Divide And Conquer.pptx
- 2. What is Divide and Conquer? ● The problem is divided into smaller subproblems and then each problem is solved independently. ● The solution of all sub-problems is finally merged in order to obtain the solution of an original problem.
- 3. 3 steps ● Divide → breaking the problem into smaller sub-problems. ● Conquer → smaller subproblems to be solved. ● Combine → recursively combines them until they formulate a solution of the original problem.
- 4. Applications ● Binary Search ● Min-Max problem ● Merge sort ● Karatsuba’s algorithm
- 5. Advantages ● Solve difficult problems ● Finding other efficient algorithms ● Uses memory caches effectively
- 6. Disadvantages ● Recursion is slow ● The same subproblem can occur many times ● One must make sure that there is sufficient memory allocated for the recursion stack
- 7. Min-Max problem We have to find the minimum and maximum element from the array Approach ● Divide the list into small groups ● Find the maximum and minimum of each group ● The max/min should be one of the max and min of all the groups Time complexity: T(n) = 2T(n/2) + 2 ,n>2 While Non-D&C takes : 2n-2 comparisons D&C takes : 3(n/2) -2 comparisons
- 8. Binary Search ● Implemented on only sorted list of elements ● The process starts with finding the element at the center of the list and comparing the key value with the element within the middle of the list, if the worth of the key's but the element at the center then the search process the key to the list up to the center element and if the worth of the key's greater than element at the center then search the key from middle element to the top of the list.
- 9. ● Best case - O (1) comparisons : In the best case, the item is the middle of array. . constant no.of comparisons (actually just 1) are required. ● Worst case - O (log n) comparisons :when the item is not in the array. Through each recursion or iteration of Binary Search, the size of the admissible range is halved. This halving can be done ceiling(log n ) times. Thus, ceiling(log n ) comparisons are required.
- 10. Merge Sort ● Was Invented by John von Neumann (1903-1957) ● Here, we divide the problem into two halves until all the elements in the subproblems would be sorted . then we start merging the individual sorted array into the final array. ● Time complexity → O(n log n)
- 11. Karatsuba Multiplication ● Reduces the no.of multiplication Conventional method Karatsuba method
- 12. ● Time complexity : T(n) = 3T(n/2) +O(n) → O(n1.584) ● Time complexity: O(n2) → conventional method Let's say we want to multiply 1234 & 5678 xH = 12 & xL = 34 yH = 56 & yL = 78 Xy = xHyH * 104 + (xH yL + xL yH) * 102 + xLyL 12*56*104 + (12*78 +34*56)*102 + 34*78 =7006652
- 13. Thank you!
Editor's Notes
- #12: This algorithm reduces the number of multiplication comparing to conventional method of multiplication.