kd-Trees: Data Structure for Efficient Range and Nearest Neighbor Searches, Study notes of Data Structures and Algorithms

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kd-Trees

CMSC 420

kd-Trees

• Invented in 1970s by Jon Bentley
• Name originally meant “3d-trees, 4d-trees, etc”

where k was the # of dimensions

• Now, people say “kd-tree of dimension d”
• Idea: Each level of the tree compares against 1

dimension.

• Let’s us have only^ two children^ at each node

(instead of 2

d

kd-tree example

x

y

x

y

insert: (30,40), (5,25), (10,12), (70,70), (50,30), (35,45)

insert(Point x, KDNode t, int cd) { if t == null t = new KDNode(x) else if (x == t.data) // error! duplicate else if (x[cd] < t.data[cd]) t.left = insert(x, t.left, (cd+1) % DIM) else t.right = insert(x, t.right, (cd+1) % DIM) return t }

Insert Code

FindMin

x

y

x

y

(51,75)

(25,40) (10,30) (55,1) (1,10) (70,70) (60,80) (35,90)

FindMin(x-dimension):

(50,50)

FindMin

x

y

x

y

(51,75)

(25,40) (10,30) (55,1) (1,10) (70,70) (60,80) (35,90)

FindMin(y-dimension):

(50,50)

Point findmin(Node T, int dim, int cd): // empty tree if T == NULL : return NULL // T splits on the dimension we’re searching // => only visit left subtree if cd == dim: if t.left == NULL : return t.data else return findmin(T.left, dim, (cd+1)%DIM) // T splits on a different dimension // => have to search both subtrees else : return minimum( findmin(T.left, dim, (cd+1)%DIM), findmin(T.right, dim, (cd+1)%DIM) T.data )

FindMin Code

Delete in kd-trees

Q (^) P

A

Want to delete node A.

Assume cutting

dimension of A is cd

In BST, we’d

findmin (A.right).

Here, we have to

findmin (A.right, cd)

cd

cd B

Everything in Q has

cd-coord < B, and

everything in P has cd-

coord ≥ B

No right subtree --- Solution

- Swap the subtrees of node to

be deleted

- B = find min (T.left) - Replace deleted node by B Q

x^ (x,y)

cd (a,b)

(a,c)

Now, if there is another

point with x=a, it

appears in the right

subtree, where it should

Point delete(Point x, Node T, int cd): if T == NULL : error point not found! next_cd = (cd+1)%DIM // This is the point to delete: if x = T.data: // use min(cd) from right subtree: if t.right != NULL: t.data = findmin(T.right, cd, next_cd) t.right = delete(t.data, t.right, next_cd) // swap subtrees and use min(cd) from new right: else if T.left != NULL: t.data = findmin(T.left, cd, next_cd) t.right = delete(t.data, t.left, next_cd) else t = null // we’re a leaf: just remove // this is not the point, so search for it: else if x[cd] < t.data[cd]: t.left = delete(x, t.left, next_cd) else t.right = delete(x, t.right, next_cd) return t

kd-Trees Nearest Neighbor

• Idea: traverse the whole tree,^ BUT make two

modifications to prune to search space:

1. Keep variable of closest point C found so far.

Prune subtrees once their bounding boxes say

that they can’t contain any point closer than C

2. Search the subtrees in order that maximizes the

chance for pruning

Nearest Neighbor: Ideas, continued

d

Query

Point Q

T

Bounding box

of subtree

rooted at T

If d > dist(C, Q), then no

point in BB(T) can be

closer to Q than C.

Hence, no reason to search

subtree rooted at T.

Recurse, but start with the subtree “closer” to Q:

First search the subtree that would contain Q if we were

inserting Q below T.

Update the best point so far, if T is better:

if dist(C, Q) > dist(T.data, Q), C := T.data

Nearest Neighbor Facts

• Might have to search close to the whole tree in the

worst case. [O(n)]

• In practice, runtime is closer to:

O(

d

• log n)

• log n to find cells “near” the query point

d to search around cells in that neighborhood

• Three important concepts that reoccur in range /

nearest neighbor searching:

storing partial results : keep best so far, and update

pruning : reduce search space by eliminating irrelevant trees.

traversal order : visit the most promising subtree first.