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Reading time: 30 minutes | Coding time: 15 minutes

**Van Emde Boas tree** (or Van Emde Boas priority queue or vEB tree) is a tree data structure which implements an associative array with m-bit integer keys. It performs all operations (*insert, delete, lookup, maximum, minimum, successor and predecessor*) in **O(log log M)** time, where M is the maximum number of elements that can be stored in the tree.

It was formulated by a team led by Dutch computer scientist **Peter van Emde Boas** in 1975.

Van Emde Boas tree is an associated array implemented through a multiway tree structure.

It is expected to perform five operations:

**find(k):**Find the key k and the value it points to.**predecessor(k):**Return the largest key in domain less than or equal to k.**successor(k):**Return the smallest key in domain less greater or equal to k.**insert(k):**Insert a key k into the tree.**delete(k):**Delete the key k from tree.

Van Emde Boas tree or vEB stores the keys in a bounded universe and performs above operations in double logarithmic time complexity with respect to universe size.

vEB tree's other than being able to **solve predecessor problem extremely fast**, can also be used as an **implementation of priority queue** (which is **faster than a heap implementation**).

vEB tree for a universe U containing **[0 ... U-1]** is implemented by dividing the universe into **âˆšU** vEB trees, where the i^{th} subtree would be responsible for universe of size âˆšU containing the keys [(i)âˆšU ... (i+1)âˆšU-1]. The data structure is implemented recursively, where it shrinks by a factor of âˆšU at every level.

The vEB tree is a modification of proto-vEB structure. A vEB tree stores following information:

**children**array of size âˆšU that points to children vEB trees. child i points to vEB tree responsible for the range

[(i)âˆšU ... (i+1)âˆšU-1].**min**to store minimum element in the universe of the tree.**max**to store maximum element in the universe of the tree.**summary**, an auxillary vEB tree to keep track of which subtree/child is empty.- A variable
**u**to keep track of universe size.

When tree is empty, **min** and **max** are both **NULL**.

The element stored in **min** does not appear in any children trees, it is only stored at **min**, while element stored at **max** does.

When the universe size is 2, vEB does not need an array. It can just store the values in min and max attributes.

At the base level, a vEB tree is just a direct-addressing with added attributes.

A vEB tree can ve represented as:

A vEB that stores values 10, 12, 13, 14.

### Algorithm

- Because of the way tree is indexed, a key
`k`

belongs to any one of âˆšU subtrees. Three function are used:

```
high(k, U) = floor(k/√U)
low(k, U) = k mod(√U)
index(k, k', U) = k√U + k'
```

The function **high** tells what subtree **k** belongs to

**low** tells the position of **k** in that subtree

The function **index** tells what is the actual key given the subtree number and position in the subtree.

The value of **U** is the universe size of tree in which the function is called. Also, it should be noted that universe size must be of form 2^{2x}.

### Successor operation

**successor(T, k):**This is a recursive funtion where T is the vEB tree to find the successor of k in T. The algorithm will be similar for predecessor.

```
function successor(T, k):
if k < T.min:
return T.min
else if k > T.max:
return T.u
# universe size used as null value
i = high(k, T.u) # subtree index
j = low(k, T.u) # key for the subtree
if j < T.children[i].max:
return k - j + successor(T.children[i], j)
# otherwise find next subtree for successor
return k - j + T.children[successor(T.summary, i)].min
```

predecessor function is implmented similarly.

### Insert operation

**insert(k):**

```
function insert(T, k):
# if tree is empty, i.e. first insertion
if T.min > T.max:
T.min = k
T.max = k
return
# if key is new T.min, instead of k insert old T.min in subtrees
# set k as new T.min
if k < T.min:
swap(k, T.min)
# since T.max is also stored in subtrees, no need to swap
if k > T.max:
T.max = k
# if at lowest vEB, i.e. the leaf tree
if T.u == 2:
# note that k can be 0 or 1
T.max = k
return
i = high(k, T.u) # subtree index
j = low(k, T.u) # key for the subtree
insert(children[i], j)
# if the recursive insertion created a new subtree, add it to summary vEB
if children[i].max == children[i].min:
insert(T.summary, i)
```

### Delete operation

**delete(k):**

```
function delete(T, k):
# if only one element in tree
if T.min == T.max:
T.min = inf
T.max = -inf
return
# if k is T.min, assign new value to it and return since min is not
# stored in children
if k == T.min:
i = T.summary.min # index of first subtree
T.min = index(i, T.children[i].min, T.u) # declared above
return
i = high(k, T.u) # index of subtree that contains k
j = low(k, T.u) # key for the subtree
# if last subtree
if u == 2:
if k == 1:
if min == 1:
min = T.u
max = -1
else if min == 0:
max = 0
else:
# i.e. k == 0
if max == 0:
min = T.u
max = -1
else if max == 1
min = 1
return
delete(T.children[i], j)
if T.children[i] is empty:
delete(T.summary, i)
# if deleted element was T.max
if k == T.max:
# if no subtrees exist, tree just has one element stored at t.min
if T.summary is empty:
T.max = T.min
else:
# else find new max element
i = T.summary.max # index of last subtree
T.max = index(i, T.children[i].max, T.u)
```

### Complexity

- Time complexity:
- Find:
**O(log**_{2}log_{2}U) - Successor, Predecessor:
**O(log**_{2}log_{2}U) - Insert:
**O(log**_{2}log_{2}U) - Delete:
**O(log**_{2}log_{2}U)

- Find:
- Space complexity:
**O(U)**

Where U is universe size.

### Implementations

**C++ 11**

```
/*
* Code for van emde boas tree. Try to implement function for predecessor as
* an exercise.
*/
#include <cmath>
#include <iostream>
#include <vector>
class vEB
{
int u;
int min;
int max;
vEB *summary;
std::vector<vEB *> children;
int high(int k){
int x = std::ceil(std::sqrt(u));
return std::floor(k / x);
}
int low(int k){
int x = std::ceil(std::sqrt(u));
return k % x;
}
int index(int k, int kk){
return (k * (int)std::sqrt(u)) + kk;
}
public:
vEB(int U){
// u = std::pow(std::sqrt(U), 2);
u = U;
min = U;
max = -1;
if (u <= 2){
summary = nullptr;
children = std::vector<vEB *> (0, nullptr);
}
else{
int children_count = std::ceil(std::sqrt(u));
int children_size = std::ceil(u / std::sqrt(u));
summary = new vEB(children_count);
children = std::vector<vEB *> (children_count, nullptr);
for(int i = 0; i < children_count; ++i){
children[i] = new vEB(children_count);
}
}
}
void insert(int k){
if(min > max){
min = max = k;
return;
}
if(k < min){
int temp;
temp = min;
min = k;
k = temp;
}
if(k > max)
max = k;
if(u == 2){
max = k;
return;
}
int i = high(k);
int j = low(k);
children[i]->insert(j);
if(children[i]->max == children[i]->min)
summary->insert(i);
}
void remove(int k){
if(min > max)
return;
if(min == max){
min = u;
max = -1;
return;
}
if(u == 2){
if(k == 1){
if(min == 1){
min = u;
max = -1;
}
else if(min == 0)
max = 0;
}
else
if(max == 0){
min = u;
max = -1;
}
else if(max == 1)
min = 1;
return;
}
if(k == min){
int i = summary->min;
min = index(i, children[i]->min);
return;
}
int i = high(k);
int j = low(k);
children[i]->remove(j);
if(children[i]->min > children[i]->max){
summary->remove(i);
}
if(k == max){
if(summary->min > summary->max){
max = min;
}
else{
i = summary->min;
max = index(i, children[i]->max);
}
}
}
int getMin(){
return min;
}
int getMax(){
return max;
}
int universe(){
return u;
}
int successor(int k){
if(k <= min)
return min;
else if(k > max)
return u;
int i = high(k);
int j = low(k);
if(j <= children[i]->max){
int res = children[i]->successor(j);
if(res >= (u / (int)std::sqrt(u)))
return u;
return k - j + res;
}
else{
int res = children[summary->successor(i)]->min;
if(res >= children[summary->min]->u)
return u;
return index(summary->successor(i), res);
}
}
};
int main(){
vEB tree(1 << 4);
std::cout << "Insert 12, 10, 13, 14\n";
tree.insert(12);
tree.insert(10);
tree.insert(13);
tree.insert(14);
std::cout << "Successor of 2 is\n";
std::cout << tree.successor(2) << '\n';
tree.remove(13);
std::cout << "Removing 13. Now successor of 13 is\n";
std::cout << tree.successor(13) << '\n';
tree.remove(14);
std::cout << "Removing 14. Now successor of 13 is\n";
std::cout << tree.successor(13) << '\n';
std::cout << "16, which is universe size, means no successor.\n";
return 0;
}
```

### Applications

- vEB trees were made specefically to solve predecessor problem in double logarithmic time.
- vEB trees can be used as extremely fast priority queues and find uses in routing applications.
- If universe size is reasonable, vEB trees are basically faster binary search trees.

### References/ Further reading

- CLRS chapter 20 to know more about vEB trees and prototype structure, and motivations beind it.
- Lecture slides from stanford cs166.
- Lecture by Eric Demaine on youtube