Algoeithm Design, Analysis and Implementation Assignment

Algoeithm Design, Analysis and Implementation - Assignment Example This is done by choosing a comparison element and placing all the elements that are less than the comparison element in the first group and the rest of the elements in the second group. This procedure is repeated recursively until the elements are sorted (a part consist of only one element). T(n) = (n-1) + ?1 ? i ? n ti As 1,2,....k-elements are already sorted, we can say that ti =0, where i = 1,2, 3... k. Then, the contribution of quick sort when early stopping is used can be given by, T(n)=(n+1)( ?k ? i ? n ti + ?(1)) = (n+1)( n lg +?(1)) =2n lg +?(n) Thus, T(n) for quick sort =O(nlg(n/k)). Given that, insertion sort is done on a partially sorted array (unsorted k-elements). In general, running time of insertion sort is O(n2 ), where n is the length of the array (total number of elements). In order to provide a solution to this problem, the total array is divided into subarrays of k-elements each, such that k/2? n ? k, then n = O(k) and the running time of insertion sort is O(k2). The total number of such subarrays (m) would then be n/k ? m ? 2n/k., which implies m = O(n/k). The total time spent on insertion sort would then be O(k2)* O(n/k) = O(nk). T(n) for insertion sort = O(nk). Therefore, the total time for this sorting algorithm is as follows: T(n) = O(nk + nlg(n/k) ). ... Solution: From the above problem (1), we find that quick sort sorts k-elements of an n-element array O(n log(n/k)) time. Quick sort sorts by partitioning the given array A[p...r] into two sub-arrays A[p...q] and A[q+1... r] such that every element in A[p...q] is less than, or equal to, elements in A[q+1... r]. This process is repeated until all the elements are sorted. Algorithm for quick sort is given by: A[P] is the pivot key upon which the comparison is made. P is chosen as the median value of the array at each step. If the element is less than, or equal to, the pivot key value, it is moved left. Otherwise, it is moved right. Assuming the best case scenario where each step produces two equal partitions, then T(n)=T(n/2)+T(n/2)+?(n) =2T(n/2)+ ?(n) By Master’s Theorem case 2, T(n) = O(n lg n) In other words, the depth of recursion is log n and at each level/step, the number of elements to be treated is n. If only k-elements are sorted, then the depth of recursion would be n/k and the number of elements would be n at each level, the time taken by this sorting algorithm is given by T(n) = O(n lg (n/k)). 2.2 Show that we can sort a k-well-sorted array of length n in O(n log k) time. As the array is already sorted for k-elements, the remaining steps required to complete the sort would be k (using the results from 1), then T(n) = O(n lg k). 3. Computing the k-th smallest element in the union of the two lists m and n using O(lg m +lg n) time algorithm: Approach 1: Merge sort can be used in this case. It splits the list into two halves, recursively sorts each half, and then merges the two sorted sub-lists. In the given problem, the lists are already sorted; hence, the

Discussion questions Coursework Example | Topics and Well Written Essays - 250 words - 1

Discussion questions - Coursework Example ather encourage employers to put selection system favoring women in the hiring process as an affirmative action in order to boost their participation in national building. This implies that the discrimination could be justified as a mechanism of smoothening out the employment sector on the basis of quotas and proportional presentation in the employment sector (Coral & Practising Law Institute, et al. 2009). Employment discrimination has often been responsible for enhanced equality in job placements especially in across Canada where inequality is highly integrated. However, discrimination is based upon diverse variables or grounds. Certain grounds are unacceptable while others are mutually accepted. In the Canadian province of Alberta, discrimination on the basis of sexual orientation is prohibited. In particular, either gay or lesbians facing discrimination on the basis of their sexual orientation are prohibited from recourse via apt mechanisms laid down in the IRPA enabling them to subject their experiences of discrimination besides incapacitating them to uphold a legal remedy (Koral & Practising Law Institute, et al. 2009). A Canadian employer may justify the adverse implications of the procedure used in employment selection on several grounds. For instance, according to the Alberta Human Rights Act, there are exemptions to discrimination. This means that some grounds of discrimination are justifiable. In section 7 of the Act, an employer may be justified to discriminate if such discrimination is based on occupational requirement. Subsequently, section 11 justifies discrimination if such action is both ‘reasonable and permissible within the prevailing circumstances’ (Koral & Practising Law Institute, et al. 2009). In this regard, employer’s discriminatory practices demonstrate the fact that their standards do not contravene the law. To accommodate an individual to a point of undue hardship basically entails a provision under the Supreme Court of Canada