Van Emde Boas Trees

Presented by: Lokesh N. Jaliminche(M.tech 1st year cse) Reg.no:-12MCS0058

Outline IntroductionPriliminary approaches Direct addressing Superimposing a binary tree structure Recursive structure Proto of van emde boas treeVan emde boas treeSome operations.comparisons

IntroductionThis was invented by a team led by

Peter van Emde Boas in 1975.data structures that support the

operations of a priority queue,binary heaps, red-black trees, Fibonacci heaps In each of these data structures, at least one important operation take O(log n) time

van Emde Boas trees support the each of these operations, in O(log logn)

Search O(log(logm)) Insert O(log(logm))Delete O(log(logm))

Terminologies and assumptionsn = number of elements

currently in setu = range of possible values.(size

of universe.universe={0,1,2,3,4……u-1)u= 2^k. for some integer k>=1No duplicate elements are

allowed.

Preliminary approaches Direct addressing

A binary tree imposed on bit vector

Superimposing tree of a constant height

Recursive structure previously , we used a summary structure of size u^1/2 on top of bit vectore with each entry pointing to another stucture of size .

Now, we make the structure recursive, shrinking the universe size by the square root at each level of recursion.

Proto of VEB

Proto of VEB

Follwing are some functions which helps us to find the position of elements within the tree

High(x) gives the index of x’s cluster Low(x) gives the index of x within the

cluster The function index(x,y) builds an element

number from x and y and identify the position of x

Van Emde Boas Tree

Some operations Finding MIN and MAX MemberInsertDelete

When to use this tree structurewe should not use a van Emde Boas tree when we perform only a small number of operations,

since the time to create the data structure would exceed the time saved in the individual operations.

use a simple data structure, such as an array or linked list, to represent a set with only a few elements

Some comparisons

Implementation

Insert Extract-min Ease of programming

Sorted linked list

O(n) O(1) Easy

Unsorted linked list

O(1) O(n) Easy

Red/Black Tree O(logn) O(logn) Hard

Binary Heap O(logn) O(logn) Easy

Van Emde Boas Tree

O(log logn) O(log logn) VERY HARD

References

1.Introduction to algorithms third edition

Thomas H Cormen.

Thank you