Menerapkan Konsep Rekursi - Berpikir Komputasional - Rekursi

Sinau Maning

22 Aug 202317:11

Summary

TLDRThis video focuses on computational thinking and recursion strategies in algorithmic programming. It presents two problems: the first involves determining how many ways tiles can be arranged on a 2xN floor using recursion. For N = 8, there are 34 ways, following a Fibonacci-like sequence. The second problem discusses stacking pancakes of varying sizes and moving them between plates according to specific rules. The recursive solution finds the minimal steps required to move six pancakes, resulting in 63 moves. The video demonstrates these concepts through step-by-step examples.

Takeaways

🧱 The problem involves placing tiles on a 2xN floor using 1x2 tiles, which can be placed either horizontally or vertically.
🔢 For N=8, the number of ways to place the tiles can be calculated using a recursive relation.
🔄 The recursive formula is based on the two previous terms: K(n) = K(n-1) + K(n-2).
📊 Example calculations: K(1) = 1, K(2) = 2, K(3) = 3, K(4) = 5, K(5) = 8, K(6) = 13, K(7) = 21, K(8) = 34.
🎯 The final answer for N=8 is 34 ways to arrange the tiles.
🥞 The second problem involves moving a stack of pancakes from one plate to another, following a size-order rule.
💡 The pancake problem is similar to the Tower of Hanoi problem, requiring a helper plate to transfer pancakes while maintaining the size-order.
⏳ The minimum steps to transfer pancakes follow a recursive pattern: P(n) = 2 * P(n-1) + 1.
📈 For example, 1 pancake takes 1 step, 2 pancakes take 3 steps, 3 pancakes take 7 steps, and 6 pancakes require 63 steps.
🚀 Both problems showcase the power of recursive thinking in solving algorithmic challenges.

Q & A

What is the first problem described in the script?
-The first problem is about tiling a 2xN floor with N tiles, where each tile is of size 1x2. The tiles can be placed either horizontally or vertically, and the task is to determine how many different ways the tiles can be arranged on the floor for N=8.
How does the script explain the solution to the tile placement problem?
-The script explains that for N=1, there is 1 way to place the tile (vertically), and for N=2, there are 2 ways (either both vertical or both horizontal). For higher values of N, the number of ways is calculated using a recursive relation: K(n) = K(n-1) + K(n-2), where K(n) represents the number of ways to tile the floor. This relation is similar to the Fibonacci sequence.
What is the recursive formula provided to solve the tile placement problem?
-The recursive formula is K(n) = K(n-1) + K(n-2), with base cases K(1) = 1 and K(2) = 2. This allows for calculating the number of ways to place tiles for any value of N.
How many ways can the tiles be arranged for N=8?
-For N=8, the number of ways to arrange the tiles is 34, as calculated by the recursive formula using the previously computed values for smaller N.
What is the second problem described in the script?
-The second problem is about moving a stack of pancakes from one plate to another. The largest pancake must always be at the bottom, and only one pancake can be moved at a time. Budi, the person in the problem, wants to know the minimum number of steps required to move 6 pancakes while following these rules.
What is the minimum number of steps required to move 6 pancakes?
-The minimum number of steps required to move 6 pancakes is 63, as derived from a recursive relation similar to the Tower of Hanoi problem.
How does the recursive relation for the pancake problem work?
-The recursive relation is P(n) = 2 * P(n-1) + 1, where P(n) is the minimum number of steps required to move n pancakes. The base case is P(1) = 1.
How many steps are required to move 3 pancakes?
-To move 3 pancakes, the minimum number of steps required is 7, following the recursive process described in the script.
What is the general strategy for solving the pancake problem?
-The strategy is to move the smallest pancake first, then progressively move larger pancakes while using an auxiliary plate to temporarily hold pancakes during the process. The recursive solution builds on smaller subproblems, similar to the Tower of Hanoi.
What are the key concepts covered in the script related to recursion?
-The key concepts covered include recursive problem-solving techniques, the use of base cases, and recursive relations. The tile placement problem uses a Fibonacci-like sequence, while the pancake problem uses a Tower of Hanoi-like approach.

Outlines

00:00

📚 Recursive Tile Arrangement Problem

The speaker introduces a problem where tiles need to be arranged on a 2xN floor using N tiles, each measuring 1x2. The challenge is to find the number of ways to arrange the tiles either horizontally or vertically. Examples with different values of N are given, including an illustration for N=4, where five distinct ways of arranging the tiles are possible. A formula for calculating the number of tile arrangements for larger values of N is outlined.

05:02

🔢 Recursive Solution for Tile Arrangements

This section provides further explanation on how the recursive relation works for calculating the number of tile arrangements. The base cases, K1=1 and K2=2, are presented, followed by the recursive relation that combines the two previous values to calculate the number of arrangements for larger N. The speaker demonstrates how this recursive relation is applied to calculate K8, which results in 34 ways to arrange tiles.

10:02

🥞 Pancake Stacking Problem

The speaker introduces the second problem about moving a stack of pancakes from one plate to another. The key rule is that larger pancakes must always remain beneath smaller ones. The example involves moving two pancakes with three steps, and the problem scales up to six pancakes. A step-by-step breakdown is provided for moving three pancakes, showing how the pancakes are transferred between plates.

15:05

⚙️ Recursive Formula for Pancake Moving

In this final section, a recursive relation for the number of steps required to move pancakes is introduced. The formula PN = 2*P(N-1) + 1 is derived, explaining how the process scales as more pancakes are added. The example progresses through 1, 2, 3, and 4 pancakes, where the steps increase to 63 for six pancakes. The speaker concludes with a motivational note for learners, encouraging them to continue practicing recursion.

Mindmap

Keywords

💡Recursion

Recursion is a computational concept where a problem is solved by breaking it down into smaller instances of the same problem. In the video, recursion is applied to solve problems like arranging tiles or moving pancakes. Each solution builds on the previous ones, with each step depending on the results of smaller cases.

💡Algorithmic strategy

An algorithmic strategy refers to a step-by-step approach for solving a problem efficiently. In the video, the strategy is demonstrated through recursive methods to determine the number of ways to arrange tiles or the number of steps to move pancakes, showing the importance of clear logical steps.

💡Tile arrangement problem

This problem involves arranging tiles on a 2xN floor, where each tile is 1x2 and can be placed either horizontally or vertically. The video shows how recursion is used to find the number of ways to arrange the tiles, with the solution based on smaller floor sizes (n-1 and n-2).

💡Base case

The base case is the simplest instance of a recursive problem, which provides the foundation for the recursion. In the video, for the tile problem, the base cases are when N=1 (1 way) and N=2 (2 ways). These are used to build the solution for larger N values.

💡Recursive relation

A recursive relation is an equation that defines each term in a sequence as a function of the preceding terms. For the tile arrangement problem, the recursive relation is KN = K(N-1) + K(N-2), which helps calculate the number of ways to arrange the tiles for larger values of N.

💡Pancake stacking problem

The pancake stacking problem involves moving a stack of pancakes from one plate to another, with the rule that a larger pancake must always be under a smaller one. The video shows how recursion is applied to find the minimum number of moves required for different numbers of pancakes.

💡Tower of Hanoi

The Tower of Hanoi is a classic recursive puzzle similar to the pancake stacking problem. It involves moving disks from one peg to another with the restriction that larger disks cannot be placed on top of smaller ones. In the video, the pancake problem is modeled similarly, where each step builds on the previous ones.

💡PN recursive formula

The PN recursive formula is used to calculate the number of steps needed to move N pancakes. The formula is PN = 2 * P(N-1) + 1, which shows how the number of steps doubles with each additional pancake. This is crucial in determining the total steps required in the pancake problem.

💡Computational thinking

Computational thinking is a problem-solving process that involves breaking down complex problems into manageable parts and creating algorithms to solve them. In the video, the focus is on recursion as a key element of computational thinking, applied to both tile arrangement and pancake stacking.

💡Fibonacci sequence

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. In the tile arrangement problem, the number of ways to arrange tiles follows a Fibonacci-like pattern, with the sequence starting from K1 = 1 and K2 = 2, then progressing with the sum of the previous two terms.

Highlights

Introduction to solving problems using recursion with computational thinking and algorithmic strategies.

Explanation of the first problem: Tiling a 2xN floor with 1x2 tiles, emphasizing the recursive approach.

Illustration of tile placement: Horizontal and vertical arrangements for different values of N.

For N=4, five distinct ways to arrange the tiles are demonstrated.

A recursive formula is derived for determining the number of ways to tile a 2xN floor, where KN = K(N-1) + K(N-2).

Base cases are defined: K1 = 1 and K2 = 2 for N = 1 and N = 2 respectively.

For N=8, the recursive approach gives 34 ways to tile the floor, illustrating the growth in complexity.

Transition to the second problem: Moving pancakes between plates while maintaining the order based on size.

The pancake stacking problem introduces three plates (A, B, and C) and the rule that larger pancakes must always be below smaller ones.

A recursive strategy is used to determine the minimal number of moves required to transfer all pancakes between plates.

For 3 pancakes, the minimum number of moves is 7, and for 4 pancakes, it is 15.

The recursive formula for the pancake problem is PN = 2 * P(N-1) + 1, where P1 = 1.

For 6 pancakes, the total number of moves required is 63, following the recursive solution.

Both problems showcase the power of recursion in breaking down complex tasks into simpler, repeatable steps.

The session concludes with a reinforcement of computational thinking concepts through recursive problem-solving.