What Is Dynamic Programming? | Dynamic Programming Explained | Programming For Beginners|Simplilearn

What’s in It For You?
Real-Life Example of Dynamic
Programming
Introduction to Dynamic
Programming
How Does Dynamic Programming Work?
Dynamic Programming Interpretation of
Fibonacci Series Program

Real-Life Example of Dynamic
Programming

This is Rachael. She loves solving complex puzzles.

While searching through her puzzle book, she came across a
tic-tac-toe puzzle.

Rachael began playing this game with her friend Alex, who was
already familiar with it.

While playing with Alex, Rachael kept losing the game. As a
result, she got frustrated!

After losing a few games, Rachael began to recall the outcomes of each of her
moves, which led her towards failure. Now she began playing Tic-tac-toe
intelligently, keeping those moves in mind.

Rachael used her memory to recall the results of her prior decisions. As a result of her
commitment to learn from her past, she went on a winning streak against Alex.

The notion behind the dynamic programming paradigm is that those who do not remember the
past are condemned to repeat it.

If we can handle and remember smaller problems, their learnings can be memorized to
solve the bigger problems. This general principle is considered as a building block of
dynamic programming.

Introduction to Dynamic
Programming

 Dynamic programming is an algorithmic paradigm for
solving a given complex problem by breaking it down
into subproblems and memorizing the outcomes of
those subproblems to prevent repeating
computations.
 Dynamic programming can only be applied to the given
problem if it follows the properties of dynamic
programming.
What Is Dynamic Programming?

Properties of Dynamic Programming
A problem is said to have an optimal substructure if we can formulate a
recurrence relation for it.
1. Optimal Substructure
Consider the coin change problem, in which you have to construct coin combinations with
the least potential number of coins.
50$ coin 20$ coin 10$ coin 5$ coin

Properties of Dynamic Programming
A problem is said to have an overlapping subproblem if the subproblems reoccur
when implementing a solution to the larger problem.
2. Overlapping Subproblem
Overlapping Subproblem can be understood by developing recurrence relationships.
For example, Fibonacci Series.
𝒇𝒊𝒃 𝒏 = 𝒇𝒊𝒃 𝒏 − 𝟏 + 𝒇𝒊𝒃(𝒏 − 𝟐)

What Should We Cover Next?
What Dynamic Programming
problems would you like us to
cover in our upcoming videos?

Dynamic Programming Interpretation of
Fibonacci Series Program

Fibonacci Series
Fibonacci series is the set of numbers which appear in nature all the time. Each
number in this series is equal to the sum of previous numbers before it.
𝒇𝒊𝒃(𝒏) = 𝒇𝒊𝒃(𝒏 − 𝟏) + 𝒇𝒊𝒃(𝒏 − 𝟐)
Fibonacci Series: 0, 1, 1, 2, 3, 5, 8, 13, .....
When n <=1
Fib(0) = 0
Fib(1) = 1
Otherwise

Fibonacci Series
Fibonacci series is the set of numbers which appear in nature all the time. Each
number in this series is equal to the sum of previous numbers before it.
Fibonacci Series: 0, 1, 1, 2, 3, 5, 8, 13, .....
n 0 1 2 3 4 5
Fib(n) 0 1 1 2 3 5

Optimal Substructure: Fibonacci Series
If we can establish a recurrence relation for a problem, we say it has
an optimal substructure.
int Fib(int z)
{
if(z<=1){
return z;
}
else{
return Fib(n-1) +
Fib(n-2);
}
}
𝒇𝒊𝒃 𝒏 = 𝒇𝒊𝒃 𝒏 − 𝟏 + 𝒇𝒊𝒃(𝒏 − 𝟐)
Recurring Relation

Overlapping Subproblem: Fibonacci Series
If the subproblems recur while implementing a solution to the bigger
problem, the problem is said to have an overlapping subproblem.
Let’s say we want to calculate Fibonacci numbers for n = 5.

Fib(5)
Fib(4)
Fib(3)
Fib(2)
Fib(1) Fib(0)
Fib(1)
Fib(2)
Fib(1) Fib(0)
Fib(3)
Fib(2)
Fib(1) Fib(0)
Fib(1)

Fib(5)
Fib(4)
Fib(3)
Fib(2)
Fib(1) Fib(0)
Fib(1)
Fib(2)
Fib(1) Fib(0)
Fib(3)
Fib(2)
Fib(1) Fib(0)
Fib(1)
The recurring problem is considered to have
Overlapping Subproblems if it solves the same
subproblem again and again.

Time Complexity: Recursion
Fib(5)
Fib(4)
Fib(3)
Fib(2)
Fib(1) Fib(0)
Fib(1)
Fib(2)
Fib(1) Fib(0)
Fib(3)
Fib(2)
Fib(1) Fib(0)
Fib(1)
T(n) = T(n-1) + T(n-2) + O(1)
T(n<=1) = O(1)
Overall, T(n) = O(𝟐𝒏)
Depth = 5
Depth = 4

Time Complexity: Recursion
0
2
4
6
8
10
12
14
16
18
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Value of n
Time
O(n) = 𝟐𝒏

How Does Dynamic
Programming Work?

Solution: Utilize Memory
Remember the result for each Sub-Problem.
Solving a Subproblem Memory Area

Solution: Utilize Memory
Remember the result for each Sub-Problem.
Memory Area Recurred Subproblem

Dynamic programming stores the results of subproblems in memory and
recalls it whenever the recurrence of calculated subproblem occurs.
Two methods of storing the results in memory.
 Memorization: In this method, we store the results in memory whenever
we solve a particular subproblem for first time.
 Tabulation: In this method, we precompute the solutions in a linear
fashion and store it in a tabular format.

Ways to Handle
Overlapping
Subproblems
Memorization Tabulation
 Also called as Top-Down
Approach
 A Lookup table is maintained and
checked before computation of
any state
 Recursion is involved
 Also called as Bottom-Up
Approach
 In this method, the solution is built
from the base or bottom-most
state
 This process is iterative

Overlapping handling
for Fibonacci Series
Program
Memorization Tabulation
int fib(int n)
{
if(n<=1){
return n;
}
if(fib(n) != -1)
return fib(n);
int res = fib(n-1) + fib(n-2);
fib(n) = res;
return res;
}
int fib(int n)
{
int t[n+1];
int i;
t[0] = 0;
t[1] = 1;
for(i=2; i <= n; i++)
{
t(i) = t(i-1) + t(i-2);
}
return t(n);
}
Memorization
Tabulation

When to Use Dynamic Programming?
1. When we need an exhaustive solution, we can use Dynamic
programming to address minimization and maximization problems.
2. Permutation problems: find the number of ways problems can be
solved using DP.

What Is Dynamic Programming? | Dynamic Programming Explained | Programming For Beginners|Simplilearn

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