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K-th Smallest in Range - Wavelet Trees | Data Structures | Repovive

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Data Structures19 sections · 729 units49h total

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K-th Smallest in Range

Range quantile query

Find the $k$ -th smallest value in $A[l..r]$ .

This is where wavelet trees excel. No other structure matches their efficiency for this query.

function kthSmallest(node, l, r, k)
 if node.lo = node.hi then
 return node.lo

 // Count elements going left
 leftCount := rank0(node.bitvector, r + 1) - rank0(node.bitvector, l)

 if k <= leftCount then
 // Answer is in left subtree
 newL := rank0(node.bitvector, l)
 newR := rank0(node.bitvector, r + 1) - 1
 return kthSmallest(node.left, newL, newR, k)
 else
 // Answer is in right subtree
 newL := rank1(node.bitvector, l)
 newR := rank1(node.bitvector, r + 1) - 1
 return kthSmallest(node.right, newL, newR, k - leftCount)

Time: $O(\log \sigma)$ .