Post on 18-Oct-2020
Discrete and Algorithmic GeometryFacultat de Matematiques i Estadıstica
Universitat Politecnica de Catalunya
Using the appropriatedata structure
Vera Sacristan
Depending on what we want to do with our data,it may be convenient to store them in different ways
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
USING THE APPROPRIATE DATA STRUCTURE
Depending on what we want to do with our data,it may be convenient to store them in different ways
Example
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
USING THE APPROPRIATE DATA STRUCTURE
Depending on what we want to do with our data,it may be convenient to store them in different ways
Example
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Consider the following graph G:
1 2
34 5
USING THE APPROPRIATE DATA STRUCTURE
Depending on what we want to do with our data,it may be convenient to store them in different ways
Example
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Consider the following graph G:
1 2
34 5
Incidence matrix0 1 1 1 01 0 1 0 11 1 0 0 01 0 0 0 00 1 0 0 0
USING THE APPROPRIATE DATA STRUCTURE
Depending on what we want to do with our data,it may be convenient to store them in different ways
Example
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Consider the following graph G:
1 2
34 5
Incidence matrix0 1 1 1 01 0 1 0 11 1 0 0 01 0 0 0 00 1 0 0 0
Adjacency list
((2, 3, 4), (1, 3, 5), (1, 2), (1), (2))
USING THE APPROPRIATE DATA STRUCTURE
Depending on what we want to do with our data,it may be convenient to store them in different ways
Example
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Consider the following graph G:
1 2
34 5
Incidence matrix0 1 1 1 01 0 1 0 11 1 0 0 01 0 0 0 00 1 0 0 0
Adjacency list
((2, 3, 4), (1, 3, 5), (1, 2), (1), (2))
Are vi and vjconnected?
USING THE APPROPRIATE DATA STRUCTURE
Depending on what we want to do with our data,it may be convenient to store them in different ways
Example
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Consider the following graph G:
1 2
34 5
Incidence matrix0 1 1 1 01 0 1 0 11 1 0 0 01 0 0 0 00 1 0 0 0
Adjacency list
((2, 3, 4), (1, 3, 5), (1, 2), (1), (2))
Are vi and vjconnected?
O(1)
O(n)
USING THE APPROPRIATE DATA STRUCTURE
Depending on what we want to do with our data,it may be convenient to store them in different ways
Example
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Consider the following graph G:
1 2
34 5
Incidence matrix0 1 1 1 01 0 1 0 11 1 0 0 01 0 0 0 00 1 0 0 0
Adjacency list
((2, 3, 4), (1, 3, 5), (1, 2), (1), (2))
Are vi and vjconnected?
O(1)
O(n)
What isdegree(vi)?
USING THE APPROPRIATE DATA STRUCTURE
Depending on what we want to do with our data,it may be convenient to store them in different ways
Example
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Consider the following graph G:
1 2
34 5
Incidence matrix0 1 1 1 01 0 1 0 11 1 0 0 01 0 0 0 00 1 0 0 0
Adjacency list
((2, 3, 4), (1, 3, 5), (1, 2), (1), (2))
Are vi and vjconnected?
O(1)
O(n)
What isdegree(vi)?
O(n)
O(degree)
USING THE APPROPRIATE DATA STRUCTURE
Depending on what we want to do with our data,it may be convenient to store them in different ways
Example
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Consider the following graph G:
1 2
34 5
Incidence matrix0 1 1 1 01 0 1 0 11 1 0 0 01 0 0 0 00 1 0 0 0
Adjacency list
((2, 3, 4), (1, 3, 5), (1, 2), (1), (2))
Are vi and vjconnected?
O(1)
O(n)
What isdegree(vi)?
O(n)
O(degree)
Storagespace
USING THE APPROPRIATE DATA STRUCTURE
Depending on what we want to do with our data,it may be convenient to store them in different ways
Example
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Consider the following graph G:
1 2
34 5
Incidence matrix0 1 1 1 01 0 1 0 11 1 0 0 01 0 0 0 00 1 0 0 0
Adjacency list
((2, 3, 4), (1, 3, 5), (1, 2), (1), (2))
Are vi and vjconnected?
O(1)
O(n)
What isdegree(vi)?
O(n)
O(degree)
Storagespace
O(n2)
O(edges)
USING THE APPROPRIATE DATA STRUCTURE
USING THE APPROPRIATE DATA STRUCTURE
Terminology
Let S be a set in a given universe U .
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
USING THE APPROPRIATE DATA STRUCTURE
Terminology
Let S be a set in a given universe U .
Any data structure representing S and allowing the following operations:
• query• add• delete
is called a dictionary
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
USING THE APPROPRIATE DATA STRUCTURE
Terminology
Let S be a set in a given universe U .
Any data structure representing S and allowing the following operations:
• query• add• delete
is called a dictionary
If the universe U is totally sorted:
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
USING THE APPROPRIATE DATA STRUCTURE
Terminology
Let S be a set in a given universe U .
Any data structure representing S and allowing the following operations:
• query• add• delete
is called a dictionary
If the universe U is totally sorted:
• locate• insert• delete
dictionary
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
USING THE APPROPRIATE DATA STRUCTURE
Terminology
Let S be a set in a given universe U .
Any data structure representing S and allowing the following operations:
• query• add• delete
is called a dictionary
If the universe U is totally sorted:
• locate• insert• delete
dictionary
• minimum
priority queue
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
USING THE APPROPRIATE DATA STRUCTURE
Terminology
Let S be a set in a given universe U .
Any data structure representing S and allowing the following operations:
• query• add• delete
is called a dictionary
If the universe U is totally sorted:
• locate• insert• delete
dictionary
• minimum
priority queue
• maximum• next• previous
augmented dictionary
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Data structures to implement these operations
USING THE APPROPRIATE DATA STRUCTURE
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Data structures to implement these operations
Linked list
start x1 x2 . . . xn end
USING THE APPROPRIATE DATA STRUCTURE
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Data structures to implement these operations
Linked list
start x1 x2 . . . xn end
Doubly linked list
x1 x2 . . . xn start/endstart/end
USING THE APPROPRIATE DATA STRUCTURE
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Data structures to implement these operations
Linked list
start x1 x2 . . . xn end
Doubly linked list
x1 x2 . . . xn start/endstart/end
How are the operations done?
• Locate: explore the list in O(i) time• Insert/ delete: Once located, change pointers in O(1) time• Min/max: O(1) time• Next/previous: O(1) time
USING THE APPROPRIATE DATA STRUCTURE
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Data structures to implement these operations
Linked list
start x1 x2 . . . xn end
Doubly linked list
x1 x2 . . . xn start/endstart/end
How are the operations done?
• Locate: explore the list in O(i) time• Insert/ delete: Once located, change pointers in O(1) time• Min/max: O(1) time• Next/previous: O(1) time
Two particular lists are:
Stack: LIFO list (insert/delete only at the end)
Queue: LILO list (insert at the end, delete at the beginning)
USING THE APPROPRIATE DATA STRUCTURE
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Data structures to implement these operations
Binary-search trees
Balanced binary trees
USING THE APPROPRIATE DATA STRUCTURE
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Data structures to implement these operations
Binary-search trees
Balanced binary trees
acyclic connected graphs
USING THE APPROPRIATE DATA STRUCTURE
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Data structures to implement these operations
Binary-search trees
Balanced binary trees
acyclic connected graphs
each node has at most two children
USING THE APPROPRIATE DATA STRUCTURE
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Data structures to implement these operations
Binary-search trees
Balanced binary trees
acyclic connected graphs
each node has at most two children
all branches have approximately the same length
USING THE APPROPRIATE DATA STRUCTURE
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Data structures to implement these operations
Binary-search trees
Balanced binary trees
acyclic connected graphs
each node has at most two children
all branches have approximately the same length
Example
S = {1, 2, 3, 4, 5, 6, 7, 8} in U = (R ≤)
USING THE APPROPRIATE DATA STRUCTURE
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Data structures to implement these operations
Binary-search trees
Balanced binary trees
acyclic connected graphs
each node has at most two children
all branches have approximately the same length
Example
S = {1, 2, 3, 4, 5, 6, 7, 8} in U = (R ≤)
≤ 4
≤ 1
≤ 6≤ 2
≤ 3 ≤ 5 ≤ 7
1 2 3 4 5 6 7 8
USING THE APPROPRIATE DATA STRUCTURE
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Data structures to implement these operations
Binary-search trees
Balanced binary trees
acyclic connected graphs
each node has at most two children
all branches have approximately the same length
Example
S = {1, 2, 3, 4, 5, 6, 7, 8} in U = (R ≤)
≤ 4
≤ 1
≤ 6≤ 2
≤ 3 ≤ 5 ≤ 7
1 2 3 4 5 6 7 8
• Locate(3)• Locate(3.5)• Min(S)• Insert(3.5)• Delete(3)
USING THE APPROPRIATE DATA STRUCTURE
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
Data structures to implement these operations
Binary-search trees
Balanced binary trees
acyclic connected graphs
each node has at most two children
all branches have approximately the same length
Theorem
Given a set S of n elements ot a totally ordered univers U , it is possible to build abalanced binary search tree representing S using O(n) space and O(n log n) time.
Then: • memberQ• locate• min/max• insert/delete
run in O(log n) time
• next/previous runs in O(1) time
USING THE APPROPRIATE DATA STRUCTURE
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
INITIAL READING
F. P.Preparata and M. I ShamosComputational Geometry: An IntroductionSpringer-Verlag, 1985, pp. 6-15 and 26-35.
J-D. Boissonnat and M. YvinecAlgorithmic GeometryCambridge University Press, 1997, pp. 3-31.
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
USING THE APPROPRIATE DATA STRUCTURE
INITIAL READING
F. P.Preparata and M. I ShamosComputational Geometry: An IntroductionSpringer-Verlag, 1985, pp. 6-15 and 26-35.
J-D. Boissonnat and M. YvinecAlgorithmic GeometryCambridge University Press, 1997, pp. 3-31.
Discrete and Algorithmic Geometry, Facultat de Matematiques i Estadıstica, UPC
FURTHER READING
T. H. Cormen, C. E. Leiserson, and R. L. RivestIntroduction to AlgorithmsThe MIT Press - McGraw Hill Book Company, 1990 (3rd ed. 2009)
A. V. Aho, J. E. Hopcroft, and J. D. UllmanData structures and algorithmsAddison-Wesley, 1983
USING THE APPROPRIATE DATA STRUCTURE