Persistent Segment Tree

[putul03]

The persistent segment tree allows us to efficiently query and update an array while preserving previous versions of the array. The primary operations include:

1. Build Operation: Constructs the segment tree from the given array. Each node represents a range of the array and stores the sum of that range.

2. Update Operation: When an update is performed, a new version of the segment tree is created without modifying the existing tree. This is achieved by copying the necessary nodes and updating the affected ranges.

3. Query Operation: To retrieve the sum of a given range, the segment tree is traversed based on the requested indices. The operation can access multiple versions of the tree to return the correct sum for each version.

Time and Space Complexity:

• Time Complexity:

  • Build Operation: O(n log n), where n is the number of elements in the array. This is because we need to recursively build the tree.
  • Update Operation: O(log n) per update, since we only traverse the necessary nodes to create a new version.
  • Query Operation: O(log n) per query, as we traverse the segment tree to find the required sums.

• Space Complexity:

  • The space complexity is O(n * log n) for storing multiple versions of the segment tree, where each version contains log n nodes and there can be up to n versions depending on the number of updates.

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