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| courses:cs211:winter2012:journals:jeanpaul:chapter_fivesection_i [2012/03/06 15:13] – created mugabej | courses:cs211:winter2012:journals:jeanpaul:chapter_fivesection_i [2012/03/06 15:25] – mugabej | ||
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| Divide and conquer algorithms are a class of algorithmic techniques in which one breaks the input into several parts, solves the problem in each part recursively, | Divide and conquer algorithms are a class of algorithmic techniques in which one breaks the input into several parts, solves the problem in each part recursively, | ||
| - | Analyzing these problems usually involves solving recurrence relation that bounds the running time recursively in terms of the running time on smaller instances. Mergesort uses the same template as other divide and conquer algorithms to sort a given list of items. | + | Analyzing these problems usually involves solving recurrence relation that bounds the running time recursively in terms of the running time on smaller instances. Mergesort uses the same template as other divide and conquer algorithms to sort a given list of items:\\ |
| + | * Break up problem of size n into two equal parts of size ½n | ||
| + | * Solve two parts recursively | ||
| + | * Combine two solutions into overall solution | ||
| + | \\ | ||
| + | \\ | ||
| + | So, | ||
| + | * Supposing that n is even, the algorithm spends O(n) time to divide the input into two pieces of size n/2 each | ||
| + | * It then spends T(n/2) to solve each one since T(n/2) is the worst-case running time for an input of size n/2 | ||
| + | * Finally, it spends O(n) time to combine the solutions from the 2 recursive calls | ||
| + | * Thus, the following ** recurrence relation ** is satisfied by the running time: | ||
| + | * **T(n) ≤ 2T(n/2) + cn** for n >2 | ||
| + | * And T(2) ≤ c | ||
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| + | T(n/2) + T | ||
