SLIDE 1
INF4130 – Algoritmer: Design og effektivitet
30th August 2018 Petter Kristiansen
SLIDE 2 Praktisk info
Gruppe 1: tirsdag 12:15-14:00 Gruppe 2: torsdag 12:15-14:00, siste gang 4/10
- Obliger (ca datoer, ikke fastlagt)
Oblig 1: tidlig i oktober Oblig 2: sist i oktober Oblig 3: sent i november
- Eksamen 18. desember kl. 09:00 (4 timer)
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What is This Course Really All About?
Problems Algorithms
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Problems
SLIDE 5 Modeling
“Given a set of numbers, sort them.” Arranging elements / Sorting numbers 4, 8, 3, 13, 1, 5, 2, 3 1, 2, 3, 3, 4, 5, 8, 13 “Given a set of positive integers, sort them in ascending
Given a set I of n positive integers, I = {i1, i2, … in}, sort i1, i2, … in in ascending order.
INPUT: A set I ϵ Z+of n positive integers I = {i1, i2, … in}. OUTPUT: A permutation Iꞌ of I such that iꞌj ≤ iꞌj+1, 1 ≤ j ≤ n-1. SORTING INSTANCE: A set I ϵ Z+of n positive integers I = {i1, i2, … in}. QUESTION: Is I sorted in ascending order?
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Modeling
Real world
Mathematical representation
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Modeling
Delivery routes / Hamiltonian cycles INPUT: A graph G=(V, E), with V = {v1, v2, … vn}, and E ={e1, e2, … em}, OUTPUT: A permutation Vꞌ of V such that vꞌi vꞌi+1 ϵ E for all 1 ≤ i ≤ n-1, and vꞌn vꞌ1 ϵ E. HAMILTONICITY INSTANCE: A graph G=(V, E). QUESTION: Is there a Hamiltonian cycle in G?
SLIDE 8
Modeling
Software verification / The halting problem
HALTING INSTANCE: A computer program P, and input to the program I. QUESTION: Does program P halt when run on input I?
SLIDE 9 Problems, Formal Languages
Computer Science Problems “Interesting”, “Natural” Problems SORTING, HAMILTONICITY, HALTING, …
graphs, numbers, logical expressions, …
Functions Sets of I/O-pairs
input
Formal Languages Sets of YES-instances (strings)
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Example, Formal Languages
SORTING
{ϵ; 1; 2; … 1,2; 1,3; 1,4; … 1,2,3; 1,2,4; 1,2,5; … …}
HAMILTONICITY
{C3; C4; … K3; K4; … …}
SLIDE 11 Problems
Problems can be very different Problems can be modeled mathematically and represented as formal languages All formal languages are objects
– A common theory and common methods become possible
Set of all Problems (Formal languages)
SLIDE 12
Exercise: Describe Algorithms
Sorting integers INPUT: A set I ϵ Z+of n positive integers I = {i1, i2, … in}. OUTPUT: A permutation Iꞌ of I such that iꞌj ≤ iꞌj+1, 1 ≤ j ≤ n-1. INPUT: {4, 8, 3, 13, 1, 5, 2, 3}
SLIDE 13
Exercise: Describe Algorithms
HAMILTONICITY INSTANCE: A graph G=(V, E). QUESTION: Is there a Hamiltonian cycle in G? INSTANCE1: INSTANCE2:
SLIDE 14 Exercise: Describe Algorithms
HALTING INSTANCE: A computer program P, and input to the program I. QUESTION: Does program P halt when run on input I?
program StringMatcher (P [0:m -1], T [0:n -1]) i ← 0 j ← 0 CreateNext(P [0:m - 1], Next [n - 1]) while i < n do if P [ j ] = T [ i ] then if j = m – 1 then return(i – m + 1) endif i ← i + 1 j ← j + 1 else j ← Next [ j ] if j = 0 then if T [ i ] ≠ P [0] then i ← i + 1 endif endif endif endwhile return(-1) end StringMatcher T = Long string to search for a pattern in P = pattern program HelloWorld() start: print (“Hello World !”) goto start end HelloWorld
P1: I1: P2: I2:
SLIDE 15 Problems and Algorithms
SORTING, HAMILTONICITY and HALTING are very different problems.
- We found (knew?) a fast algorithm for SORTING.
- We came up with a naive algorithm for HAMILTONICITY.
- We really didn’t get anywhere whith HALTING.
SLIDE 16 Problems can be divided into classes depending on how effieciently (fast, running time) they can be solved.
Problems and Algorithms
Problems
SLIDE 17 Undecidability
1900s: Metamathematics (Mathematical studies of mathematics itself, its possibilities and limitations ) 1930s: Results on the existence and non-existence of algorithms for certain problems
– Kurt Gödel (1931): Incompleteness results. – Alan Turing (1936): «On computable numbers, with an application to the Entscheidungsproblem».
Turing’s results and techniques
SLIDE 18 Complexity
John von Neumann (ca 1945): modern computer Jack Edmonds (1965): Paths, Trees, and Flowers Cook / Levin (1972): NP-completeness
Cook / Levin’s results and techniques
SLIDE 19 Algorithms and Turing Machines
…
b 1 b b
… …
b b 1 b b
…
http://aturingmachine.com/
SLIDE 20 Church’s Thesis
Algorithm ≈ Turing machine Turing machines can calculate any function that can be calculated by an algorithm, a computer program or a computer. Expressive power of programming languages A programming language (Java, C, Lisp, …) is Turing complete if it can do the same calculations as a Turing machine. If we can implement a Turing machine with our programming language, it is Turing complete. Universal computer models A computer model is Turing complete if it can do the same calculations as a Turing machine. McCulloch and Pitts have shown that neural nets can simulate Turing machines. Uncomputability If a function f can not be computed by a Turing machine, then no computer can compute f.
SLIDE 21 Course Topics
Undecidability Complexity Problems
String Search Matching Dynamic Programming Search Strategies (A*) Game trees (Alpha – Beta, Chess) Priority Queues
