Skiena algorithm 2007 lecture16 introduction to dynamic programming

Lecture 16:
Introduction to Dynamic Programming
Steven Skiena

Department of Computer Science
State University of New York
Stony Brook, NY 11794–4400

http://www.cs.sunysb.edu/∼skiena

Problem of the Day
Multisets are allowed to have repeated elements. A multiset
of n items may thus have fewer than n! distinct permutations.
For example, {1, 1, 2, 2} has only six different permutations:
{1, 1, 2, 2}, {1, 2, 1, 2}, {1, 2, 2, 1}, {2, 1, 1, 2}, {2, 1, 2, 1},
and {2, 2, 1, 1}. Design and implement an efﬁcient algorithm
for constructing all permutations of a multiset.

Dynamic Programming
Dynamic programming is a very powerful, general tool for
solving optimization problems on left-right-ordered items
such as character strings.
Once understood it is relatively easy to apply, it looks like
magic until you have seen enough examples.
Floyd’s all-pairs shortest-path algorithm was an example of
dynamic programming.

Greedy vs. Exhaustive Search
Greedy algorithms focus on making the best local choice at
each decision point. In the absence of a correctness proof
such greedy algorithms are very likely to fail.
Dynamic programming gives us a way to design custom
algorithms which systematically search all possibilities (thus
guaranteeing correctness) while storing results to avoid
recomputing (thus providing efﬁciency).

Recurrence Relations
A recurrence relation is an equation which is deﬁned in terms
of itself. They are useful because many natural functions are
easily expressed as recurrences:
Polynomials: an = an−1 + 1, a1 = 1 −→ an = n
Exponentials: an = 2an−1 , a1 = 2 −→ an = 2n
Weird: an = nan−1 , a1 = 1 −→ an = n!
Computer programs can easily evaluate the value of a given
recurrence even without the existence of a nice closed form.

Computing Fibonacci Numbers

Fn = Fn−1 + Fn−2, F0 = 0, F1 = 1
Implementing this as a recursive procedure is easy, but slow
because we keep calculating the same value over and over.
F(6)=13

F(5) F(4)

F(4) F(3) F(3) F(2)

F(2) F(1) F(2) F(1) F(1) F(0)
F(3) F(2)

F(2) F(1) F(1) F(1) F(0)
F(0) F(1) F(0)

F(1) F(0)

How Slow?
√
Fn+1/Fn ≈ φ = (1 + 5)/2 ≈ 1.61803
Thus Fn ≈ 1.6n .
Since our recursion tree has 0 and 1 as leaves, computing Fn
requires ≈ 1.6n calls!

What about Dynamic Programming?
We can calculate Fn in linear time by storing small values:
F0 = 0
F1 = 1
For i = 1 to n
Fi = Fi−1 + Fi−2
Moral: we traded space for time.

Why I Love Dynamic Programming
Dynamic programming is a technique for efﬁciently comput-
ing recurrences by storing partial results.
Once you understand dynamic programming, it is usually
easier to reinvent certain algorithms than try to look them up!
I have found dynamic programming to be one of the most
useful algorithmic techniques in practice:
• Morphing in computer graphics.
• Data compression for high density bar codes.
• Designing genes to avoid or contain speciﬁed patterns.

Avoiding Recomputation by Storing Partial
Results
The trick to dynamic program is to see that the naive recursive
algorithm repeatedly computes the same subproblems over
and over and over again. If so, storing the answers to them
in a table instead of recomputing can lead to an efﬁcient
algorithm.
Thus we must ﬁrst hunt for a correct recursive algorithm –
later we can worry about speeding it up by using a results
matrix.

Binomial Coefficients
The most important class of counting numbers are the
binomial coefficients, where ( ) counts the number of ways
n
k

to choose k things out of n possibilities.
• Committees – How many ways are there to form a k-
member committee from n people? By definition, ( ). n
k

• Paths Across a Grid – How many ways are there to travel
from the upper-left corner of an n × m grid to the lower-
right corner by walking only down and to the right? Every
path must consist of n + m steps, n downward and m to
the right, so there are ( ) such sets/paths.
n+m
n

Computing Binomial Coefficients
Since ( ) = n!/((n − k)!k!), in principle you can compute
n
k

them straight from factorials.
However, intermediate calculations can easily cause arith-
metic overflow even when the final coefficient fits comfort-
ably within an integer.

Pascal’s Triangle
No doubt you played with this arrangement of numbers in
high school. Each number is the sum of the two numbers
directly above it:
1
11
121
1331
14641
1 5 10 10 5 1

Pascal’s Recurrence
A more stable way to compute binomial coefﬁcients is using
the recurrence relation implicit in the construction of Pascal’s
triangle, namely, that
()=( )+( )
n
k
n−1
k−1
n−1
k

It works because the nth element either appears or does not
appear in one of the ( ) subsets of k elements.
n
k

Basis Case
No recurrence is complete without basis cases.
How many ways are there to choose 0 things from a set?
Exactly one, the empty set.
The right term of the sum drives us up to ( ). How many ways
k
k

are there to choose k things from a k-element set? Exactly
one, the complete set.

Binomial Coefficients Implementation

long binomial coefficient(n,m)
int n,m; (* compute n choose m *)
{
int i,j; (* counters *)
long bc[MAXN][MAXN]; (* table of binomial coefficients *)

for (i=0; i<=n; i++) bc[i][0] = 1;

for (j=0; j<=n; j++) bc[j][j] = 1;

for (i=1; i<=n; i++)
for (j=1; j<i; j++)
bc[i][j] = bc[i-1][j-1] + bc[i-1][j];

return( bc[n][m] );
}

Three Steps to Dynamic Programming

1. Formulate the answer as a recurrence relation or recursive
algorithm.
2. Show that the number of different instances of your
recurrence is bounded by a polynomial.
3. Specify an order of evaluation for the recurrence so you
always have what you need.

Skiena algorithm 2007 lecture16 introduction to dynamic programming

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