Computational Complexity,
Physical Mapping III + Perl
CIS 667 March 4, 2004
Computational Complexity

An
Overview
•
We are primarily interested in efficient
algorithms
Efficient means that the running time of the
algorithm is bounded by some polynomial
function
p(n)
The size of the problem is measured by
n
We use
big

oh notation
, e.g.
O(n
2
)
, in which lower
order terms are ignored
Thus for small problem sizes, an
O(n
2
)
algorithm
may run slower than an
O(n)
one
Computational Complexity

An
Overview
•
This means that we are talking about
asymptotic
behavior
•
An inefficient algorithm is one whose asymptotic
efficiency is exponential

e.g.
O(2
n
)
•
Problems for which efficient algorithms exist
belong to a class P
•
Problems for which no efficient algorithms are
known to exist belong to class NP
NP

complete Problems
•
An important subset of these problems is called
NP

complete
The solutions to problems in NP, once found, can be
checked in polynomial time
NP includes the class P as a subset
Any NP

complete problem can be transformed in
polynomial time to an instance of any other NP

complete problem
So all NP

complete problems are equivalent under
polynomial transformation
NP

complete Problems
•
So, if a polynomial time algorithm is found
for one NP

complete problem, there are
polynomial time algorithms for
all
NP

complete problems
If so, then P=NP
Most researchers believe that P
NP
The model of computation that is used in
defining NP

complete problems is the
Nondeterministic Turing Machine
NP

hard Problems
•
Classes P and NP include only
decision
problems

the answer is yes or no
•
An
NP

hard
problem is one which is at
least as hard as NP

complete problems
If an NP

hard problem can be solved in
polynomial time, then so can all NP

complete
problems
NP

hard problem is not necessarily a decision
problem
NP

hard Problems
•
NP

complete
NP

hard
•
Example: does there exist a solution to the
Traveling Salesman problem is NP

hard
and NP

complete.
Find a solution to the Traveling Salesman is
NP

hard, but not NP

complete (not decision
form)
But if we have a polynomial solution for the
2nd, we can use it to solve the 1st (and hence
all NP

complete problems)
NP

completeness
•
Initially, several hard problems were shown to
solvable in polynomial time on a
nondeterministic TM
Polynomial time reductions between the problems
were also shown
Nowadays, to show a problem is NP

complete
Verify the problem is in NP (solution can be verified in
polynomial time)
Show a polynomial time reduction of
any
NP

complete to your
problem
NP

completeness
•
So when faced with an NP

complete or NP

hard
problem

what to do?
See if a meaningful restriction of the problem can be
solved in polynomial time
See if the size of the problem in practice is always
small
Devise a polynomial time approximation algorithm

guaranteed to find a near optimal solution
Devise heuristics
Algorithmic Implications
•
We are trying to solve a real

life problem
The models we use may give us many
solutions, but we want to find the one solution
which corresponds to the real ordering of the
clones in the target DNA
Use the algorithmic results in an iterative
fashion with the experimental biologist
Algorithmic Implications
•
A mapping algorithm should
Work better with more data, assuming a
constant error rate
Give a solution which makes it clear how it
was obtained and tell which parts of the
solution are good and which bad
Give all candidate solutions
An algorithm for C1P
•
This algorithm determines whether an
n
m
matrix has the C1P for rows
Assume
All rows different
No row is all zero
Let
S
i
be the set of columns of row
i
with value
1 then
i
and
j
we can have
S
i
S
j
=
.
S
i
S
j
or
S
j
S
i
S
i
S
j
and neither of them is a subset of the
other
An algorithm for C1P
•
In the first case, we don’t need to consider
the two rows together, so we separate
them into two
components
Deal with them separately
•
For non

empty intersection
Suppose there is a row that is either a subset
or has empty intersection with every row in
the component

move it out of the component
An algorithm for C1P
•
To see if two rows belong to the same
component
Build a graph
G
c
using
M
Each vertex of
G
c
will be a row from
M
•
There will be an undirected edge from
i
to
j
if
S
i
S
j
and neither of them is a subset of the other
So the components we want are the connected
components of
G
c
Basic Algorithm
•
The algorithm will have the following
phases
Separate rows into components according to
above rules
Permute the columns of each component to
achieve C1P for component
Join components together
Example Matrix
c
1
c
2
c
3
c
4
c
5
c
6
c
7
c
8
c
9
l
1
1
1
0
1
1
0
1
0
1
l
2
0
1
1
1
1
1
1
1
1
l
3
0
1
0
1
1
0
1
0
1
l
4
0
0
1
0
0
0
0
1
0
l
5
0
0
1
0
0
1
0
0
0
l
6
0
0
0
1
0
0
1
0
0
l
7
0
1
0
0
0
0
1
0
0
l
8
0
0
0
1
1
0
0
0
1
Example Graph
l
1
l
2
l
3
l
5
l
4
l
7
l
6
l
8
a
b
g
d
Placing Rows in a Component
c
1
c
2
c
3
c
4
c
5
c
6
c
7
c
8
l
1
0
1
0
0
0
0
1
1
l
2
0
1
0
0
1
0
1
0
l
3
1
0
0
1
0
0
1
1
How can the first row (by itself) be arranged? (Keep track of all possibilities)
l
1
… 0 1 1
1 0 …
{2, 7, 8} {2, 7, 8} {2, 7, 8}
Now add the second row

it can go to right or left of first
l
1
… 0 0 1 1
1 0 …
l
2
… 0 1 1
1 0 0…
{5} {2, 7} {2, 7} {8}
Placing Rows in a Component
•
How do we place the third row?
In the graph, there are edges for both rows already
placed. Let’s place the third with respect to the
second
Does it go to the right or to the left?
If l
1
l
3
<min(l
1
l
2
, l
2
l
3
)

same direction second
w.r.t. first, else opposite direction
In our case, we have to place in the opposite (right
direction) as shown on the next slide
Placing Rows in a Component
l
1
… 0 0 1 1
1 0 0 0…
l
2
… 0 1 1
1 0 0 0 0…
{5} {2} {7} {8} {1, 4} {1, 4}
l
3
… 0 0 0
1 1 1 1 0…
Placing Rows in a Component
•
All of the other rows in the component are
placed in the same way, using two
previously place rows:
One which has an edge to the row to be
placed in the graph
Second has an edge to the previous row in
the graph
Joining Components Together
•
For the next part of the solution, we use a
graph
G
M
which tells us how the
components fit together
Each component of the original matrix will be
a vertex in
G
M
A directed edge is added between
a
and
b
if
the sets
S
i
for all i in
b
are contained in at least
one set
S
j
of component
a
Example Graph
a
b
g
d
Joining Components Together
•
We process components not contained in
any other component first
So process the components in the topological
order of the graph
•
We may come up with multiple solutions if
one or more columns is not constrained to
one value
•
The algorithm is polynomial
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