TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
Lecture 5 – Overview
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Implementation of Set and Dictionary using tables
+ Analysis of binary search
J. Maluszynski, IDA, Linköpings Universitet, 2004.
[Lewis/Denenberg p.181-184]
[Lewis/Denenberg 4.1]
J. Maluszynski, IDA, Linköpings Universitet, 2004.
Implementation of Set and Dictionary using linked lists
Trees: Basic terminology
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TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
Lecture 3)
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Implementations of ADTs Set and Dictionary
Table (
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Lecture 7)
Lecture 5, 6)
Lecture 3)
– Unordered table
– Ordered table
Linked lists (
Lecture 4)
– Unordered lists
– Ordered lists
Hashing (
(Binary) search trees (
AVL-trees, Splay trees (
TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
0 elements, key k to search for
J. Maluszynski, IDA, Linköpings Universitet, 2004.
[Lewis/Denenberg 6.3]
[Lewis/Denenberg 6.2]
[Lewis/Denenberg 6.4]
J. Maluszynski, IDA, Linköpings Universitet, 2004.
Analysis of binary search in an ordered table (1)
Table representations for ADTs Set and Dictionary
Given: table T with n
For unordered keys:
LookUp by linear search
unsuccessful lookup: n comparisons
O n time
successful lookup, worst case: n comparisons
O n time
successful lookup, average case with uniform distribution of requests:
1
n 1
1 2
n 1 n
O n time
comparisons
n
2
Insert in O 1
LookUp by binary search
worst case and average case: O log n time
Insert requires reconstruction of the table
O n worst case time
function BinSearchLookU p table T 0 n 1 key k : pointer
(1) l 0; r n 1
(2) while true do
search in interval l
r for k:
(3)
if r l then return Λ
l r 2
(4)
mid
(5)
if k key T mid then return T mid
(6)
else if k key T mid then r mid 1
(7)
else l mid 1
For ordered keys:
TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
Page 5
r=9
J. Maluszynski, IDA, Linköpings Universitet, 2004.
depth = 3
depth = 2
depth = 1
depth = 0
TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
v from the root to each tree node v.
Maximum depth of an internal node is log2 n
Maximum length of a path is log2 n or log2 n
log2 n 1 comparisons
1 comparisons
1
Depth dv of a node v of the tree:
number of nodes (comparisons) on path from root to v, excluding v
Length of a path π:
number of edges on path from root to v
root
Analysis of binary search in an ordered table (3): Worst case
k>T[4]
l=5
m=7
k>T[7]
Analysis of binary search in an ordered table (2)
l=0
r=9
m=4
k<T[7]
k<T[4]
r=3
k>T[1]
m=8
k>T[8]
There is exactly one path π
l=0
m=1
l=8
r=9
Possible executions represented by a binary tree (a decision tree):
k<T[1]
k>T[5]
r=9
successful search: at least 1, at most
unsuccessful search: log2 n or log2 n
r=6
l=9
10 9 depth = 4
m=9
7
8
m=5
8
9
l=5
6
k>T[2]
r=6
m=6
l=6
7
r=3
5 4
5
m=2
r=3
m=3
l=3
6
l=2
3
r=0
2 1
4
m=0
1 0
2
l=0
0 -1
3
internal nodes correspond to comparisons and successful returns
external (leaf) nodes represent unsuccessful termination points
edges correspond to a backward branch in the loop
1
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2n
1 log2 n
1
J. Maluszynski, IDA, Linköpings Universitet, 2004.
E
n
n
1
n
Ext dv
E
E
log2 n
log2 n
1
log2 n
1
Σv
n
and E
n
Int dv
C
C
not worse than unsuccesful search
Σv
1
Expected number of comparisons:
1
I
I n
1
n
n
Search succeeds at internal node v with depth dv.
log2 n
1 comparisons.
1 external nodes).
Depth of an external node: dv
C
Needs then dv
The tree has n internal nodes (thus n
Fact: E I 2n
(prove by induction)
TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
J. Maluszynski, IDA, Linköpings Universitet, 2004.
TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
Analysis of binary search in an ordered table (5): Average case cont.
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Analysis of binary search in an ordered table (4): Average case
Recall I
Table with n items. Succesful search. Uniform distribution.
Denote
I Σv Int dv
E Σv Ext dv
move to front heuristic
can improve average time.
+ expression trees
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
+ structured documents
(book: chapters, sections, subsections, paragraphs, ...)
+ hierarchical organization diagrams
(company: departments, divisions, groups, employees)
+ hierarchical classification systems in science and engineering
+ genealogic trees
(successors of a person)
J. Maluszynski, IDA, Linköpings Universitet, 2004.
TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
on the keys.
binary search impossible.
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TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
Unordered list representation of ADTs Set and Dictionary
In practice some keys are accessed more often than others:
J. Maluszynski, IDA, Linköpings Universitet, 2004.
Analysis of binary search in an ordered table (6): Average case cont.
1
J. Maluszynski, IDA, Linköpings Universitet, 2004.
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By the analysis we obtained:
binary search trees
Binary Search Expected Time Theorem:
Insert in constant time
LookUp and Delete require list traversal
unsuccessful and worst case successful needs n length L comparisons
average case with uniform distribution of requests:
1
n 1
1 2
n 1 n
n
2
log2 n
Given a table of n elements,
assuming uniform distribution of keys,
the expected number C of comparisons in a successful search is
C
TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
log2 n 1
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The list items are ordered wrt ordering
combine the flexibility of linked lists (insertion, deletion)
with fast access to the middle elements.
TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
Trees: examples of use
Ordered list representation for ADTs Ordered Set, Dictionary
Unsuccessful LookUp
needs on average n 2 comparisons
Successful LookUp
needs on average n 2 comparison
no direct access to the middle element
Idea:
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
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V:
u
w
V:
V:
1
J. Maluszynski, IDA, Linköpings Universitet, 2004.
path from v to w in T
path from u to v in T
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height h v of a node v V
length of longest path from v to a successor of v
height h T of tree T = height of the root of T
Complete binary trees
1
J. Maluszynski, IDA, Linköpings Universitet, 2004.
vl in T
V E from v1 to vl with length l
i l, and vi vi 1
E i 1 i l
An ordered binary tree T is complete iff:
TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
Trees: Basic terminology (2)
v1 v2
V i 1
TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
Trees: Basic terminology (1)
path π
if vi
ancestors of a node v
V:
successors of a node v
VE .
V (store data items)
depth d v of a node v V
length of longest path from the root to v
A Tree is a graph T
Nodes v
Edges E V V
E referred to as parent-child relation
E: w V
0, or
TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
There is exactly one node that has no parent: the root of T .
vw
Each non-root node has exactly one parent; may have siblings
The degree of a node v V is the number of its children:
J. Maluszynski, IDA, Linköpings Universitet, 2004.
A node that has no children is called a leaf node.
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TDDB57 DALG – Lecture 5: ADT Dictionary and Its Implementations
Ordered tree: linear order among the children of each node
Special kinds of trees
height T
1 and T has left child, (may have both) or
height T
height T
n 1 and
either
left subtree of T is a perfect tree of height n 1
right subtree of T is a complete tree of height n
or
2 for each node
left subtree of T is a complete tree of height n 1
right subtree of T is a perfect tree of height n 2
Binary tree: ordered tree with degree
left child, right child
Empty binary tree (Λ): binary tree with no nodes
Full binary tree: nonempty; degree is either 0 or 2 for each node
Fact: number of leaves = 1 + number of interior nodes (proof by induction)
Perfect binary tree: full, all leaves have the same depth
Fact: number of nodes = 2h 1 1 (2h leaves) for a perfect tree of height h
(proof by induction on h)
Complete binary tree: approximation to perfect, see next slide