divide and conquer - data structured and algorithms

divide and conquer - data structured and algorithms

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  DATA STRUCTURES AND ALGORITHMS Divideand Conquer Algorithm Lecturer: Diem Cong Hoang
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  02 Meet The Team Nguyễn Quốc Vượng Vũ ThịThu Hường Đỗ Thị Cẩm Ly Nguyễn Đức Anh
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  1. ABOUT THEALGORITHM Divide and Conquer Algorithm Divide and Conquer is an algorithmic technique that solves complex problems by breaking them into smaller subproblems, solving them independently, and combining their solutions.
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  2. ALGORITHM TARGET Divideand Conquer Algorithm Reduce computational complexity. Take advantage of recursion to break down and process each part more efficiently. Make parallelization easier in big data processing
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  3. WHEN WEUSE THIS ALGORITHM? The problem can be divided into independent or nearly independent subproblems. The subproblems can be solved in a similar way to the original problem. The results of the subproblems can be combined to solve the entire problem.
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  4. TIME COMPLEXITY Let T(n)be the computational complexity, it is the stopping case. According to the general algorithm T(n) = O(1) if n = n0 T(n) = aT(n/b) + D(n) + C(n) nế u n>no In there a: Number of subproblems n/b: Subproblem size D(n): Time complexity of the Devide part C(n): Time complexity of the Combine part
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  Example with b=2. Accordingto Master's theorem: T(n) = 2T(n/2) + n then the complexity will be O(n log n)
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  5. THE ADVANTAGEAND DISADVANTAGE OF ALGORITHM. Overhead Function Calls: Recursion introduces overhead due to function calls, which can be significant for problems with a very large number of recursive calls. Space Complexity StackUsage:Recursivealgorithmsmayuseadditionalstackspacepro portionaltothedepthofrecursion. For instance, Merge Sort uses 0(n) auxiliary space for temporary arrays, Complexity in Combining Solutions Merging Subproblems: Combining solutions from subproblems can be complex and may require additional computational resources. For example, the merge step in Merge Sort requires merging two sorted arrays, which is non-trivial. Base Case Handling Special Cases: Identifying and handling base cases properly is crucial. Incorrect base case handling can lead to incorrect results or infinite recursion. Efficiency TimeComplexity:Manyproblemscanbesolvedmoreefficientlyus ingdivideandconqueralgorithms. Parallelism Independent Subproblems: Since subproblems are solved independently, divide and conquer algorithms can be easily parallelized. This can lead to significant performance improvements on multi-core processors or distributed systems. Simplicity Problem Decomposition: Breaking down complex problems into simpler subproblems can make them easier to understand and solve. This step-by-step approach is often easier to implement and debug. Modularity Reusable Components: Components of the solution can be reused for different problems. For instance, the merge function in Merge Sort can be used independently to merge two sorted arrays. Advantage Disadvantage
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  6. PROBLEMS USINGALGORITHMS The Divide and Conquer algorithm is versatile and can be applied to a wide range of problems, especially those that can be broken down into smaller, independent subproblems. Here are some common problems where Divide and Conquer is particularly effective: Sorting Merge Sort Quick Sort Searching Binary Search Mathematical Computations Karatsuba Algorithm Strassen's Algorithm Computational Geometry Closest Pair of Points Convex Hull Dynamic Programming Matrix Chain Multiplication Longest Common Subsequence Optimization Problems Maximum Subarray Problem Median of Medians Recursion-Based Problems Tower of Hanoi Fibonacci Sequence Graph Algorithms Divide and Conquer for Closest Pair of Points in 2D Fast Fourier Transform (FFT
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  7. SOLVING REALPROBLEM Maximum Subarray Sum Idea: Divide the array into two halves. Find the maximum subarray in the left half, the right half, and the maximum subarray crossing the middle. Return the maximum of the three.
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  Merge Sort (Divideand Conquer) Idea: Recursively divide the array into halves. Sort each half. Merge the sorted halves
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  THANK YOU Group 2 FORYOUR ATTENTION