TDDB57 DALG – Lecture 1: Basics.
Main concepts (1)
Page 1
Algorithmic problem
+ describes input data and corresponding output
J. Maluszynski, IDA, Linköpings Universitet, 2004.
Abstract data type (ADT)
+ machine-independent, high-level description
of data and operations on them
+ e.g.: Dictionary, Stack, Queue, PriorityQueue, Set, ...
Data structure
+ logical organization of computer memory for storing data
Algorithm
+ high-level description of concrete operations on data structures
ADT implemented by suitable data structures and algorithms
J. Maluszynski, IDA, Linköpings Universitet, 2004.
TDDB57 DALG – Lecture 1: Basics.
Main concepts (2)
Algorithm
Page 2
J. Maluszynski, IDA, Linköpings Universitet, 2004.
+ Concrete but machine- and language-independent specification
of all necessary steps (simple operations on data)
to compute a solution (output) from a given problem instance (input)
+ Correctness
+ Complexity analysis
time and space consumption, scalability, efficiency
worst case, best case, average case, amortized analysis
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
+ Algorithmic paradigms
commonly used problem solving strategies
e.g., divide&conquer, dynamic programming, greedy strategy, ...
TDDB57 DALG – Lecture 1: Basics.
Example problem (2): Describing Data
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Example algorithmic problem (1)
TDDB57 DALG – Lecture 1: Basics.
Checking the exam result of a student:
First step:
Choose a suitable Abstract Data Type: objects and operations
Here, ADT Set would be also appropriate.
In the example: ADT Dictionary
Domains:
K: linearly ordered set of keys (here: names)
I: information (here: just information passed yes / not passed no)
Operations on a dictionary D:
LookUp D k K returns “yes” iff k D
...
Details of the operations are hidden from the user.
input data:
exam result list, name
ADT)
data structure)
desired output:
yes or no
What are the data and necessary operations? (
How to represent the data in the computer? (
How to realize the operations with that data structure?
Next step:
Find suitable representation of the ADT as data structure
logical organization of computer memory.
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1028
1032
....
222
224
226
228
230
232
...
412
N
N
A
J. Maluszynski, IDA, Linköpings Universitet, 2004.
A
1032
D
B
Info
224
000
232
Next
D
J. Maluszynski, IDA, Linköpings Universitet, 2004.
A
222
C
C
...
228
P
B
TDDB57 DALG – Lecture 1: Basics.
Page 6
Memory (2): RAM model
static:
program memory
current instruction
Page 8
clock
load
CPU
register 1
store
dynamic:
amount of data can be changed
e.g., singly linked lists
amount of data cannot be changed
e.g. tables
Static vs. Dynamic Data Structures
TDDB57 DALG – Lecture 1: Basics.
J. Maluszynski, IDA, Linköpings Universitet, 2004.
data memory
M[3]
.....
M[2]
M[1]
M[0]
J. Maluszynski, IDA, Linköpings Universitet, 2004.
RAM model assumes: All primitive operations take 1 unit of time.
arithmetic operations (add, sub, mul, ...) on word-sized operands
load/store of word-sized operands
jumps (conditional, non-conditional)
Primitive operations in the RAM model:
PC
register 2
....
ALU
An abstract model of a simple but general computer architecture
with such a flat, random-access memory model is called a
Random Access Machine (RAM).
TDDB57 DALG – Lecture 1: Basics.
Memory (1)
basic addressable units (bytes)
cell: a number of units with an address
to store a simple data type
random access:
the same amount of time is taken
for retrieving / storing at any address
abstracts from memory hierarchy
Page 7
pointer:
a cell containing an address;
Λ null pointer
TDDB57 DALG – Lecture 1: Basics.
Memory (3): Basic Memory Structures
record:
cell(s) containing a structured object;
the components are called fields
table:
memory cells of equal size
in a contiguous block
linked structure:
built with pointers
A
J. Maluszynski, IDA, Linköpings Universitet, 2004.
TDDB57 DALG – Lecture 1: Basics.
Page 10
Dictionary lookup algorithms: linear vs. binary search
Page 9
Efficiency of Algorithms
Search for the name (key) k in an ordered table T .
TDDB57 DALG – Lecture 1: Basics.
How many steps (primitive RAM operations) are needed to obtain the result?
Linear search:
Binary search:
How many comparisons are needed?
Compare k with the keys in consecutive cells of T .
This depends on:
- the size of input
express / approximate the number of steps as a function of input size
- the kind of input: worst-case / average case analysis
Compare k with the key m in the middle cell of T
k m: name found, stop
k m: search in the first half of T
k m: search in the second half of T
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
J. Maluszynski, IDA, Linköpings Universitet, 2004.
to abstract from too low-level (primitive RAM) operations
to discuss efficiency of the algorithms
to present commonly used data structures and algorithms
Objectives:
Straightforward to translate into source code for high-level programming
languages such as Ada, Pascal, C, Java
Easy description of typical memory operations
Reflecting memory organization
As simple as possible
Pseudocode-Notation for Data Structures and Algorithms
TDDB57 DALG – Lecture 1: Basics.
How many comparisons are needed?
- the data structure used
J. Maluszynski, IDA, Linköpings Universitet, 2004.
(more ambitious: search phone number in a telephone directory with n entries)
Page 11
Example: Dictionary search (LookUp operation)
search a student’s name in an exam-passed list with n entries
+ Linear search
+ Binary search
TDDB57 DALG – Lecture 1: Basics.
Suming-up the Example
The same data can be stored in a different way
table vs. records linked with pointers
The same problem can be solved with different algorithms
linear search vs. binary search
The efficiency of the algorithms
depends on
the size of data and on
the kind of data e.g. worst case, best case
can be characterized/approximated by a function on the size
TDDB57 DALG – Lecture 1: Basics.
Page 14
Y
Y
Z
X
Y
X
J. Maluszynski, IDA, Linköpings Universitet, 2004.
X
Y
Z
X
Y
X
TableSearch D V
procedure Link pointer P, Q
...
J. Maluszynski, IDA, Linköpings Universitet, 2004.
J. Maluszynski, IDA, Linköpings Universitet, 2004.
Page 13
Pseudocode Notation: Assignment
Y means:
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TDDB57 DALG – Lecture 1: Basics.
Pseudocode Notation: Data Types
X
Simple assignment
atomic: integer, boolean, ...
where X is a variable, a table element (e.g. T i ), a field (e.g. Next L )
and Y is an expression representing a value
X
Simultaneous assignment:
Abbreviation:
TDDB57 DALG – Lecture 1: Basics.
pointer values are addresses, null pointer Λ
J. Maluszynski, IDA, Linköpings Universitet, 2004.
key
a generic name for totally ordered data
table:
declaration: table boolean T a b
table of booleans with integer indices ranging from a to b
the i-th element, for a i b
access: T i
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node with fields
occupies a single block of memory (cell)
fields accessed by names (dot or macro notation) anode.name, name anode
created by NewNode(nodetype), which returns a pointer to the node
TDDB57 DALG – Lecture 1: Basics.
Pseudocode Notation: Subroutines
Pseudocode Notation: Control
Functions:
Conditional statements:
if ... then ... / if ... then ... else ...
function TableSearch table key T 1 n , key K : integer
a local variable
integer x;
...
stops execution and returns the value x
... return x;
...
Call:
this is a comment
Loops:
for ... from ... to ... do
while ... do
scope of do shown by indentation
Comments
Procedures (functions not returning values)
Recursion is allowed.
Short-cut evaluation of Boolean expressions:
evaluated from left to right
execution stops as soon as the final value is determined
e.g., X and Y and ...
evaluation stops if false obtained
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J. Maluszynski, IDA, Linköpings Universitet, 2004.
TDDB57 DALG – Lecture 1: Basics.
Page 18
J. Maluszynski, IDA, Linköpings Universitet, 2004.
Example: Dictionary Search, Variant 2
TDDB57 DALG – Lecture 1: Basics.
Example: Dictionary Search, Variant 1
Dictionary ADT represented as an ordered, singly linked list L.
The LookUp operation is realized by ListSearch:
D
J. Maluszynski, IDA, Linköpings Universitet, 2004.
Dictionary ADT represented as an ordered table T ,
where T i gives the contents of the i-th cell.
The LookUp operation is realized by TableSearch:
C
function ListSearch (pointer List, key k ) : boolean
pointer P List
while P Λ and Key P
k do
if Key P
k then return true
P Next P
return false
B
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function TableSearch ( table key T 1 n , key k ): boolean
for i from 1 to n do
if T i k then return true
if T i k then return false
return false
P
A
Alternatively, the binary search algorithm may be applied.
List
TDDB57 DALG – Lecture 1: Basics.
Using Locatives
J. Maluszynski, IDA, Linköpings Universitet, 2004.
Page 19
600
procedure LLInsert ( key k, locative P )
insert a cell containing key k in list P if none exists already
while P Λ and Key P
k do
P Next P
if P Λ and Key P
k then return
P NewNode k P
TDDB57 DALG – Lecture 1: Basics.
400
Locative: an extended pointer
Locative P
600
The course book Lewis/Denenberg
uses so-called locatives
to simplify algorithm descriptions:
400
P <- Next(P)
Locative P
600
+ locative = extended pointer
that memoizes the “source”
of a link automatically
500
+ eliminates some special cases
400
P <= NewNode(500, P)
“syntactic sugar”
Locative P
+ eliminates an auxiliary pointer
TDDB57 DALG – Lecture 1: Basics.
Summary
Page 21
ADT tells what to do: set of operations on data.
J. Maluszynski, IDA, Linköpings Universitet, 2004.
To describe how to do that, we must
choose a data structure (representation in memory)
find algorithms that realize the ADT operations
and describe them as functions on that data structure
The same ADT can be implemented with different data structures.
The data structure chosen may restrict the choice of the algorithms to be
used for the ADT operations and thus the efficiency that can be achieved.
We use pseudocode notation for precise and concise description of algorithms.