Computability - Tractable, Intractable and Non-computable Function

Computability
Tractable, Intractable and
Non-computable functions
(C) 2010 Thomas J Cortina, Carnegie Mellon University

Efficiency
●A computer program should be totally correct, but
it should also
oexecute as quickly as possible (time-efficiency)
ouse memory wisely (storage-efficiency)
●How do we compare programs (or algorithms in
general) with respect to execution time?
ovarious computers run at different speeds due to different
processors
ocompilers optimize code before execution
othe same algorithm can be written differently depending on the
programming language used

Order of Complexity
●For very large n, we express the number of operations
as the order of complexity.
●Order of complexity for worst-case behavior is often
expressed using Big-O notation:
Number of operations Order of Complexity
n O(n)
n/2 + 6 O(n)
2n + 9 O(n)
Usually doesn't
matter what the
constants are...
we are only
concerned about
the highest power
of n.(C) 2010 Thomas J Cortina, Carnegie Mellon University

O(n) ("Linear")
n
(amount of data)
Number of
Operations
n
n/2 + 6
2n + 9

Order of Complexity
n2 O(n2)
2n2 + 7 O(n2)
n2/2 + 5n + 2 O(n2)
Usually doesn't
matter what the
constants are...
we are only
concerned about
the highest power
of n.

O(n2) ("Quadratic")
n
(amount of data)
Number of
Operations
n2/2 + 5n + 22n2 + 7
n2

Order of Complexity
log2n O(log n)
log10n O(log n)
2(log2n) + 5 O(log n)
The logarithm base
is not written in
big O notation
since all that matters
is that the function
is logarithmic.

O(log n) ("Logarithmic")
n
(amount of data)
Number of
Operations
log2 n
log10 n
2(log2 n) + 5

Comparing Big O Functions
n
(amount of data)
Number of
Operations
O(2n)
O(1)
O(n log n)
O(log n)
O(n2)
O(n)

Searching & Sorting
WORST CASE Order Of Complexity on N data elements
●Linear Search O(N)
●Binary Search O(log N)
●Selection Sort O(N2)
●Bubble Sort O(N2)
●Merge Sort O(N log N)
●Quick Sort O(N2)
●Sort + Binary Search O(N log N) + O(log N)
= O(N log N)

Comparing Algorithms
●Assume an algorithm processes n data values. If each
operation takes 1 s to execute, how many s will it take
to run the algorithm on 100 data values if the algorithm
has the following number of computations?
Number of Computations Execution Time
n 100 s
n • log2 n 665 s
n2 10,000 s
n3 1,000,000 s = 1 sec
2n > 1030 s
n! > 10160 s

Decision Problems
●A specific set of computations are classified as
decision problems.
●An algorithm describes a decision problem if its
output is simply YES or NO, depending on
whether a certain property holds for its input.
●Example:
Given a set of n shapes,
can these shapes be
arranged into a rectangle?

Monkey Puzzle Problem
●Given:
oA set of n square cards whose sides are imprinted with
the upper and lower halves of colored monkeys.
on is a square number, such that n = m2.
oCards cannot be rotated.
●Problem:
oDetermine if an arrangement of the n cards in an
m X m grid exists such that each adjacent pair of cards
display the upper and lower half of a monkey of the
same color.
Source: www.dwheeler.com (2002)(C) 2010 Thomas J Cortina, Carnegie Mellon University

Example
Images from: Simonas Šaltenis, Aalborg University, simas@cs.auc.dk
1
2
3
4
5
6
7
8
9

Analysis
Simple algorithm:
●Pick one card for each cell of m X m grid.
●Verify if each pair of touching edges make a full
monkey of the same color.
●If not, try another arrangement until a solution is
found or all possible arrangements are checked.
●Answer "YES" if a solution is found. Otherwise,
answer "NO" if all arrangements are analyzed and
no solution is found.

Analysis
If there are n = 9 cards (m = 3):
To fill the first cell, we have 9 card choices.
To fill the second cell, we have 8 card
choices left.
To fill the third cell, we have 7 card choices
remaining.
etc.The total number of unique arrangements for n = 9 cards is:
9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362,880

Analysis
For n cards, the number of arrangements to examine is
n! (n factorial)
If we can analyze one arrangement in a microsecond:
n Time to analyze all arrangements
9 362,880 s = 0.36288 s
16 20,922,789,888,000 s
≈ 242 days
25 15,511,210,043,330,985,984,000,000 s
≈ 491,520,585,955 years(C) 2010 Thomas J Cortina, Carnegie Mellon University

Map Coloring
●Given a map of n territories, can the map be
colored using k colors such that no two
adjacent territories are colored with the same
color?
●k=4: Answer is always yes.
●k=2: Only if the map contains no point that is
the junction of an odd number of territories.

Map Coloring

Map Coloring
●Given a map of 48 territories, can the map be colored
using 3 colors such that no two adjacent territories are
colored with the same color?
oPick a color for California (3 choices)
oPick a color for Nevada (3 choices)
o...
●There are 348 = 79,766,443,076,872,509,863,361
possible colorings.
●No one has come up with a better algorithmic solution
that works in general for any map, so far.(C) 2010 Thomas J Cortina, Carnegie Mellon University

Classifications
●Algorithms that are O(nk) for some
fixed k are polynomial-time algorithms.
oO(1), O(log n), O(n), O(n log n), O(n2)
oreasonable, tractable
●All other algorithms are super-polynomial-
time algorithms.
oO(2n), O(n!), O(nn)
ounreasonable, intractable

Traveling Salesperson
●Given: a weighted graph of nodes
representing cities and edges representing
flight paths (weights represent cost)
●Is there a route that takes the salesperson
through every city and back to the starting
city with cost no more than k?
oThe salesperson can visit a city only once (except
for the start and end of the trip).

A
B
D
C
G
E
F
12
6
4
5
9
8
10
7 11
3
7
7
Is there a route with cost at most 52? YES (Route above costs 50.)
Is there a route with cost at most 48? YES? NO?

●If there are n cities, what is the maximum number of
routes that we might need to compute?
●Worst-case: There is a flight available between
every pair of cities.
●Compute cost of every possible route.
oPick a starting city
oPick the next city (n-1 choices remaining)
oPick the next city (n-2 choices remaining)
o...
●Maximum number of routes: (n-1)! = O(n!)
how to
build
a
route

P and NP
●The class P consists of all those decision problems
that can be solved on a deterministic sequential
machine (e.g. a computer) in an amount of time that
is polynomial with respect to the size of the input
●The class NP consists of all those decision problems
whose positive solutions can be verified in
polynomial time given the right information.
from Wikipedia

NP Complete
●The class NPC consists of all those problems in NP
that are least likely to be in P.
oEach of these problems is called NP Complete.
oMonkey puzzle, Traveling salesperson, Hamiltonian path,
map coloring, satisfiability are all in NPC.
●Every problem in NPC can be transformed to another
problem in NPC.
oIf there were some way to solve one of these problems in
polynomial time, we should be able to solve all of these
problems in polynomial time.

Complexity Classes
NP Problems
P Problems
NP Complete
Problems
But does P = NP?
If P ≠ NP, then all decision problems can be broken
down into this classification scheme.
If P = NP, then all three classes are one and the same.
The Clay Mathematics Institute is offering a $1M prize
for the first person to prove P = NP or P ≠ NP.
(http://www.claymath.org/millennium/P_vs_NP/)
We know that P < NP, since
any problem that can be solved
in polynomial time can certainly
have a solution verified in
polynomial time.

It gets worse...
●Tractable Problems
oProblems that have reasonable, polynomial-time
solutions
●Intractable Problems
oProblems that have no reasonable, polynomial-time
solutions
●Noncomputable Problems
oProblems that have no algorithms at all to solve
them

Noncomputability and
Undecidability
●An algorithmic problem that has no algorithm is
called noncomputable.
●If the noncomputable algorithm requires a yes/no
answer, the problem is called undecidable.
●Example:
oGiven any set of any number of different tile designs
(examples shown above), with an infinite number of
each type of tile, can we tile any area with these tiles so
that like colored edges touch?
oThis problem is undecidable!

Tiling Problem
YES
Note the periodicity in the tiling.

Tiling Problem
NO
For this 3 X 3 room, if we try all 39
tiling configurations, no tiling works.

Tiling Problem
●Possible algorithm:
oIf we find a repeating pattern, report YES.
oIf we find a floor we cannot tile, report NO.
●BUT: there are some tilings which have no
repeating pattern!

Tiling Problem
●The only way to know if this set of tiles
can tile every finite-sized floor is to evaluate every
possible floor.
●BUT: there are an infinite number of finite-sized
floors!
oSo we could never answer YES in this case.
●This problem is undecidable.

Another Undecidable Problem:
The Barber Paradox
Suppose there is a town with one
male barber; and that every man in
the town keeps himself clean-shaven:
some shave themselves and some
are shaved by the barber. Only the
barber can shave another man. The
barber shaves all and only those men
who do not shave themselves.
Does the barber shave himself?(C) 2010 Thomas J Cortina, Carnegie Mellon University

Program Termination
●Can we determine if a program will terminate given a
valid input?
●Example:
1. Input x
2. While x is not equal to 1, do the following:
(a) Subtract 2 from x.
●Does this algorithm terminate when x = 15105?
●Does this algorithm terminate when x = 2008?

Program Termination
●Another Example:
1. Input x
2. While x is not equal to 1, do the following:
(a) If x is even, divide x by 2.
(b) Otherwise, Set x to 3x + 1.
●Does this algorithm terminate for x = 15?
●Does this algorithm terminate for x = 105?
●Does this algorithm terminate for any positive x?

The Halting Problem
●Can we write a general program Q that takes
as its input any program P and an input I and determines if
program P will terminate (halt) when run with input I?
oIt will answer YES if P terminates successfully on input I.
oIt will answer NO if P never terminates on input I.
●This computational problem is undecidable!
oNo such general program Q can exist!
oIt doesn’t matter how powerful the computer is.
oIt doesn’t matter how much time we devote to the computation.
oThe proof of this involves contradiction.

start
end
Does this
algorithm end?
yes
no
Contradiction

Contradiction isn't just for
computer scientists...

Computability - Tractable, Intractable and Non-computable Function

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