBuilding($heights, $queries)); // [2, 5, -1, 5, 2]
   

   This approach efficiently handles large constraints by leveraging a monotonic stack and binary search. It ensures optimal query processing while maintaining correctness.

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