A Lecture on Cartesian Trees

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1 A Lecture o Cartesia Trees Kaylee Kutschera, Pavel Kodratyev, Ralph Sarkis February 28, 207 This is the augmeted trascript of a lecture give by Luc Devroye o the 23rd of February 207 for a Data Structures ad Algorithms class (COMP 252). The subject was Cartesia trees ad quicksort. Cartesia Trees Defiitio. A Cartesia tree is a biary tree with the followig Vuillemi [980] properties:. The data are poits i R 2 : (x, y ),..., (x, y ). We will refer to the x i s as keys ad the y i s as time stamps. 2. Each ode cotais a sigle pair of coordiates. (0,) 3. It is a biary search tree with respect to the x-coordiates. (0,2) 4. All paths from the root dow have icreasig y-coordiate. (3,4) (5,3) The Cartesia tree was itroduced i 980 by Jea Vuillemi i his paper A Uifyig Look at Data Structures. It is easy to see that if all x i s ad y i s are distict, the the Cartestia tree is uique its form is completely determied by the data. Ordiary Biary Search Tree (7,8) Figure : Example of a Cartesia tree with 5 odes. No other cofiguratio for these odes will satisfy the properties of the Cartesia tree. A ordiary biary search tree for x,..., x ca be obtaied by usig the values to i the y-coordiate i the order that data was give: (x, ),..., (x, ). Ufortuately, this ca result i a very ubalaced search tree with height =. Radom Biary Search Tree (RBST) Let (σ,..., σ ) be a radom uiform permutatio of (,..., ). The the data of a RBST is (x σ, ),..., (x σ, ) or equivaletly (x, τ ),..., (x, τ ), where (τ,..., τ ) is the reverse permutatio of (σ,..., σ ). This iverse permutatio will also be radom ad uiform as there is a -to- relatioship with the origial permutatio. Treap Suppose we have data x,..., x that we wat to store. To build a treap, we geerate radom umbers T,..., T called radom time stamps ad pair each x i with a T i, thus givig data pairs (x i, T i ).

2 a lecture o cartesia trees 2 A treap is a combiatio of a biary search tree ad a heap where the x i s are keys to the biary search tree ad T i s are the keys of the heap. Although treaps are ot fully balaced trees, they have expected height O(l()) ad form a typical Cartesia tree. The treap was first described i Arago ad Seidel [989]. Treaps are radom biary search trees, but are easier to maitai as data are added ad deleted. Operatios o Cartesia Trees Compared to other data structures ad especially other search trees that we have see i class, a Cartesia tree has a ice structure that lets us defie complex operatios i terms of two atomic operatios. We will first see the two FIX operatios ad the look at how they are used to costruct the others. 2 Atomic operatios Firstly, we iformally defie what we mea by a fix. The iput is v a Cartesia tree i which oly oe ode, v, does ot satisfy the heap property. We fix the Cartesia tree by movig v iside the tree, keepig the ivariat that oly v does ot satisfy the heap property. Durig this procedure, the ode v ca oly move oe way, either upwards (FIX-UP) or dowwards (FIX-DOWN). Sice a Cartesia tree is fiite, the procedure must halt ad v must be fixed whe this happes (it is ot clear why at the momet but it will be explaied i the followig paragraphs). Whe we say that we are movig a ode i the tree, we caot do this i a arbitrary maer. Each chage we make i the tree should keep the ivariat metioed above. We will use a simple move called a tree rotatio 3 i order to either push a ode above its paret or below its childre. The particularity of the rotatio is that it maitais all the properties of the Cartesia tree. It is easier to describe this pictorially so we have to refer to Figure 2. 2 I defiitio, the fourth property comes from the heap data structure. We give two simpler formulatios for this property (heceforth called the heap property): For ay ode, the paret has a smaller y coordiate. For ay ode, the childre have a greater y coordiate. We will use these simpler formulatios because they correspod more to the operatios we will describe. 3 More iformatio o rotatios i Sleator et al. [988] Figure 2: Example of a tree rotatio that would be doe durig a FIX- UP operatio. The black ode is the misplaced ad should be above the gray ode. I this diagram represetig oe iteratio of a FIX-UP operatio, we have draw two odes ad three subtrees which would be part of a bigger tree. The tagged ode (i black) is misplaced, amely, the y-coordiate of its paret is bigger, however, all of its descedats are well-sorted. We will use a rotatio to try to fix it without messig the

3 a lecture o cartesia trees 3 x-wise order. We put the black ode above the gray oe by replacig the left child of the tagged ode with its paret. The, we put subtree 2 as the right child of the gray ode. Hece, sub-tree 2 is to the left of the black ode ad to the right of the gray ode ad the previous orderig is maitaied. This is called a right rotatio because the tagged ode is the right child of its paret. Nevertheless, by chagig the directio of the arrow ad the colors of the ode i the diagram above, we obtai the completely symmetric left rotatio. Secodly, we will describe the FIX-UP operatio i more depth ad argue about the correctess of the operatio. As we metioed above, the iput is a ode v that has a paret with a larger y-coordiate. Moreover, it is the oly usorted ode i the sese that if you list all y-coordiates from the root to a descedat leaf of v, you obtai a list i icreasig order with oe usorted elemet. Viewig it as a list of y coordiates, the procedure is very similar to a isertio sort. We will loop ad halt whe the y-coordiate of v is greater tha that of its paret or whe v is the root of the tree. At each iteratio, we either do a right rotatio or a left rotatio to make the ode v go up. There are two possibilities for the loop to stop. If v.y > v.paret.y, the v must satisfy the heap property because the y-coordiate of v is smaller tha all the y-coordiates of its descedats (which form a well ordered Cartesia tree) ad it is greater tha all the y- coordiates of its acestors 4 (well ordered as well). If the loop stops because v is at the root, the this meas its y-coordiate is smaller tha every other ode i the tree ad hece v also satisfies the heap property. 4 Recall the isertio sort compariso FIX-UP(ode) while ode = root && ode.y < ode.paret.y 2 // We assume the paret poiters are updated as well 3 if ode = = ode.paret.right the 4 ode. paret. right = ode. left 5 ode. left = ode. paret 6 else // Completely symmetric process to above 7 ode. paret. left = ode. right 8 ode. right = ode. paret For the FIX-DOWN operatio, we will use the same rotatio as described above but we wat to make the tagged ode move dow, as show i Figure 3.

4 a lecture o cartesia trees Figure 3: Example of a tree rotatio that would be doe durig a FIX-DOWN operatio. Note that the oly differece is that the tagged ode is the paret at first ad goes dow i the tree. The algorithm for this operatio is quite similar but it loops util v becomes a leaf or util its y-coordiate becomes smaller tha that of its childre. Furthermore, there is a slight catch whe choosig to do a left rotatio or right rotatio. Sice we will put oe of the childre above the other, we must choose to rotate at the child with the smallest y-coordiate i order to maitai the heap property. FIX-DOWN(ode) while ode is ot leaf && (ode.y > ode.left.y ode.y > ode.right.y) 2 // We assume that the y coordiate of a o-existig ode is 3 if ode.right.y > ode.left.y the 4 ode. leftt = ode. left. right 5 ode. left. right = ode 6 else // Completely symmetric process to above 7 ode. right = ode. right. left 8 ode. right. left = ode Other operatios We will briefly describe four operatios that follow almost immediately from the two atomic operatios. We leave the details of the pseudocode as a exercise.. INSERT(t,(x, y)): Iputs are a Cartesia tree t ad a ode (x, y) to be added to the tree. First, isert the ode (x, ) just as you would do it i a biary search tree, implyig the ode will be a leaf sice the heap property must be satisfied. Secod, chage the ode to (x, y) ad use FIX-UP to fix the Cartesia tree. 2. DELETE(t, ode): Iputs are a Cartesia tree t ad a poiter ode to a ode that will be removed. First, chage ode.y to ad use FIX-DOWN to move the ode dow the tree. It will ed up as a leaf so the secod step is just to remove the ode. 3. JOIN(t,t 2 ): Iputs are two Cartesia trees that eed to be joied ito a sigle tree. It is uderstood that all keys i t are smaller tha all keys i t 2. First, create a temporary ode (k, ) such that MAX X (t ) < k < MIN X (t 2 ) ad let its left child be the root of t while its right child is the root of t 2. Secod, delete the ode ad the tree will fix itself up.

5 a lecture o cartesia trees 5 4. SPLIT(t,k): Iputs are a Cartesia tree t ad a value k that will split the tree ito two trees that have x coordiates smaller tha k ad bigger tha k, respectively. First, isert a temporary ode (k, ) (it will ed up as a root) ad after the procedure, the left subtree ad right subtree of the ode are the trees we are lookig for. Probability review To provide a probabilistic aalysis of treaps, we have to provide some backgroud o probability. 5 We defie a sample space (usually 5 Geoffrey R. Grimmett [200] deoted as Ω) as a set of elemetary evets, or possible outcomes of a experimet. We also deote S as the set of all evets o Ω, amely all subsets of Ω. Example 2. If a coi is flipped: Ω = {H, T}, S = {{H}, {T}, {H, T}, } Defiitio 3. A probability (P) is a mappig from a sample space to R satisfyig the followig properties: A S, P(A) 0 2 P(Ω) = 3 If A i S; i N; ad i = j, A i A j = the P ( i A i ) = i P(A i ) Example 4. Give the coi flip example from above, P({H}) = /2, P({H, T}) =, P( ) = 0. Defiitio 5. A radom variable is a fuctio that associates each outcome i the sample space with a real umber. Example 6. If we wat to model a experimet with two outcomes we ca express this as P(X = ) = p ad P(X = 0) = p. I other words the probability of the first evet happeig is p, ad the probability of the secod evet is -p. This particular example is ofte referred to as the Beroulli radom variable. Lastly, we eed to develop the idea of a expected value. Defiitio 7. The expectatio (we ca thik of this as the "mea" or "average") of a radom variable X is defied to be E[X] = x P(X = x ). = Example 8. If X is the Beroulli radom variable(as above) (p (0, )), X {0, } the E[X] = 0( p) + p = p.

6 a lecture o cartesia trees 6 Expected value aalysis for treaps I a treap, data is represeted as (x, T ),..., (x, T ), where the T i are radom time stamps. Equivaletly, ad for simplicity, let the data be (, T ),..., (, T ) where,..., will be refereed to as the rak of the ode/pair. Defie D k to be the depth of the ode with rak k, amely (k, T k ). Sice the depth of a ode is equivalet to the umber of acestors it has, let D k = X jk where X jk = j =k, if j acestor of k 0, otherwise So ext we wish to calculate the expected value for D k. E[D k ] = P((j, T j ) is a acestor of (i, T i )) j =k = P(T j is the smallest of T j, T j+,, T k ) j =k = = j<k k j + + j k + j>k ) + ( k =(H k ) + (H k+ ) 2 l().39 log 2 (). ( k ) We will deote H k to be the harmoic umber, k = = k. It ca be approximated by l(k) but more importatly, it ca be bouded as follows H k [l(k + ), l(k) + ]. If we set T k =, the E[D k ] = H k+ + H k. So ext we wish to see how may FIX steps are expected whe isertig (k, T k. Usig the two results above, we fid that it is equal to (H k + H k+ 2) (H k+ + H k ) 2 Usig Figure 4 ad the above argumet, we ca see that it is expected to take two steps to fix the Cartesia tree. We have show that search, isert, ad delete take O(log()) expected umber of steps, makig the treap a viable data structure for the abstract data type "dictioary". Figure 4: Represetatio of a treap with the expected height differece whe isertig a ew ode. Quicksort Give a set of umbers, Quicksort 6. picks a pivot at radom from the set, partitios the remaiig umbers ito elemets smaller tha 6 This algorithm was first itroduced by Toy Hoare [962] ad a full aalysis of the algorithm was doe by Robert Sedgewick [977] for his Ph.D. thesis

7 a lecture o cartesia trees 7 the pivot ad elemets bigger tha the pivot ad recurses ito the two ew sets. There are may differet algorithm to partitio the elemets of the set but it must be doe i O() time for a set of elemets. A simple pseudocode follows. Quicksort(list, low, high) i = radom idex betwee low ad high 2 a = list[i] 3 if high > low the 4 Partitio list[low...high] so elemets before idex j are smaller tha a ad elemets after idex j are greater tha a 5 Quicksort(list, low, j ) 6 Quicksort(list, j +, high) Oe ca see that the Quicksort procedure is equal to buildig a radom biary search tree. I fact, the umber of comparisos eeded to build a radom biary search tree is the same eeded to Quicksort. Compariso Complexity Let C = # of comparisos = i= D i. E[C ] = = i= i= i= i= i= j= j= E[D i ] (H i + H i+ 2) (H i ) H i 2 i j= j j 2 i=j 2 ( j + ) 2 j ( + )H 4 2 l() = log 2 (). It is oteworthy that this is about 39% worse tha if we had used mergesort, or ideed, the best possible compariso based sortig method.

8 a lecture o cartesia trees 8 Improvemets Oe ca improve Quicksort, by pickig three radom umbers ad takig the media of the three as the pivot. The resultig umber of comparisos would be approximately 2 7 l(). If we take five rather tha three radom umbers, the umber of comparisos would be approximately l(). Exercise 9. Show that the expected umber of FIX-DOWN rotatios eeded for the operatio JOIN o two treaps of size m ad, respectively, is H + H m.

9 a lecture o cartesia trees 9 Refereces C. R. Arago ad R. G. Seidel. Radomized search trees. 30th Aual Symposium o Foudatios of Computer Sciece, pages , Oct 989. doi: 0.09/SFCS URL washigto.edu/arago/pubs/rst89.pdf. David R. Stirzaker Geoffrey R. Grimmett. Probability ad Radom Processes. Oxford Uiversity Press, 3rd editio, 200. ISBN C. A. R. Hoare. Quicksort. The Computer Joural, 5:0 6, Ja 962. doi: 0.093/comjl/5..0. URL comjl/5..0. Robert Sedgewick. The aalysis of quicksort programs. Acta Iformatica, 7(4): , 977. ISSN doi: 0.007/BF Daiel D. Sleator, Robert E. Tarja, ad William P. Thursto. Rotatio distace, triagulatios, ad hyperbolic geometry. Joural of the America Mathematical Society, (3):647 68, 988. ISSN , URL Jea Vuillemi. A uifyig look at data structures. Commuicatios of the ACM, 23: , Apr 980. doi: 0.45/ URL org/742/95ba5043ec45345d6a9c9ee6545d345ecd.pdf.

CSE 4095/5095 Topics in Big Data Analytics Spring 2017; Homework 1 Solutions

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