Table 3.

The algorithm to execute interval query using range tree with fractional cascading. Input T is the range tree; I is the query interval; R is the relationship. Return the result interval set S corresponding to all the intervals in range tree T satisfying the relationship R with interval I.

Algorithm QueryRTFC(T, I, R)
1.	Transform interval query with respect to interval I and relationship R to range query with x range [x1, x2] and y range [y1, y2] according to Table 1.
2.	Initialize the result set S = {}
3.	Find the split node vsplit in range tree T where the paths to x1 and x2 split, or the leaf where both paths end.
4.	if vsplit is a leaf node then
5.	if vsplit. interval. x ∈ [x1, x2] and vsplit. interval. y ∈ [y1, y2] then
6.	Report the interval in vsplit, S = S ∪ {vsplit. interval}
7.	end if
8.	return S
9.	Perform binary search on vsplit. data for y1 by y-value, find the index i of the smallest element no less than y1 in vsplit. data.
10.	vsplit = vsplit. lchild, i = vsplit. lfc[i]
11.	while v is not a leaf and i ≠ 1 do
12.	if x1 ≤ v. x then
13.	j = v. rfc[i]
14.	while j ≠ −1 and v rchild. data[j]. y ≤ y2 and j ≤ v. rchild. data.size do
15.	Report interval, S = S ∪ {v. rchild. data[j]}
16.	j = j + 1
17.	end while
18.	i = v. lfc[i], v = v. lchild
19.	else
20.	i = v. rfc[i], v = v. rchild
21.	end if
22.	end while
23.	if v is a leaf and v. interval. x ∈ [x1, x2] and v. interval. y ∈ [y1, y2] then
24.	Report the interval in v, S = S ∪ {v. interval}
25.	end if
26..	v = vsplit. rchild, i = vsplit. rcf[i]
27.	while v is not a leaf and i ≠ −1 do
28.	if x2 ≥ v. x then
29.	j = v. lfc[i]
30.	while j ≠ −1 and v. lchild. data[j]. y ≤ y2 and j ≤ v. lchild. data. size do
31.	Report interval, S = S ∪ {v. lchild. data[j]}
32.	j = j + 1
33.	end while
34.	i = v. rfc[i], v = v. rchild
35.	else
36.	i = v. lfc[i], v = v. lchild
37.	end if
38.	end while
39.	if v is a leaf and v. interval. x ∈ [x1, x2] and v. interval. y ∈ [y1, y2] then
40.	Report the interval in v, S = S ∪ {v. interval}
41.	end if
42.	return S