Dynamic Programming Intro in Algorithm Design

Nov 6, 20230 likes43 views

This document provides an introduction to dynamic programming. It discusses how dynamic programming can be used to solve complex problems by breaking them down into smaller subproblems. It explains the two fundamental principles of dynamic programming as optimal substructure, meaning optimal solutions can be constructed from optimal solutions to subproblems, and overlapping subproblems, where the same subproblems are solved repeatedly. It describes how dynamic programming handles overlapping subproblems through memoization and tabulation and provides examples. Potential limitations and advantages of dynamic programming are also outlined.

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EXAMPLE(Fibonacci Sequence/Series)
f(n)
{
if(n==0)
return 0;
if(n==1)
return 1;
if(n>1)
return (f(n-1)+f(n-2);
}
n : 1 2 3 4 5 6 7 8 9 1 0
f(n): 0 1 1 2 3 5 8 13 21 34 55
F(5)
F(3)
F(4)
F(3)
F(1)
F(2)
F(2)
F(0)
F(1)
F(1)
F(2)
F(0)
F(1)
F(0)
F(1)

• It's a powerful algorithm
technique.
• •Used for Optimization
problems
• Solve complex problems by
breaking them into smaller
subproblems.
• Optimal substructure and
overlapping subproblems.
INTRODUCTION

Two Fundamental
Principles
• Optimal Substructure
• Overlapping Subproblems
FUNDAMENTALS

Optimal Substructure means we
can break a big problem into smaller
subproblems, and if we solve those
smaller subproblems optimally, we
can combine their solutions to solve
the big problem optimally.
OPTIMAL SUBSTRUCTURE:
EXAMPLE: JIGSAW PUZZLE

Overlapping Subproblems refers to
the situation where the same smaller
problems are encountered and solved
multiple times when solving a larger
problem.It's about reusing solutions to
avoid redundant work.
OVERLAPPING SUBPROBLEMS

Memoization(Top-Down)
A lookup table is maintained and
checked before computation of any
state.Recursion is involved.
WAYS TO HANDLE OVERLAPPING SUBPROBLEMS
Tabulation(Bottom-Up)
Solution is built from base.It is an
iterative process.

Dynamic programming, specifically memoization or
tabulation, helps here by storing the results of these
subproblems the first time you solve them.When you
need the same result again, you simply look it up,
avoiding redundant calculations.This significantly
improves efficiency, especially for larger problems.

• Space complexity
• Complexity of identifying subproblems
• Inability to handle problems without
optimal substructure:
LIMITATIONS
LIMITATIONS

• Efficiency: It can improve the efficiency of
algorithms by avoiding redundant computations.
• Optimality: The algorithms are guaranteed to
find optimal solutions to problems with optimal
substructure.
• Generality: Dynamic programming can be
applied to a wide variety of problems.
ADVANTAGES
ADVANTAGES

Dynamic programming is a method for solving complex problems by breaking them down into simpler overlapping subproblems with optimal substructure. It is often preferred over greedy methods for optimization problems due to its efficient use of previously solved subproblems through approaches like memoization (top-down) and tabulation (bottom-up). While it can reduce time complexity and avoid redundant calculations, dynamic programming may require significant memory and necessitates the identification of suitable subproblems.

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Dynamic Programming Intro in Algorithm Design

1. Dynamic Programming Ahsan Raza

2. EXAMPLE(Fibonacci Sequence/Series) f(n) { if(n==0) return 0; if(n==1) return 1; if(n>1) return (f(n-1)+f(n-2); } n : 1 2 3 4 5 6 7 8 9 1 0 f(n): 0 1 1 2 3 5 8 13 21 34 55 F(5) F(3) F(4) F(3) F(1) F(2) F(2) F(0) F(1) F(1) F(2) F(0) F(1) F(0) F(1)

3. • It's a powerful algorithm technique. • •Used for Optimization problems • Solve complex problems by breaking them into smaller subproblems. • Optimal substructure and overlapping subproblems. INTRODUCTION

4. Two Fundamental Principles • Optimal Substructure • Overlapping Subproblems FUNDAMENTALS

5. Optimal Substructure means we can break a big problem into smaller subproblems, and if we solve those smaller subproblems optimally, we can combine their solutions to solve the big problem optimally. OPTIMAL SUBSTRUCTURE: EXAMPLE: JIGSAW PUZZLE

6. Overlapping Subproblems refers to the situation where the same smaller problems are encountered and solved multiple times when solving a larger problem.It's about reusing solutions to avoid redundant work. OVERLAPPING SUBPROBLEMS

7. Memoization(Top-Down) A lookup table is maintained and checked before computation of any state.Recursion is involved. WAYS TO HANDLE OVERLAPPING SUBPROBLEMS Tabulation(Bottom-Up) Solution is built from base.It is an iterative process.

8. Dynamic programming, specifically memoization or tabulation, helps here by storing the results of these subproblems the first time you solve them.When you need the same result again, you simply look it up, avoiding redundant calculations.This significantly improves efficiency, especially for larger problems.

9. • Space complexity • Complexity of identifying subproblems • Inability to handle problems without optimal substructure: LIMITATIONS LIMITATIONS

10. • Efficiency: It can improve the efficiency of algorithms by avoiding redundant computations. • Optimality: The algorithms are guaranteed to find optimal solutions to problems with optimal substructure. • Generality: Dynamic programming can be applied to a wide variety of problems. ADVANTAGES ADVANTAGES

11. Thank you.

Dynamic Programming Intro in Algorithm Design

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