Difference between revisions of "The Monad.Reader/Issue4/On Treaps And Randomization"
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−  '''This article needs reformatting! Please help tidy it up.'''[[User:WouterSwierstraWouterSwierstra]] 14:23, 9 May 2008 (UTC) 

+  =Treaps and Randomization in Haskell= 

+  :''by Jesper Louis Andersen <jlouis@mongers.org> for [[The Monad.Reader]], Issue Four'', 05/07 2005 

−  = Treaps and Randomization in Haskell = 

+  We give an example implementation of treaps (tree heaps) in Haskell. The emphasis is partly on treaps, partly on the System.Random module from the hierarchical libraries. We show how to derive the code and explain it in an informal style. 

−  ''by Jesper Louis Andersen <jlouis@mongers.org> for The Monad.Reader IssueFour'' 

−  [[BR]] 

−  ''05/07  2005'' 

−  
−  '''Abstract.''' 

−  We give an example implementation of Treaps in Haskell. The emphasis is partly on treaps, partly on the System.Random module from the hierachial libraries. We show how to derive the code and explain it in an informal style. 

−  
−  == Introduction == 

+  ==Introduction== 

I have, a number of times, warned people that I ought to do a TMR 
I have, a number of times, warned people that I ought to do a TMR 

article. The world had its way, and I had to wait 
article. The world had its way, and I had to wait 

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it. Originally, I considered playing around with the 
it. Originally, I considered playing around with the 

ALLPAIRSSHORTESTPATH algorithms, but for some reason I was 
ALLPAIRSSHORTESTPATH algorithms, but for some reason I was 

−  not really satisfied with it. Also, with the upcoming Matrix library in the 
+  not really satisfied with it. Also, with the upcoming Matrix library in the hierarchical libraries, this might prove to be a better solution. 
Instead I will provide a treatise on the ''Treap'' data structure, 
Instead I will provide a treatise on the ''Treap'' data structure, 

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16(4/5):464497, 1996). This document owes about 90% to the mentioned article. 
16(4/5):464497, 1996). This document owes about 90% to the mentioned article. 

−  I also advise you to check out Oleg 
+  I also advise you to check out Oleg Kiselyov's work on treaps for 
Scheme. He does a number of optimizations on the data structure which 
Scheme. He does a number of optimizations on the data structure which 

−  I have skipped over here. Take a look at [http://okmij.org/ftp/Scheme/lib/treap.scm 
+  I have skipped over here. Take a look at [http://okmij.org/ftp/Scheme/lib/treap.scm Oleg's Scheme Treap implementation] 
−  
−  == Search trees == 

+  ==Search trees== 

The classic problem of computer science is how to express and 
The classic problem of computer science is how to express and 

represent a finite map in a programming language. Formally a finite 
represent a finite map in a programming language. Formally a finite 

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finally, ''delete(f, k)'', which removes the association of ''k'' from ''f''. 
finally, ''delete(f, k)'', which removes the association of ''k'' from ''f''. 

−  One such representation is the binary search tree (In much 
+  One such representation is the binary search tree (In much literature, 
the acronym BST is used). I assume most readers of TMR are familiar 
the acronym BST is used). I assume most readers of TMR are familiar 

with binary search trees, and especially the pathological case degenerating the worst case search time bounds to ''O(n)''. 
with binary search trees, and especially the pathological case degenerating the worst case search time bounds to ''O(n)''. 

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1. This ensures the AVLtree is always balanced and the pathological 
1. This ensures the AVLtree is always balanced and the pathological 

case where a tree is actually a list is ruled out. As a side note, 
case where a tree is actually a list is ruled out. As a side note, 

−  an AVL tree will never be worse in structure than a 
+  an AVL tree will never be worse in structure than a Fibonacci tree (Knuth's ''The Art of Computer Programming'' Volume 3 has a good treatment of this tree type). 
−  Another famous example is the RedBlack tree, which provides a less 
+  Another famous example is the RedBlack tree, which provides a less 
−  strict balance invariant than the AVL tree. The invariant is harder 
+  strict balance invariant than the AVL tree. The invariant is harder 
to describe in a single paragraph  but involves colouring nodes either 
to describe in a single paragraph  but involves colouring nodes either 

−  red or black and adding invariants such that the tree always stays 
+  red or black and adding invariants such that the tree always stays 
reasonably balanced. See Introduction to algorithms by Cormen, Leiserson, Rivest and Stein if you want to read the hard, incomprehensible imperative 
reasonably balanced. See Introduction to algorithms by Cormen, Leiserson, Rivest and Stein if you want to read the hard, incomprehensible imperative 

version of this data structure, or Purely Functional Data Structures by Chris Okasaki if you want the functional approach to this (the functional code is a mere 12 lines without the '''delete''' operation). 
version of this data structure, or Purely Functional Data Structures by Chris Okasaki if you want the functional approach to this (the functional code is a mere 12 lines without the '''delete''' operation). 

−  The '''Data.Map''' module in the 
+  The '''Data.Map''' module in the hierarchical libraries of Haskell 
use another type of tree known as the ''23 tree''. The ''23 tree'' is selfbalancing but uses a different trick. A ''23 tree'' node contains 
use another type of tree known as the ''23 tree''. The ''23 tree'' is selfbalancing but uses a different trick. A ''23 tree'' node contains 

either one or two ''(k, v)''pairs and thus has either 2 or 3 
either one or two ''(k, v)''pairs and thus has either 2 or 3 

−  children. We call a node with 2 children a 2node and a node with 3 children a 3node. Insertions are always done into leaf nodes which can grow from 2nodes to 3nodes in a natural way. Growth of a 3node is then done by splitting the node into two 2nodes; the least ''(k, v)''pair, the greatest pair and the middle pair is inserted into the parent node 
+  children. We call a node with 2 children a 2node and a node with 3 children a 3node. Insertions are always done into leaf nodes which can grow from 2nodes to 3nodes in a natural way. Growth of a 3node is then done by splitting the node into two 2nodes; the least ''(k, v)''pair, the greatest pair and the middle pair is inserted into the parent node (there is a reference inside the documentation of the Data.Map). 
−  +  <blockquote>Sadly, the above is not true. Data.Map has binary trees which are balanced according to the size of the left and right subtrees. If one subtree grows beyond a certain constant factor it is rebalanced  Jesper Andersen</blockquote> 

Yet another variant of the finite map is the splay tree. In the splay tree 
Yet another variant of the finite map is the splay tree. In the splay tree 

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worst case running time. Splay trees are not that good for purely 
worst case running time. Splay trees are not that good for purely 

functional languages, since they change the tree for all 
functional languages, since they change the tree for all 

−  operations, including '''lookup'''. Thus our type for a '''lookup''' 
+  operations, including '''lookup'''. Thus our type for a '''lookup''' 
function would be: 
function would be: 

−  +  <haskell> 

Splay_lookup :: key > SplayTree key value > (Maybe value, SplayTree) 
Splay_lookup :: key > SplayTree key value > (Maybe value, SplayTree) 

−  }}} 

+  </haskell> 

As a consequence, the programmer has to ''thread'' his splay tree 
As a consequence, the programmer has to ''thread'' his splay tree 

around where she wants to use it. This tends to clutter the code a 
around where she wants to use it. This tends to clutter the code a 

great deal. Further, splay trees are not friendly to a cache or page 
great deal. Further, splay trees are not friendly to a cache or page 

−  +  hierarchy, since the constant updating of nodes tends to dirty more 

−  pages/cache lines than necessary  but it hurts an imperative language 
+  pages/cache lines than necessary  but it hurts an imperative language 
−  more than a functional one which already has a fair deal of copying to do, due 
+  more than a functional one which already has a fair deal of copying to do, due 
to persistence. 
to persistence. 

+  * '''Side Note''': a paging hierarchy can be seen as a cache 

+  hierarchy, if you take the swap space as the lowest level, page table mapped pages as the second level and TLB mapped pages as the third (and fastest) level. 

−  '''Side Note''': a paging hierachy can be seen as a cache 

+  ==Heaps== 

−  hierachy, if you take the swap space as the lowest level, page table mapped pages as the second level and TLB mapped pages as the third (and fastest) level. 

−  
−  == Heaps == 

−  
A basic Queue (FIFO) is something I assume all know. A priority queue 
A basic Queue (FIFO) is something I assume all know. A priority queue 

is a queue, where each element is assigned a priority from a totally 
is a queue, where each element is assigned a priority from a totally 

ordered set P. Elements in the priority queue are extracted 
ordered set P. Elements in the priority queue are extracted 

according to the order of the priorities. For the case where the order is 
according to the order of the priorities. For the case where the order is 

−  increasing, the queue is often called a minpriq, since the minimum priority 
+  increasing, the queue is often called a "minpriq", since the minimum priority 
−  element is extracted first. Of course, a maxpriq is also possible. 
+  element is extracted first. Of course, a "maxpriq" is also possible. 
Priority queues are often implemented as heaps. In a functional 
Priority queues are often implemented as heaps. In a functional 

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priority less than the priorities of its children. If a node is placed 
priority less than the priorities of its children. If a node is placed 

at the leaf of such a tree, it can be ''floated'' up by comparing it 
at the leaf of such a tree, it can be ''floated'' up by comparing it 

−  and its parent, eventually exchanging their places until the priority invariant has been fulfilled. 
+  and its parent, eventually exchanging their places until the priority invariant has been fulfilled. Similarly, a node can be floated down by comparing the children priorities to each other, and exchanging the node for the child with the least priority. 
−  can be floated down by comparing the children priorities to each 

−  other, and exchanging the node for the child with the least priority. 

−  
−  == Treaps == 

+  ==Treaps== 

So, why attempt another data structure for the finite map problem? 
So, why attempt another data structure for the finite map problem? 

One, it is fun. Two, this algorithm is so simple, it can be explained in a single, tiny(??), TMR article. Third, we need more TMR articles. Simplicity usually means a fast algorithm. Benchmarking treaps against 
One, it is fun. Two, this algorithm is so simple, it can be explained in a single, tiny(??), TMR article. Third, we need more TMR articles. Simplicity usually means a fast algorithm. Benchmarking treaps against 

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will do carry out this benchmarking. 
will do carry out this benchmarking. 

−  While the introduction mentions finite maps, we will explore the simpler case where V is the singleton {True} set. The map ''f'' then represents a set of keys ''K'', since a key is either mapped to '''True''' or it is not, in which case we can return '''False'''. Thus, we do not even bother storing the singleton {True} set in the Treap structure. However, extending the treap to also posses arbitrary value data at each node is trivial and left as an exercise to the (interested, practically oriented) reader. 
+  While the introduction mentions finite maps, we will explore the simpler case where V is the singleton {True} set. The map ''f'' then represents a set of keys ''K'', since a key is either mapped to '''True''' or it is not, in which case we can return '''False'''. Thus, we do not even bother storing the singleton {True} set in the Treap structure. However, extending the treap to also posses arbitrary value data at each node is trivial and left as an exercise to the (interested, practically oriented) reader. 
Let ''K'' be a totally ordered space of keys. It is clear a binary 
Let ''K'' be a totally ordered space of keys. It is clear a binary 

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actually achieve a balanced tree (!). It might be wise to try to draw 
actually achieve a balanced tree (!). It might be wise to try to draw 

such a tree. In fact it is unique. To see this, construct the tree by 
such a tree. In fact it is unique. To see this, construct the tree by 

−  inserting ''K''s in increasing order of priorities, by using the binary 
+  inserting ''K''s in increasing order of priorities, by using the binary 
search tree '''insert''' algorithm. 
search tree '''insert''' algorithm. 

−  == 
+  ==Show me the code!== 
−  
Enough talk. Haskell! A module representing treaps is first defined: 
Enough talk. Haskell! A module representing treaps is first defined: 

−  +  <haskell> 

module Treap ( 
module Treap ( 

−  +  RTreap 

−  +  , empty 

−  +  , null 

−  +  , insert 

−  +  , delete 

−  +  , member 

−  +  , stdGenTreap 

−  +  , splitTreap 

−  +  , joinTreap 

) where 
) where 

import System.Random 
import System.Random 

import Prelude hiding (null) 
import Prelude hiding (null) 

−  }}} 

+  </haskell> 

A treap is a binary search tree, where each node 
A treap is a binary search tree, where each node 

has a key and a priority: 
has a key and a priority: 

−  +  <haskell> 

data Treap k p = Leaf  Branch (Treap k p) k p (Treap k p) 
data Treap k p = Leaf  Branch (Treap k p) k p (Treap k p) 

−  +  deriving (Show, Read) 

−  +  </haskell> 

The empty tree and the null predicate are simple. They are copied 
The empty tree and the null predicate are simple. They are copied 

verbatim from the binary search tree: 
verbatim from the binary search tree: 

−  +  <haskell> 

treap_Empty :: Treap k p 
treap_Empty :: Treap k p 

treap_Empty = Leaf 
treap_Empty = Leaf 

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treap_Null :: Treap k p > Bool 
treap_Null :: Treap k p > Bool 

treap_Null Leaf = True 
treap_Null Leaf = True 

−  treap_Null _ 
+  treap_Null _ = False 
−  +  </haskell> 

Insertion into a treap works by inserting the node, as if inserting 
Insertion into a treap works by inserting the node, as if inserting 

into a binary search tree. Then we use the famous left and 
into a binary search tree. Then we use the famous left and 

−  rightrotations to float the node up, until it 
+  rightrotations to float the node up, until it fulfills the 
heapproperty on its priority. If you are not familiar with left and 
heapproperty on its priority. If you are not familiar with left and 

right rotations, they are just restructurings of a binary search tree, 
right rotations, they are just restructurings of a binary search tree, 

maintaining the ordering property. What is important is they alter the 
maintaining the ordering property. What is important is they alter the 

−  heights of the subtrees and so can help balance the tree more. They are easily 
+  heights of the subtrees and so can help balance the tree more. They are easily 
−  +  definable in Haskell by pattern matching. Drawing them on paper is a good 

exercise: 
exercise: 

−  +  <haskell> 

rotateLeft :: Treap k p > Treap k p 
rotateLeft :: Treap k p > Treap k p 

rotateLeft (Branch a k p (Branch b1 k' p' b2)) = 
rotateLeft (Branch a k p (Branch b1 k' p' b2)) = 

−  +  Branch (Branch a k p b1) k' p' b2 

rotateLeft _ = error "Wrong rotation (rotateLeft)" 
rotateLeft _ = error "Wrong rotation (rotateLeft)" 

rotateRight :: Treap k p > Treap k p 
rotateRight :: Treap k p > Treap k p 

rotateRight (Branch (Branch a1 k' p' a2) k p b) = 
rotateRight (Branch (Branch a1 k' p' a2) k p b) = 

−  +  Branch a1 k' p' (Branch a2 k p b) 

rotateRight _ = error "Wrong rotation (rotateRight)" 
rotateRight _ = error "Wrong rotation (rotateRight)" 

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treap_Insert k p Leaf = Branch Leaf k p Leaf 
treap_Insert k p Leaf = Branch Leaf k p Leaf 

treap_Insert k p (Branch left k' p' right) = 
treap_Insert k p (Branch left k' p' right) = 

−  +  case compare k k' of 

−  +  EQ > Branch left k' p' right  Node is already there, ignore 

−  +  LT > case Branch (treap_Insert k p left) k' p' right of 

−  +  (t @ (Branch (Branch l' k p r') k' p' right)) > 

−  +  if p' > p 

−  +  then rotateRight t 

−  +  else t 

−  +  t > t 

−  +  GT > case Branch left k' p' (treap_Insert k p right) of 

−  +  (t @ (Branch left k' p' (Branch l' k p r'))) > 

−  +  if p' > p 

−  +  then rotateLeft t 

−  +  else t 

−  +  t > t 

−  +  </haskell> 

When coding structures based upon binary trees it can be convenient to ''forget'' the deletion case. It is often the hardest 
When coding structures based upon binary trees it can be convenient to ''forget'' the deletion case. It is often the hardest 

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off the leaf (Notice the nice metaphors, please). 
off the leaf (Notice the nice metaphors, please). 

−  +  <haskell> 

treap_Delete :: (Ord k, Ord p) => k > Treap k p > Treap k p 
treap_Delete :: (Ord k, Ord p) => k > Treap k p > Treap k p 

treap_Delete k treap = recDelete k treap 
treap_Delete k treap = recDelete k treap 

−  +  where recDelete k Leaf = error "Key does not exist in tree (delete)" 

−  +  recDelete k (t @ (Branch left k' p right)) = 

−  +  case compare k k' of 

−  +  LT > Branch (recDelete k left) k' p right 

−  +  GT > Branch left k' p (recDelete k right) 

−  +  EQ > rootDelete t 

−  +  priorityCompare Leaf (Branch _ _ _ _) = False 

−  +  priorityCompare (Branch _ _ _ _) Leaf = True 

−  +  priorityCompare (Branch _ _ x _) (Branch _ _ y _) = x < y 

−  +  rootDelete Leaf = Leaf 

−  +  rootDelete (Branch Leaf _ _ Leaf) = Leaf 

−  +  rootDelete (t @ (Branch left k p right)) = 

−  +  if priorityCompare left right 

−  +  then let Branch left k p right = rotateRight t 

−  +  in Branch left k p (rootDelete right) 

−  +  else let Branch left k p right = rotateLeft t 

−  +  in Branch (rootDelete left) k p right 

−  +  </haskell> 

We must not forget the '''member''' function. This is simple, as it 
We must not forget the '''member''' function. This is simple, as it 

is nothing but the original binary search tree function: 
is nothing but the original binary search tree function: 

−  +  <haskell> 

treap_Member :: (Ord k, Ord p) => k > Treap k p > Bool 
treap_Member :: (Ord k, Ord p) => k > Treap k p > Bool 

treap_Member e Leaf = False 
treap_Member e Leaf = False 

treap_Member e (Branch left k _ right) = 
treap_Member e (Branch left k _ right) = 

−  +  case compare e k of 

−  +  LT > treap_Member e left 

−  +  GT > treap_Member e right 

−  +  EQ > True 

−  +  </haskell> 

−  
−  == Providing random priorities == 

+  ==Providing random priorities== 

The premise of the Treap algorithm is the provision of a good random 
The premise of the Treap algorithm is the provision of a good random 

number generator. If the priorities are randomly assigned, the tree 
number generator. If the priorities are randomly assigned, the tree 

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structure an '''RTreap''': 
structure an '''RTreap''': 

−  +  <haskell> 

newtype RTReap g k p = RT (g, Treap k p) 
newtype RTReap g k p = RT (g, Treap k p) 

−  +  deriving (Show, Read) 

−  +  </haskell> 

The empty treap is then an initialization of the random number 
The empty treap is then an initialization of the random number 

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the function above: 
the function above: 

−  +  <haskell> 

empty :: RandomGen g => g > RTreap g k p 
empty :: RandomGen g => g > RTreap g k p 

empty g = RT (g, treap_Empty) 
empty g = RT (g, treap_Empty) 

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null :: RandomGen g => RTreap g k p > Bool 
null :: RandomGen g => RTreap g k p > Bool 

null (RT (g, t)) = treap_Null t 
null (RT (g, t)) = treap_Null t 

−  }}} 

+  </haskell> 

Insertion into the treap is done by requesting a new random number 
Insertion into the treap is done by requesting a new random number 

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can be pulled with these values. 
can be pulled with these values. 

−  +  <haskell> 

insert :: (RandomGen g, Ord k, Ord p, Num p, Random p) 
insert :: (RandomGen g, Ord k, Ord p, Num p, Random p) 

−  +  => k > RTreap g k p > RTreap g k p 

insert k (RT (g, tr)) = 
insert k (RT (g, tr)) = 

−  +  let (p, g') = randomR (2000000000, 2000000000) g 

−  +  in RT (g', treap_Insert k p tr) 

delete :: (RandomGen g, Ord k, Ord p) => k > RTreap g k p 
delete :: (RandomGen g, Ord k, Ord p) => k > RTreap g k p 

−  +  > RTreap g k p 

delete k (RT (g, tr)) = RT (g, treap_Delete k tr) 
delete k (RT (g, tr)) = RT (g, treap_Delete k tr) 

member :: (RandomGen g, Ord k, Ord p) => k > RTreap g k p 
member :: (RandomGen g, Ord k, Ord p) => k > RTreap g k p 

−  +  > Bool 

member k (RT (g, tr)) = treap_Member k tr 
member k (RT (g, tr)) = treap_Member k tr 

−  }}} 

+  </haskell> 

The initialization of the '''RTreap''' will then be something like: 
The initialization of the '''RTreap''' will then be something like: 

−  +  <haskell> 

stdGenTreap :: Int > RTreap StdGen k p 
stdGenTreap :: Int > RTreap StdGen k p 

stdGenTreap = (empty . mkStdGen) 
stdGenTreap = (empty . mkStdGen) 

−  }}} 

+  </haskell> 

The ''Int'' type one has to provide is an initialization seed. We can get one such inside an '''IO''' monad when starting our program and then use it to seed the Treaps we need afterwards. The functions needed are defined inside the '''System.Random''' module. 
The ''Int'' type one has to provide is an initialization seed. We can get one such inside an '''IO''' monad when starting our program and then use it to seed the Treaps we need afterwards. The functions needed are defined inside the '''System.Random''' module. 

−  == 
+  ==Cool additions== 
−  
If we wish to split a treap at a certain node k in K, we can do so, 
If we wish to split a treap at a certain node k in K, we can do so, 

by inserting k with the minimum priority. Assuming p are in the 
by inserting k with the minimum priority. Assuming p are in the 

'''Bounded''' class: 
'''Bounded''' class: 

−  +  <haskell> 

splitTreap :: (RandomGen g, Bounded p, Ord k, Ord p) 
splitTreap :: (RandomGen g, Bounded p, Ord k, Ord p) 

−  +  => k > RTreap g k p > (RTreap g k p, RTreap g k p) 

splitTreap k (RT (g, tr)) = 
splitTreap k (RT (g, tr)) = 

−  +  let (g', g'') = split g 

−  +  Branch left _ _ right = treap_Insert k minBound tr 

−  +  in (RT (g', left), RT (g'', right)) 

−  +  </haskell> 

−  +  Similarly to join two ''disjoint'' treaps with key spaces K1 and K2, where the keys in K1 are smaller than the keys in K2 (formally: max K1 < min K2), we can 

choose a key k not in the union (K1, K2) and form the tree where k is the 
choose a key k not in the union (K1, K2) and form the tree where k is the 

root and the treaps are left and right children. We then proceed by 
root and the treaps are left and right children. We then proceed by 

deleting the node k: 
deleting the node k: 

−  +  <haskell> 

joinTreap :: (Bounded p, Ord p, Ord k) 
joinTreap :: (Bounded p, Ord p, Ord k) 

−  +  => k > RTreap g k p > RTreap g k p > RTreap g k p 

joinTreap k (RT (g, tr1)) (RT (_, tr2)) = 
joinTreap k (RT (g, tr1)) (RT (_, tr2)) = 

−  +  RT (g, (treap_Delete k (Branch tr1 k maxBound tr2))) 

−  +  </haskell> 

−  
−  == Optimizations == 

+  ==Optimizations== 

I will simply direct people to the article by Oleg pointed at in the introduction. There are certain optimizations possible, which he thoroughly discusses. Implementing these is an exercise. 
I will simply direct people to the article by Oleg pointed at in the introduction. There are certain optimizations possible, which he thoroughly discusses. Implementing these is an exercise. 

−   

+  
−  CategoryArticle 

+  [[Category:Article]] 
Latest revision as of 01:08, 10 May 2008
Contents
Treaps and Randomization in Haskell
 by Jesper Louis Andersen <jlouis@mongers.org> for The Monad.Reader, Issue Four, 05/07 2005
We give an example implementation of treaps (tree heaps) in Haskell. The emphasis is partly on treaps, partly on the System.Random module from the hierarchical libraries. We show how to derive the code and explain it in an informal style.
Introduction
I have, a number of times, warned people that I ought to do a TMR article. The world had its way, and I had to wait until the Summer to be able to finish an article. So this is it. Originally, I considered playing around with the ALLPAIRSSHORTESTPATH algorithms, but for some reason I was not really satisfied with it. Also, with the upcoming Matrix library in the hierarchical libraries, this might prove to be a better solution.
Instead I will provide a treatise on the Treap data structure, devised by Aragon and Seidel. I have much to thank them for in the following. Usually citations are at the back of an article, but I really advise you to read Randomized Search Trees (Algorithmica, 16(4/5):464497, 1996). This document owes about 90% to the mentioned article.
I also advise you to check out Oleg Kiselyov's work on treaps for Scheme. He does a number of optimizations on the data structure which I have skipped over here. Take a look at Oleg's Scheme Treap implementation
Search trees
The classic problem of computer science is how to express and represent a finite map in a programming language. Formally a finite map is a function f: K > V, which is said to map a finite set K, of keys, to a (thus also finite) set V, of values. The basic functions are: lookup(f, k), which will return the value f(k) in V, associated with the value k in K; insert(f, (k,v)) which extends or updates the finite map with a new key/value pair; and finally, delete(f, k), which removes the association of k from f.
One such representation is the binary search tree (In much literature, the acronym BST is used). I assume most readers of TMR are familiar with binary search trees, and especially the pathological case degenerating the worst case search time bounds to O(n).
There are a number of strategies for avoiding the degenerate case where the tree becomes a linked list in effect. One could be to add invariants to the tree, which ensures that it stays inside certain balance bounds. One example is the AVL tree, which maintains the following invariant: At each node, the childsubtrees differ in depth by at most 1. This ensures the AVLtree is always balanced and the pathological case where a tree is actually a list is ruled out. As a side note, an AVL tree will never be worse in structure than a Fibonacci tree (Knuth's The Art of Computer Programming Volume 3 has a good treatment of this tree type).
Another famous example is the RedBlack tree, which provides a less strict balance invariant than the AVL tree. The invariant is harder to describe in a single paragraph  but involves colouring nodes either red or black and adding invariants such that the tree always stays reasonably balanced. See Introduction to algorithms by Cormen, Leiserson, Rivest and Stein if you want to read the hard, incomprehensible imperative version of this data structure, or Purely Functional Data Structures by Chris Okasaki if you want the functional approach to this (the functional code is a mere 12 lines without the delete operation).
The Data.Map module in the hierarchical libraries of Haskell use another type of tree known as the 23 tree. The 23 tree is selfbalancing but uses a different trick. A 23 tree node contains either one or two (k, v)pairs and thus has either 2 or 3 children. We call a node with 2 children a 2node and a node with 3 children a 3node. Insertions are always done into leaf nodes which can grow from 2nodes to 3nodes in a natural way. Growth of a 3node is then done by splitting the node into two 2nodes; the least (k, v)pair, the greatest pair and the middle pair is inserted into the parent node (there is a reference inside the documentation of the Data.Map).
Sadly, the above is not true. Data.Map has binary trees which are balanced according to the size of the left and right subtrees. If one subtree grows beyond a certain constant factor it is rebalanced  Jesper Andersen
Yet another variant of the finite map is the splay tree. In the splay tree the rebalancing is done according to a simple heuristic which amortized over a certain number of operations yields O(lg n) worst case running time. Splay trees are not that good for purely functional languages, since they change the tree for all operations, including lookup. Thus our type for a lookup function would be:
Splay_lookup :: key > SplayTree key value > (Maybe value, SplayTree)
As a consequence, the programmer has to thread his splay tree around where she wants to use it. This tends to clutter the code a great deal. Further, splay trees are not friendly to a cache or page hierarchy, since the constant updating of nodes tends to dirty more pages/cache lines than necessary  but it hurts an imperative language more than a functional one which already has a fair deal of copying to do, due to persistence.
 Side Note: a paging hierarchy can be seen as a cache
hierarchy, if you take the swap space as the lowest level, page table mapped pages as the second level and TLB mapped pages as the third (and fastest) level.
Heaps
A basic Queue (FIFO) is something I assume all know. A priority queue is a queue, where each element is assigned a priority from a totally ordered set P. Elements in the priority queue are extracted according to the order of the priorities. For the case where the order is increasing, the queue is often called a "minpriq", since the minimum priority element is extracted first. Of course, a "maxpriq" is also possible.
Priority queues are often implemented as heaps. In a functional setting, a very simple heap to program is the pairing heap, which takes no more than 12 lines of Haskell. Unfortunately, this article is not about pairing heaps. Instead, we need the all familiar binary heap.
A binary heap is a binary tree, where each node is a queue element and a priority. For the minpriq case, each node in the tree has a priority less than the priorities of its children. If a node is placed at the leaf of such a tree, it can be floated up by comparing it and its parent, eventually exchanging their places until the priority invariant has been fulfilled. Similarly, a node can be floated down by comparing the children priorities to each other, and exchanging the node for the child with the least priority.
Treaps
So, why attempt another data structure for the finite map problem? One, it is fun. Two, this algorithm is so simple, it can be explained in a single, tiny(??), TMR article. Third, we need more TMR articles. Simplicity usually means a fast algorithm. Benchmarking treaps against Data.Map was my original idea and maybe a followup article will do carry out this benchmarking.
While the introduction mentions finite maps, we will explore the simpler case where V is the singleton {True} set. The map f then represents a set of keys K, since a key is either mapped to True or it is not, in which case we can return False. Thus, we do not even bother storing the singleton {True} set in the Treap structure. However, extending the treap to also posses arbitrary value data at each node is trivial and left as an exercise to the (interested, practically oriented) reader.
Let K be a totally ordered space of keys. It is clear a binary search tree can be formed obeying this order. Formally, for each node, the left subtree contains keys less than the key at the node and the right subtree contains keys greater than the key at the node.
Let P be a totally ordered set of priorities. It is clear we can form a binary minheap containing the elements of P. Formally, for each node, the subtrees contains keys ordering greater than the key at the node.
Associate with each key k in K a priority p in P. A Treap is then a binary tree obeying the binary search tree property with respect to the Ks as well as the minpriq property of the Ps. Now, if the priorities are chosen randomly, we will actually achieve a balanced tree (!). It might be wise to try to draw such a tree. In fact it is unique. To see this, construct the tree by inserting Ks in increasing order of priorities, by using the binary search tree insert algorithm.
Show me the code!
Enough talk. Haskell! A module representing treaps is first defined:
module Treap (
RTreap
, empty
, null
, insert
, delete
, member
, stdGenTreap
, splitTreap
, joinTreap
) where
import System.Random
import Prelude hiding (null)
A treap is a binary search tree, where each node has a key and a priority:
data Treap k p = Leaf  Branch (Treap k p) k p (Treap k p)
deriving (Show, Read)
The empty tree and the null predicate are simple. They are copied verbatim from the binary search tree:
treap_Empty :: Treap k p
treap_Empty = Leaf
treap_Null :: Treap k p > Bool
treap_Null Leaf = True
treap_Null _ = False
Insertion into a treap works by inserting the node, as if inserting into a binary search tree. Then we use the famous left and rightrotations to float the node up, until it fulfills the heapproperty on its priority. If you are not familiar with left and right rotations, they are just restructurings of a binary search tree, maintaining the ordering property. What is important is they alter the heights of the subtrees and so can help balance the tree more. They are easily definable in Haskell by pattern matching. Drawing them on paper is a good exercise:
rotateLeft :: Treap k p > Treap k p
rotateLeft (Branch a k p (Branch b1 k' p' b2)) =
Branch (Branch a k p b1) k' p' b2
rotateLeft _ = error "Wrong rotation (rotateLeft)"
rotateRight :: Treap k p > Treap k p
rotateRight (Branch (Branch a1 k' p' a2) k p b) =
Branch a1 k' p' (Branch a2 k p b)
rotateRight _ = error "Wrong rotation (rotateRight)"
treap_Insert :: (Ord k, Ord p) => k > p > Treap k p > Treap k p
treap_Insert k p Leaf = Branch Leaf k p Leaf
treap_Insert k p (Branch left k' p' right) =
case compare k k' of
EQ > Branch left k' p' right  Node is already there, ignore
LT > case Branch (treap_Insert k p left) k' p' right of
(t @ (Branch (Branch l' k p r') k' p' right)) >
if p' > p
then rotateRight t
else t
t > t
GT > case Branch left k' p' (treap_Insert k p right) of
(t @ (Branch left k' p' (Branch l' k p r'))) >
if p' > p
then rotateLeft t
else t
t > t
When coding structures based upon binary trees it can be convenient to forget the deletion case. It is often the hardest case to grasp and it can be quite hard to maintain invariants associated with the tree such as the AVLtree or Red/Blacktree. Not so for Treaps, however. We just locate the node by a binary tree search and then float it down by rotations until the node is a leaf using the heapproperties and operations. Then we cut off the leaf (Notice the nice metaphors, please).
treap_Delete :: (Ord k, Ord p) => k > Treap k p > Treap k p
treap_Delete k treap = recDelete k treap
where recDelete k Leaf = error "Key does not exist in tree (delete)"
recDelete k (t @ (Branch left k' p right)) =
case compare k k' of
LT > Branch (recDelete k left) k' p right
GT > Branch left k' p (recDelete k right)
EQ > rootDelete t
priorityCompare Leaf (Branch _ _ _ _) = False
priorityCompare (Branch _ _ _ _) Leaf = True
priorityCompare (Branch _ _ x _) (Branch _ _ y _) = x < y
rootDelete Leaf = Leaf
rootDelete (Branch Leaf _ _ Leaf) = Leaf
rootDelete (t @ (Branch left k p right)) =
if priorityCompare left right
then let Branch left k p right = rotateRight t
in Branch left k p (rootDelete right)
else let Branch left k p right = rotateLeft t
in Branch (rootDelete left) k p right
We must not forget the member function. This is simple, as it is nothing but the original binary search tree function:
treap_Member :: (Ord k, Ord p) => k > Treap k p > Bool
treap_Member e Leaf = False
treap_Member e (Branch left k _ right) =
case compare e k of
LT > treap_Member e left
GT > treap_Member e right
EQ > True
Providing random priorities
The premise of the Treap algorithm is the provision of a good random number generator. If the priorities are randomly assigned, the tree will be balanced well (with a high probability). So, our next quest is to assign priorities randomly to each node. The random assignment also makes it impossible for an evil adversary to unbalance the structure.
There are numerous possibilities, but the one shining most is the System.Random library. The library provides us with 2 type classes Random``Gen and Random. The Random``Gen class are those types g, which can be used as random number generators. The Random class on the other hand are those types a, from which random values can be drawn. That is, given a type of class Random``Gen, any value with a type of class Random can be drawn from it.
The System.Random library also provides a standard random number generator. For our purpose it has the disadvantage of being wrapped inside the IO monad and having to rely on a monad for our treap operations is bad since we then have to thread the monad around with us.
Thus the plan is the following: Initialize a treap as a random number generator and the structure above. Then maintain the random number generator while running operations in the treap. We call this structure an RTreap:
newtype RTReap g k p = RT (g, Treap k p)
deriving (Show, Read)
The empty treap is then an initialization of the random number generator, as said. The null predicate is just a simple reusage of the function above:
empty :: RandomGen g => g > RTreap g k p
empty g = RT (g, treap_Empty)
null :: RandomGen g => RTreap g k p > Bool
null (RT (g, t)) = treap_Null t
Insertion into the treap is done by requesting a new random number from our supply and using this for the node in question. Delete and member are just the same from above with some added structure.
Note we draw random values in a bounded area, such that we have a value less than every random priority in the treap and a value greater than every random priority in the heap. There are certain tricks which can be pulled with these values.
insert :: (RandomGen g, Ord k, Ord p, Num p, Random p)
=> k > RTreap g k p > RTreap g k p
insert k (RT (g, tr)) =
let (p, g') = randomR (2000000000, 2000000000) g
in RT (g', treap_Insert k p tr)
delete :: (RandomGen g, Ord k, Ord p) => k > RTreap g k p
> RTreap g k p
delete k (RT (g, tr)) = RT (g, treap_Delete k tr)
member :: (RandomGen g, Ord k, Ord p) => k > RTreap g k p
> Bool
member k (RT (g, tr)) = treap_Member k tr
The initialization of the RTreap will then be something like:
stdGenTreap :: Int > RTreap StdGen k p
stdGenTreap = (empty . mkStdGen)
The Int type one has to provide is an initialization seed. We can get one such inside an IO monad when starting our program and then use it to seed the Treaps we need afterwards. The functions needed are defined inside the System.Random module.
Cool additions
If we wish to split a treap at a certain node k in K, we can do so, by inserting k with the minimum priority. Assuming p are in the Bounded class:
splitTreap :: (RandomGen g, Bounded p, Ord k, Ord p)
=> k > RTreap g k p > (RTreap g k p, RTreap g k p)
splitTreap k (RT (g, tr)) =
let (g', g'') = split g
Branch left _ _ right = treap_Insert k minBound tr
in (RT (g', left), RT (g'', right))
Similarly to join two disjoint treaps with key spaces K1 and K2, where the keys in K1 are smaller than the keys in K2 (formally: max K1 < min K2), we can choose a key k not in the union (K1, K2) and form the tree where k is the root and the treaps are left and right children. We then proceed by deleting the node k:
joinTreap :: (Bounded p, Ord p, Ord k)
=> k > RTreap g k p > RTreap g k p > RTreap g k p
joinTreap k (RT (g, tr1)) (RT (_, tr2)) =
RT (g, (treap_Delete k (Branch tr1 k maxBound tr2)))
Optimizations
I will simply direct people to the article by Oleg pointed at in the introduction. There are certain optimizations possible, which he thoroughly discusses. Implementing these is an exercise.