PROBLEM SOLVING TECHNIQUES PART- 1

💻 Explain Problems and Problem Instances.

💡 Problem:

In computer science, a problem is a general description of a computational task to be solved. 

Imagine the Problem itself is the General Goal or the Concept of a Recipe. It's not about making one specific cake; it's about the idea of making a cake.

It specifies:

1. 📝 Input Format:  "What Ingredients Do I Need?"

Formal Definition: The type and structure of data given.

(The Sorting Problem): 

You must provide a list of numbers. It can be any number of numbers (2, 10, or 1000), but they must be numbers.

2. 🎁 Output Format: "What Should the Final Dish Look Like?"

Formal Definition: The type and structure of the expected result.

(The Sorting Problem): 

The final dish must be the exact same list of numbers, but now arranged in non-decreasing (smallest to largest) order.

3. 🔗 Relation Between Input and Output: "The Core Instruction"

This is the logical requirement that connects what you start with to what you finish with. This is the most important part of the problem.

Formal Definition: The rule that must hold true between the input and the output.

(The Sorting Problem): 

Every number in the output list must be identical to a number in the input list, and for any two numbers next to each other in the output, the first must be less than or equal to the second.

🔢 Problem Instance: 

A problem instance is a specific case of a problem where actual input values are provided. It is the data on which the algorithm will operate to produce the result.

(The Sorting Problem):

• The Problem is the concept of sorting.

• The Problem Instance is the list: [15, 3, 7, 20].

Component	Role in the Computer
Problem	Defines the Goal (e.g., Sort)
Problem Instance	Provides the Data (e.g., *[15, 3, 7, 20]*)
Algorithm	Provides the Method (e.g., Bubble Sort, Merge Sort, etc.)
Output	The Result (e.g., *[3, 7, 15, 20]*)

⚖️ Explain Generalization and Special Cases.

🌎 Generalization: Making a Universal Tool

Generalization is rewriting the recipe book so it can be used for any dessert, not just cakes.

Generalization means broadening a problem so that it can handle a wider variety of inputs or situations.

Original Problem: Add two positive integers (whole numbers). 

Restrictions: 

• Must be addition.

• Must involve only two numbers.

• The numbers must be positive.

• The numbers must be integers (no decimals).

Generalized Problem (Broad, Flexible): Combine any two mathematical objects using a basic arithmetic operation.

Restriction Removed	Original Problem (Specific)	Generalized Problem (Flexible)
Operation	Only Addition (+)	Can be Addition, Subtraction, Multiplication, or Division (+,-, \times, \div)
Number Count	Only two numbers	Can be any number of inputs (add 2+3+4+5)
Value Type	Only Positive numbers	Can handle Positive or Negative numbers
Data Type	Only Integers (whole)	Can handle Integers OR Floating Point (decimals)

🎯 Special Case: Making a Highly Efficient, Specific Tool

A special case is a restricted or simplified version of a general problem.

we deliberately re-introduce restrictions that the generalized problem removed.

Example: We're going to use new, strict rules to cut down the meal's options. This makes the cooking much simpler and faster!

• 👤 The meal must be for one person only.

• 🧀 The meal must use bread and sliced cheese.

• ⏱️ The preparation time must be under 5 minutes.

Feature	General Problem (Meal Planning) 🍝	Special Case (Cheese Sandwich) 🥪
Flexibility	High (Any food, any method)	Low (Fixed ingredients, simple assembly)
Input Restriction	None	Must be bread and cheese
Solution Speed	Slow (hours of cooking)	Fast (seconds of assembly)
Algorithm	Complex (mixing, heating, timing)	Simple (Put cheese between bread)

Explain types of Computational Problems

Computational problems can be classified into several types based on the nature of their required output: 

❓ Decision Problems (Yes/No Questions)

• The goal is to check if a given input satisfies certain conditions.

• Output is Yes/No (True/False).

Type	General Question	Example Problem Instance	Answer
Decision	Is there a path?	Is there a path in this road network from City A to City B?	Yes or No
Decision	Is it divisible?	Is the number 1024 evenly divisible by 4?	Yes or No
Decision	Is it satisfiable?	Is there an assignment of True/False values that makes this Boolean formula true?	Yes or No

🔍 Search Problems (Find a Solution)

Imagine a computer's job is not just to answer yes/no (like a Decision Problem) but to actively find something for you.

Example: The Locked Treasure Chest 🔑

Imagine you have a big treasure chest, and the computer's task is to find the correct key to open it.

1. The Input (The Data)

• Input: A collection of 100 different keys 🗝️ (The input data set).

• The Rule: Only one key will open the chest.

2. The Task (The Search Problem)

• General Search Problem: Find the key in the collection that successfully opens the treasure chest.

3. The Algorithm (The Method)

• Step 1: Pick up the first key.

• Step 2: Try to insert and turn the key in the lock.

• Step 3: If it opens, STOP and output the key.

• Step 4: If it doesn't, move to the next key.

4. The Output (The Solution)

Problem Type	Question Asked	Example Output
Decision Problem ❓	Does the correct key exist in the pile?	Yes or No
Search Problem 🔍	Which key is the correct one?	Key #42 (The specific key object)

🥇 Optimization Problems (Find the Best Solution)

Output is the best solution among all possible solutions, based on some criteria.

Example: Planning a Road Trip 🚗

Imagine you are planning a trip to visit five different cities, and you have to decide the order in which to visit them.

1. The General Search Problem (Finding a Solution)

The first step is a Search Problem: simply find any route that visits all five cities.

• Possible Route 1: City A ⇾ B ⇾ C ⇾ D ⇾ E (Total Time: 12 hours)

• Possible Route 2: City A ⇾ C ⇾ E ⇾ D ⇾ B (Total Time: 14 hours)

The search problem is solved as soon as you find one valid route.

2. The Optimization Problem (Finding the Best Solution)

• Objective: You want to minimize the total driving time.

The computer must now look at all possible routes (not just one or two) and calculate the total driving time for each one.

Solution (Route)	The Value (Total Driving Time)	Optimization Goal
Route 1	12 hours	Minimize Time
Route 2	14 hours	Minimize Time
Optimal Route	9 hours	This is the answer!

The output of the Optimization Problem is the specific route that has the lowest total time (9 hours).

🧮 Counting Problems (Count the Number of Solutions)

Output is the number of possible solutions that satisfy the given constraints.

Type	General Question	Example Problem Instance	Answer
Counting	Count the paths.	How many unique routes exist from City A to City B?	(3 routes)
Counting	Count the colorings.	How many distinct ways can this map be colored using only four colors?	(48 ways)

⚙️ Function Problems: The Recipe Conversion Tool

Think of a Function Problem as a machine designed to perform a predictable conversion—it always translates one thing into another based  on a simple, consistent rule.

Example: Converting Units 📏

Imagine a simple app or program whose only job is to convert temperature from Celsius to Fahrenheit.

1. The Input (x): What We Know

• Input: A single temperature value in Celsius (e.g., 25^o C).

• The Rule: The input must be a number representing temperature.

2. The Task (The Function): The Conversion Rule

The computer applies the fixed, known formula for temperature conversion. This formula is the "function" it executes:

F = C × 9 5 + 32

• Output: The single, precise value in Fahrenheit (e.g., 77^o F).

• The Result: The computer has successfully calculated and returned the required, definitive answer.

Problem Type	The Core Action	Output Focus
Function ⚙️	Calculate and convert the unit.	The calculated value in the new unit.