In the worst case, what is the time complexity of performing a range search of a k-d tree in terms of the number of nodes n in the tree?

Question

In the worst case, what is the time complexity of performing a range search of a k-d tree in terms of the number of nodes n in the tree?

BlackTom AI · Accepted Answer

To begin, let's lay out the scenario: we are performing a range search on a k-d tree with n nodes, and we want the worst-case time complexity.
Option 1: O(1). This would imply the search finishes in constant time regardless of n, which is not possible for a search over a data structure containing n nodes since at least information about many nodes may need to be examined.
Option 2: O(n). In the worst case, a range query may have to visit every node in the tree, for example if the query range encompasses nearly all points or if the tree is highly unbalanced with many nodes lying on the boundary of the query region. This leads to examining all nodes, yielding linear time in n.
Option 3: O(log n). This is a common target for balanced binary search-like structures for a single-point query or certain average-case scenarios, but in the worst case a range search can degrade to visiting many subtrees or nodes, which can exceed logarithmic growth.
Option 4: O(n^2). Quadratic time would indicate repeatedly traversing large portions of the tree for many reasons, which is not the standard worst-case bound for a single range query in a kd-tree; while pathological inputs could blur into higher constants, the formal worst-case bound is linear in n, not quadratic.
In summary, the only option that aligns with the established worst-case behavior of a kd-tree range search is that the operation may touch every node, leading to a linear-time bound in the number of nodes, i.e., O(n) for the worst case.

XLMC0202501 Topic 18 Quiz

In the worst case, what is the time complexity of performing a range search of a k-d tree in terms of the number of nodes n in the tree?

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