TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
Lecture 8 – Overview
Splay trees
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
[Lewis/Denenberg 7.3]
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
Splay operation
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
Splay around missing key 4
5
is the key in T where the search of K succeded or failed.
Given: BST T and key K ( may not be in T )
Splay T around K gives BST T such that:
Root T
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Concatenate
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Conc T1 T2 defined if all keys of T1 smaller than any key of T2:
Splay T1 around ∞; attach T2 as right child of the root.
Dictionary operations on splay tree (2)
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
Splay around 5
T has the same keys as T
an alternative to binary search tree balancing,
[Lewis/Denenberg 9.1]
[Lewis/Denenberg 9.2]
J. Maluszynski, IDA, Linköpings Universitet, 2004.
used for implementation of ADT Dictionary,
guarantees good amortized efficiency.
Lab 3: Binary Search Trees
Priority Queue ADT
extension of ADT Dictionary
used e.g. in shortest paths algorithm for graphs
implementations by balanced trees,
heaps, leftist trees
Up-trees
implementing union of disjoint sets
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K concatenate LC T and RC T
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TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
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Insert 4
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Dictionary operations on splay tree (1)
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up K T : Splay T around K; check if the root is K
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Look
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Insert K T : Splay T around K; add K as the root.
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Splay K T , If Root T
Delete K T : T
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
How to splay
Splay A T ,
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1. A has no g-parent:
rotate around parent B (Fig.7.21)
Three cases:
J. Maluszynski, IDA, Linköpings Universitet, 2004.
General idea:
lift A by rotating T around parent B of A (and g-parent C)
2. key A
key B
key C or key A
key B
key C :
rotate around C, rotate around B (Fig. 7.22)
B
T2
T3
C
T4
A
B
T1
C
A
A has predecessors in line : (1) gp rotation, (2)p rotation
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Repeat until A becomes the root.
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
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Splay step: case 2
T1
T1
T2
T3
T4
T2
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T3
C
T4
J. Maluszynski, IDA, Linköpings Universitet, 2004.
3. otherwise : rotate around B, rotate around C (Fig. 7.23)
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
A
T2
B
T3
T4
T3
T1
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C
T2
A
T3
B
T1
T4
T1
A predecessors not in line : (1) p rotation, (2)gp rotation
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A has no grandparent: Right Rotation
C
Splay step: case 1
T1
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
C
A
Splay step: case 3
T1
T2
C
T3
J. Maluszynski, IDA, Linköpings Universitet, 2004.
T2
A
T3
B
T4
J. Maluszynski, IDA, Linköpings Universitet, 2004.
A
C
T2
J. Maluszynski, IDA, Linköpings Universitet, 2004.
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
Lab 3 Introduction
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Splay Tree Theorem
An interactive system for manipulating BST’s with character keys.
A skeleton code of the system is provided
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
Consider ADT Dictionary implemented as splay tree.
J. Maluszynski, IDA, Linköpings Universitet, 2004.
The skeleton gives access to some operations on BST trees.
The code of the operations is missing.
The assignment: Implement the operations
Any sequence of m dictionary operations such that the splay tree
is initially empty
never has more than n nodes
uses O m log n time.
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delete with auxiliary function deletemin
preorder, postorder, inorder
Proof: Lewis & Denenberg 7.3
levelorder
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
J. Maluszynski, IDA, Linköpings Universitet, 2004.
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TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
ADT Priority Queue
ADT Priority Queue: Motivation
Linearly ordered set K of keys
Commonly encountered situation:
We store pairs k i (as in Dictionary),
but multiple pairs with same key are allowed.
Waiting list (tasks, passengers, vehicles entering a ferry, phone calls)
A frequent operation is retrieving pairs with minimal key.
If a resource is freed choose an element from the waiting list
PQ with minimal k, return i
Operations on a Priority Queue PQ:
The choice is based on some partial ordering:
some tasks are more essential to achieve the goal,
some passengers should be served before the others (children,...)
fire dept. and first aid vehicles have priority
How to organize prioritized service?
MakeEmptyPQ
IsEmpty PQ
Insert PQ k i
DeleteMin PQ delete k i
...
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
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TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
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Heap Insert Example
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
m
J. Maluszynski, IDA, Linköpings Universitet, 2004.
Insert
[L/D Fig. 9.3, Algorithm 9.1]
insert the new node as the “next” leaf
restore partial ordering by swapping the nodes (upwards)
DeleteMin = deletion of the root
[L/D Fig. 9.2, Algorithm 9.1]
replace the root by the “last” leaf
restore partial ordering by swapping the nodes (downwards)
Priority Queue operations on complete partially ordered trees
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Implementing Priority Queues
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Given a complete binary partially ordered tree:
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Searching for a minimal element in a search tree (BST, AVL, 2-3,...)
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BST: does not admit repeated keys, extension needed
Another idea: keep the minimal element as the root of the tree
partially ordered tree
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(implicit representation with size info).
A heap is a complete partially ordered tree represented as a table
Heap
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
the key of a parent is less than or equal to the key of each child
Partially ordered binary tree:
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
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DeleteMin:
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m=0
K=12
Size=4
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
m
J. Maluszynski, IDA, Linköpings Universitet, 2004.
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
Leftist Tree
T binary tree, v node
D LC v
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D RC v
DT v distance to closest external node
T is a leftist tree iff
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
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Consider partially ordered leftist trees.
Heap DeleteMin example
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TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
J. Maluszynski, IDA, Linköpings Universitet, 2004.
for every v
J. Maluszynski, IDA, Linköpings Universitet, 2004.
How to represent such data for efficient implementation of these operations?
Example:
Check to which group a student belong.
Groups may be merged.
Disjoint Sets with Union: Motivation
Implementing ADT Priority Queue with Leftist Trees
Division into disjoint sets appears often in practice.
T, T1, T2 partially ordered leftist trees.
LUnion T 1 T 2 produces a leftist tree with all keys of T 1 T 2
(definition of LUnion p.306)
LUnion LC T RC T
LUnion is logarithmic:
it is accomplished by traversing the shortest path(s) of arguments.
For a tree of size n the shortest path has at most log2 n 1 nodes.
DeleteMin T
Insert X T
LUnion TX T
TX - one node tree with record X
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
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of disjoint sets is maintained:
The ADT Operations on Disjoint Sets
A database
MakeSet X returns a new set with single element X
such that X
Up-tree Implementation
Operations:
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S
[L/D 9.2]
J. Maluszynski, IDA, Linköpings Universitet, 2004.
.
J. Maluszynski, IDA, Linköpings Universitet, 2004.
Union S T
root pointer of S directed to the root of T
or vice versa; O 1
To avoid unbalance attach smaller tree to larger. (Algorithm 9.3)
no restriction on number of children
one pointer per node
Union S T returns S T , which replaces S and T in
Find X returns S in
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
Find X
starting from pointer to X follow the pointers until the root is reached;
O log n
TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
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Up-Tree Representation of Sets
A
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B
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Elements of a set represented by nodes of a tree.
Sets identified by roots
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TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
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Union of Up-Trees
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
D
J. Maluszynski, IDA, Linköpings Universitet, 2004.
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TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
Locating element by a key
B
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H
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Create a dictionary of element pointers.
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
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TDDB57 DALG – Lecture 8: Splay trees. Priority Queues.
Good News
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
Up-trees constructed from singleton sets by Algorithm 9.3
are balanced (why?) (see p.309)
Union takes constant time,
Find takes logarithmic time,
Path compression on Find (p.311) makes amortized time sublogarithmic
(p.314)