Fractional Cascading and Its Applications G. S. Lueker. A data structure for orthogonal range queries. In Proc. 19 th annu. IEEE Sympos. Found. Comput. Sci., pages 28-34, D. E. Willard. Predicate-oriented database search algorithms. Ph.D. thesis, Aiken Comput. Lab., Harvard Univ., Cambridge, MA, 1978, Report TR B. Chazelle, L. J. Guibas: Fractional Cascading: I. A Data Structuring Technique. Algorithmica 1(2): (1986) B. Chazelle, L. J. Guibas: Fractional Cascading: II. Applications. Algorithmica 1(2): (1986) Slides by Dror Aiger

What is Fractional Cascading? A technique to speed up a sequence of binary searches for the same value in a sequence of related data structures. The first binary search in the sequence takes a logarithmic time, but successive searches in the sequence are faster.

A simple example Let A 1 and A 2 be two sorted arrays of real numbers. Problem: report all numbers of A 1 and A 2 in the range [y,y’]. Solution: binary search for first number ≥ y in A 1, traverse until number is ≥ y’. Same for A 2. Query time: O(k) + two binary searches. What if numbers in A 2 are a subset of A 1 ?

Adding pointers We add pointers from the entries in A 1 to the entries in A 2 (in the preprocess stage). Binary search for first number ≥ y in A 1. Store pointer from that number to array A 2. Traverse A 1 until number is ≥ y’. Traverse A 2 from pointer until number is ≥ y’. Query time: one binary search on A 1, plus reporting k numbers.

Adding pointers - example

Application in Range Searching In the plane the query time of range trees is O(log 2 (n)+k). Can we do better? Yes, we can obtain O(log(n)+k) query time with fractional cascading.

A reminder: a range tree

A reminder: Canonical sets We store the points in the set P in a balanced binary tree T, using the x–coordinates as keys. Each node v of T is associated with a canonical set P(v), which is the set of all the points in P that are stored in the sub tree rooted at v:

The idea When processing a query [x:x’] x [y,y’], we search some trees with the same keys. For each such tree we spend O(log(n)) time in standard range tree. P(lc(v)) and P(rc(v)) are subsets of P(v). We will keep pointers between nodes of T(v) and nodes of lc(v) and rc(v) that keep the same key, or the next smallest key. After performing a search in T(v) this will allow to perform a search in lc(v) and rc(v) in O(1) time.

The data structure Each canonical subset P(v) is stored in an array A(v). Each entry of A(v) stores two pointers: –A pointer into A(lc(v)) and a pointer into A(rc(v)). –Let A(v)[i] stores a point p - we store a pointer from A(v)[i] to the entry of A(lc(v)) such that the y-coordinate of the point stored there is the smallest one larger than or equal to p y.

Layered range tree example

Query We search with x and x’ in the main tree T to determine O(log(n)) nodes whose canonical subsets together contain the points with x- coordinate in [x:x’]. Let the path splits at v – we find the entry in A(v) whose y-coordinate is the smallest one larger than or equal to y (O(log(n) with binary search). While we search further with x and x’ in the main tree we keep track of the entry in the associated arrays. They can be maintained in constant time by following the pointers. If v is one of the O(log(n)) nodes we selected, we have to report the points stored in A(v) whose y-coordinate is in [y:y’] and this is done in O(1+k v ) by walking through the array, where k v is the number of points reported at v. The total time now becomes O(log(n)+k)

Consequences By induction, it also improves by a factor of O(log(n)) the results in d > 2. Range trees with fractional cascading in d ≥ 2 yield query time: O(k + log d−1 (n)). Space usage: O(n log d−1 n). Preprocessing time: O(n log d−1 (n)). In d = 2, the query time and preprocessing time are optimal, but space usage is not.

Another application Intersecting a polygonal path with a line CG86 Bernard Chazelle, Leonidas J. Guibas: Fractional Cascading: II. Applications. Algorithmica 1(2): (1986)

Intersecting a polygonal path with a line We are given a polygonal path P and we wish to preprocess it into a data structure so that given any query line l, we can quickly report all the intersections of P and l. The idea is based on recursive application of the following: –A straight line l intersects a polygonal line P if and only if it intersects the convex hull of P. The convex hulls is computed (recursively) in the preprocess stage. In each step we compute the CH of the first and second halves of the current polyline (F(P) and S(P)) – This takes O(n log(n)) time and space. We have a balanced binary tree T:

Query The query is simple:

Intersecting a polygonal path with a line This still gives Ω(log 2 n) query time since we need logarithmic time for each convex hull and we have at least log(n) such operations: We need some tools to be able to use fractional cascading: –Slope Sequence of convex polygon C is a (unique) circular permutation of the edges of C such that the slopes are non decreasing (it is well known that it exists). –Finding intersection can be done in constant time in this sequence if we know the positions of the slopes of l. –We view each node x, of T as containing the slope sequence of the convex polygon associated with x and apply fractional cascading to these structures. –Any time we need to decide whether to descent to a subtree, we look up the slopes of l in that subtree root’s sequence and find the answer in constant time (we still have the logarithmic time for the root of T).

Intersecting a polygonal path with a line We thus get O(log n + size of the subtree of T actually visited) query time. This can be shown [CG86] to be O((k+1)log(n/(k+1))) where k is the number of intersections. The size and preprocessing time of the structure is O(nlog(n)):