The space complexity of a binary heap is O(n) because it stores n elements in an array.

Building a binary heap from an array can be done in O(n) time using the heapify process.

Building a heap from an array can be done in O(n) time using the heapify process.

Deleting the maximum element from a max-heap involves removing the root and then re-heapifying, which takes O(log n) time.

Extracting the minimum element from a min-heap requires restructuring the heap, which takes O(log n) time.

Inserting an element into a binary heap takes O(log n) time due to the need to maintain the heap property.

Inserting an element into a binary heap requires maintaining the heap property, which takes O(log n) time.

In the average case, inserting an element into a hash table has a time complexity of O(1) due to direct indexing.

Removing the highest priority element from a binary heap has a time complexity of O(log n) due to the need to re-heapify.

Removing the highest priority element from a binary heap takes O(log n) time due to the need to maintain the heap property after removal.

In the average case, searching for an element in a hash table has a time complexity of O(1) due to direct indexing.

On average, searching for an element in a hash table has a time complexity of O(1) due to direct access via the hash function.

In a well-designed hash table, the average time complexity for searching is O(1), assuming a good hash function and low collision rate.

The time complexity of the 'find' operation in a Disjoint Set Union with path compression is O(α(n)), where α is the inverse Ackermann function.

The time complexity of the 'find' operation in a well-optimized Disjoint Set Union with path compression is O(α(n)), where α is the inverse Ackermann function.

The time complexity of the 'Find' operation with path compression and union by rank is O(α(n)), where α is the inverse Ackermann function.

The time complexity of the 'Find' operation with path compression in Disjoint Set Union is O(α(n)), where α is the inverse Ackermann function.

The time complexity of the 'Union' operation in an optimized Disjoint Set Union with path compression is O(α(n)), where α is the inverse Ackermann function.

The time complexity of the 'Union' operation in an optimized Disjoint Set Union with path compression and union by rank is O(α(n)), where α is the inverse Ackermann function.

The worst-case time complexity for a sequence of m union and find operations in Disjoint Set Union with path compression and union by rank is O(m α(n)).

The worst-case time complexity for building a heap from an array of n elements is O(n) using the heapify process.

Deleting the minimum element from a binary min-heap requires re-structuring the heap, which takes O(log n) time.

Deleting the minimum element from a binary heap takes O(log n) time as it requires re-heapifying the structure.

Deleting the minimum element from a min-heap involves removing the root and then re-heapifying, which takes O(log n) time.

In the worst case, if all keys hash to the same index, searching in a hash table with chaining can take O(n) time.

The worst-case time complexity of the 'Union' operation in Disjoint Set Union with union by rank is O(α(n)).

The worst-case time complexity of the union operation in a basic Disjoint Set Union without optimizations is O(n).

Path Compression is a technique used in Disjoint Set Union to optimize the 'Find' operation by flattening the structure of the tree whenever 'Find' is called.

Path Compression is a technique used to optimize the 'Find' operation by flattening the structure of the tree whenever 'Find' is called.

Path compression is a technique used to optimize the union operation in Disjoint Set Union, making future find operations faster.