Dynamic programming

Dynamic Programming and Applications
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Dynamic Programming
2
Dynamic Programming is a general algorithm design technique
for solving problems defined by recurrences with overlapping
subproblems
• Invented by American mathematician Richard Bellman in the
1950s to solve optimization problems and later assimilated by CS
• “Programming” here means “planning”
• Main idea:
- set up a recurrence relating a solution to a larger instance to
solutions of some smaller instances
- solve smaller instances once
- record solutions in a table
- extract solution to the initial instance from that table

Divide-and-conquer
• Divide-and-conquer method for algorithm design:
• Divide: If the input size is too large to deal with in a
straightforward manner, divide the problem into two or
more disjoint subproblems
• Conquer: conquer recursively to solve the subproblems
• Combine: Take the solutions to the subproblems and
“merge” these solutions into a solution for the original
problem

Dynamic programming
• Dynamic programming is a way of improving on inefficient divide-
and-conquer algorithms.
• By “inefficient”, we mean that the same recursive call is made
over and over.
• If same subproblem is solved several times, we can use table to
store result of a subproblem the first time it is computed and thus
never have to recompute it again.
• Dynamic programming is applicable when the subproblems are
dependent, that is, when subproblems share subsubproblems.
• “Programming” refers to a tabular method

Difference between DP and Divide-
and-Conquer
• Using Divide-and-Conquer to solve these
problems is inefficient because the same
common subproblems have to be solved many
times.
• DP will solve each of them once and their
answers are stored in a table for future use.

Dynamic Programming vs. Recursion
and Divide & Conquer

Elements of Dynamic Programming
(DP)
DP is used to solve problems with the following characteristics:
• Simple subproblems
– We should be able to break the original problem to smaller
subproblems that have the same structure
• Optimal substructure of the problems
– The optimal solution to the problem contains within optimal
solutions to its subproblems.
• Overlapping sub-problems
– there exist some places where we solve the same subproblem more
than once.

Steps to Designing a
Dynamic Programming Algorithm
1. Characterize optimal substructure
2. Recursively define the value of an optimal
solution
3. Compute the value bottom up
4. (if needed) Construct an optimal solution

Principle of Optimality
• The dynamic Programming works on a principle
of optimality.
• Principle of optimality states that in an optimal
sequence of decisions or choices, each sub
sequences must also be optimal.

Example 1: Fibonacci numbers
11
• Recall definition of Fibonacci numbers:
F(n) = F(n-1) + F(n-2)
F(0) = 0
F(1) = 1
• Computing the nth Fibonacci number recursively (top-down):
F(n)
F(n-1) + F(n-2)
F(n-2) + F(n-3) F(n-3) + F(n-4)
...

Fibonacci Numbers
• Fn= Fn-1+ Fn-2 n ≥ 2
• F0 =0, F1 =1
• 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
• Straightforward recursive procedure is slow!
• Let’s draw the recursion tree

Fibonacci Numbers
• We can calculate Fn in linear time by remembering
solutions to the solved subproblems – dynamic
programming
• Compute solution in a bottom-up fashion
• In this case, only two values need to be
remembered at any time

Example Applications of Dynamic
Programming

Knapsack Problem
1
Value
18
22
28
1
Weight
5
6
6 2
7
Item
1
3
4
5
2W = 11

Shortest path problems
A traveler wishes to minimize the length of a journey
from town A to J.

Shortest path problems(Cont..)
The length of the route A-B-F-I-J: 2+4+3+4=13.
Can we find shorter route?

Matrix Chain Multiplication
• Given : a chain of matrices {A1,A2,…,An}.
• Once all pairs of matrices are parenthesized, they can
be multiplied by using the standard algorithm as a sub-
routine.
• A product of matrices is fully parenthesized if it is either
a single matrix or the product of two fully parenthesized
matrix products, surrounded by parentheses. [Note: since
matrix multiplication is associative, all parenthesizations yield the
same product.]

Matrix Chain Multiplication cont.
• For example, if the chain of matrices is {A, B, C,
D}, the product A, B, C, D can be fully
parenthesized in 5 distinct ways:
(A ( B ( C D ))),
(A (( B C ) D )),
((A B ) ( C D )),
((A ( B C )) D),
((( A B ) C ) D ).
• The way the chain is parenthesized can have a
dramatic impact on the cost of evaluating the
product.

Matrix Chain Multiplication Optimal
Parenthesization
• Example: A[30][35], B[35][15], C[15][5]
minimum of A*B*C
A*(B*C) = 30*35*5 + 35*15*5 = 7,585
(A*B)*C = 30*35*15 + 30*15*5 = 18,000
• How to optimize:
– Brute force – look at every possible way to
parenthesize : Ω(4n/n3/2)
– Dynamic programming – time complexity of Ω(n3) and
space complexity of Θ(n2).

Matrix Chain Multiplication Structure of
Optimal Parenthesization
• For n matrices, let Ai..j be the result of AiAi+1….Aj
• An optimal parenthesization of AiAi+1…An splits
the product between Ak and Ak+1 where 1  k <
n.
• Example, k = 4 (A1A2A3A4)(A5A6)
Total cost of A1..6 = cost of A1..4 plus total
cost of multiplying these two matrices
together.

Matrix Chain Multiplication
Overlapping Sub-Problems
• Overlapping sub-problems helps in reducing the
running time considerably.
– Create a table M of minimum Costs
– Create a table S that records index k for each optimal sub-
problem
– Fill table M in a manner that corresponds to solving the
parenthesization problem on matrix chains of increasing
length.
– Compute cost for chains of length 1 (this is 0)
– Compute costs for chains of length 2
A1..2, A2..3, A3..4, …An-1…n
– Compute cost for chain of length n
A1..nEach level relies on smaller sub-strings

Answer Q1
The length of the route A-B-F-I-J:
2+4+3+4=13.

Q2. Who invited dynamic programming?
• A. Richard Ernest Bellman
• B. Edsger Dijkstra
• C. Joseph Kruskal
• D. David A. Huffman

Answer Q2
• A. Richard Ernest Bellman
• B. Edsger Dijkstra
• C. Joseph Kruskal
• D. David A. Huffman

Q3. What is meaning of programming in DP?
A. Plannig
B. Computer Programming
C. Programming languages
D. Curriculum

Dynamic programming

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