Splay tree, tree Marko Berezovský Radek Mařík PAL 2012

Transcription

1 Splay tree, --4 tree Marko erezovský Radek Mařík PL 0 p < Hi!?/ x+y To read x--y [] Weiss M.., Data Structures and lgorithm nalysis in ++, rd Ed., ddison Wesley, 4.5, pp [] Daniel D. Sleator and Robert E. Tarjan, "Self-djusting inary Search Trees", Journal of the M (), 985, pp.5-8. [] en Pfaff: Performance nalysis of STs in System Software, 004, Stanford University, Department of omputer Science, Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4 See also PL webpage for references

2 Splay Tree - Description asic idea VL trees and red-black trees are binary search trees with logarithmic height. This ensures all operations are O(ln(n)) n alternative idea is to make use of an old maxim: Data that has been recently accessed is more likely to be accessed again in the near future. ccessed nodes are splayed ( = moved by one or more rotations) to the root of the tree: Find: Find the node like in a ST and then splay it to the root. Insert: Insert the node like in a ST and then splay it to the root. Delete: Splay the node to the root and then delete it like in a ST. Invented in 985 by Daniel Dominic Sleator and Robert Endre Tarjan. Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

3 Splay Tree - Description Properties overview Splay tree - binary search tree. - No additional tree shape description (no additional memory!) is used. - Each node access or insertion splays that node to the root. - Rotations are zig, zig-zig and zig-zag, based on ST single rotation. - ll operations run times are O(n), as the tree height can be (n). - mortized run times of all operations are O(ln(n)). Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

4 Splay Tree - rotation Step-by-step scheme Zig rotation Zig rotation is the same as a rotation (L or R) in VL tree. L R R R rotation L R R L rotation Y L R R L R R fected nodes and edges Note The terms "Zig" and "Zag" are not chiral, that is, they do not describe the direction (left or right) of the actual rotations. Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

5 Splay Tree - rotation Step-by-step scheme 4 Zig - zig rotation L R R R L L L R RRrotation LLrotation L R R L R R L L Note that the topmost node might be either the tree root or the left or the right child of its parent. Only the left child case is shown. The other cases are analogous. Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

6 Splay Tree - rotation Step-by-step scheme 5 Zig - zig rotation L R R R R rotation L R R R L R rotation L R R R R R R oth simple rotations are performed at the top of the current subtree, the splayed node (with key ) is not involved in the first rotation. Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

7 Splay Tree - rotation Step-by-step scheme Zig - zag rotation L L R R L L R R LRrotation RLrotation L L R R L L R R Note that the topmost node might be either the tree root or the left or the right child of its parent. Only the left child case is shown. The other cases are analogous. Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

8 Splay Tree - rotation Step-by-step scheme 7 Zig - zag rotation L rotation L L R R L L R R R rotation L L R R L L R R Note: Zig-Zag rotation is identical to the double (LR or RL) rotation in VL tree. Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

9 Splay Tree - Insert Example 8 Insert Insert Splay Insert Splay Insert 4 Splay 4 4 Insert 5 Splay etc... Insert Splay Note the extremely inefficient shape of the resulting tree Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

10 Splay Tree - Find Example 9 Find Key is the deepest key in the tree Find operation is of (n) complexity in this case :-( Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

11 Splay Tree - Insert Example Scheme - Result of the most unfavourable Find operation Note that the tree heigth is roughly halved. H (H + ) / Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

12 Splay Tree - Find Example Find 4 5 Key is the deepest key in the tree The Find operation would be again of ~n complexity. :-( Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

13 Splay Tree - Find Example Scheme - Progress of the two most unfavourable Find operations. Note the relatively favourable shape of the resulting tree. Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

14 Splay Tree - Delete Simple scheme Delete(k). Find(k); // This splays k to the root. Remove the root; // Splits the tree into L and R subtree of the root.. y = Find max in L subtree; // This splays y to the root of L subtree 4. y.right = R subtree;.find k k. Split = remove root k L R y y = maximum key in L = closest smaller value to k. FindMax(L) y 4. y.right = R y L - {y} R L - {y} R Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

15 Splay Tree - Performance Summary 4 It is difficult to demonstrate the amortized logarithmic behaviour of splay trees using only small trees with few nodes. The original M article [] proves the balance theorem: The run time of performing a sequence of m operations on a splay tree with n nodes is O( m( + ln(n)) + n ln(n) ). Therefore, the run time for a splay tree is comparable to any balanced tree assuming at least n operations. Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

16 Splay Tree - Performance Summary 5 From the time of introducing splay trees (985) up till today the following conjecture (among others) remains unproven. Dynamic optimality conjecture [] onsider any sequence of successful accesses on an n-node search tree. Let be any algorithm that carries out each access by traversing the path from the root to the node containing the accessed item, at a cost of one plus the depth of the node containing the item, and that between accesses performs an arbitrary number of rotations anywhere in the tree, at a cost of one per rotation. Then the total time to perform all the accesses by splaying is no more than O(n) plus a constant times the time required by algorithm. Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

17 Splay Tree - Performance omparisons dvantages: The amortized run times are similar to that of VL trees and redblack trees The implementation is easier No additional information (height/colour) is required Disadvantages: The tree will change with read-only operations Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

18 --4 tree Shape search tree is structurally a -tree of min degree and max degree 4. node is a -node or a -node or a 4-node. If a node is not a leaf it has the corresponding number (,, 4) of children. ll leaves are at the same distance from the root, the tree is perfectly balanced. -node -node 4-node Example Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

19 --4 tree Operations 9 Find: s in -tree Insert: s in -tree: Find the place for the inserted key x in a leaf and store it there. If necessary, split the leaf and store the median in the parent. Splitting strategy dditional insert rule (like single phase strategy in -trees): In our way down the tree, whenever we reach a 4-node (including a leaf), we split it into two -nodes, and move the middle element up to the parent node. This strategy prevents the following from happening: fter inserting a key it might be necessary to split all the nodes going from inserted key back to the root. Such outcome is considered to be time consuming. Splitting 4-nodes on the way down results in sparse occurence of 4-nodes in the tree, thus the nodes never have to be split recursively bottom-up. Delete: s in -tree Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

20 --4 tree Splitting strategy 0 Split node is the root. Only the root splitting increases the tree height a b c d a b c d Split node is the leftmost or the rightmost child of either a -node or a -node. (Only the leftmost case is shown, the righmost case is analogous) X a b c d a b c d X X Y X Y Split node is the middle child of a -node. a b c d a b c d The node being split cannot be a child of a 4-node, due to the splitting strategy. Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

21 --4 Tree Insert example I Insert keys into initially empty --4 tree: S E R H I N G K L M Insert S Insert E Insert S E S E S Insert R E Insert E R S R S Insert H E Insert I E R H R S H I S Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

22 N Insert I Insert N Insert G Insert K E S H R I E S H R I N E S G I H N R E S G I H N R Note the seemingly unnecessary split of E,I,R 4-node during insertion of K. --4 Tree Insert example II Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4 Insert L Insert M K E G I H S L R K N E G I H S M L K R

23 --4 Tree Size example 4 Results of an experiment with N uniformly distributed random keys from range {,..., 0 9 } inserted into initially empty --4 tree: N Tree depth -nodes -nodes 4-nodes Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

24 --4 tree Relation to R- tree 5 Relation of a --4 tree to a red-black tree X X a b a b X Y a X Y a b c b c X Y Z a b c d a X b Y c Z d Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

25 Trees comparison Example en Pfaff: Performance nalysis of STs in System Software, 004, [] onclusions:...unbalanced STs are best when randomly ordered input can be relied upon; if random ordering is the norm but occasional runs of sorted order are expected, then red-black trees should be chosen. On the other hand, if insertions often occur in a sorted order, VL trees excel when later accesses tend to be random, and splay trees perform best when later accesses are sequential or clustered. Some consequences: Managing virtual memory areas in OS kernel:... Many kernels use STs for keeping track of virtual memory areas (VMs) : Linux before.4.0 used VL trees, OpenSD and later versions of Linux use red-black trees, FreeSD uses splay trees, and so does Windows NT for its VM equivalents... Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4

26 Trees comparison Example (excerpt) 7 tree / time in msec / order Memory management supporting web browser ST VL R splay rtificial uniformly random data ST VL R splay Secondary peer cache tree ST VL R splay ompilation identifiers cross-references ST VL R splay Pokročilá lgoritmizace, 4MPL, ZS 0/0, FEL ČVUT, /4