محدثين كے ہاں قراء ثلاثہ كا مقام و مرتبہ اور علم حديث ميں ان كى خدمات

Allah Himself has taken the responsibility to protect the Holy Quran and the Hadith of the Holy Prophet. He Himself has provided the sources of their protection. One of the means of the protection that was the creation of such a group of the Qura who not only served the Holy Quran but also provided worth mentioning services in Ahadith of the Holy Prophet. But their services are hidden from us. By Qura the researcher means those Qura whose recitation styles and narrations are studied and taught in the different quarters of the world who are known as Qura Saba & Ashra (سبعہ وعشرہ). They are ten imams each with two Ravi’s. They are thirty Qura in total. I have selected only last three Imam & their two narrators in this Article. These Qurra are known as Qurra Thlathah (قراء ثلاثہ).  The services of these imams have been highlighted in the light of the following eleven Ahadith books. Sihah:   Sahih Bukhari, Sahih Muslim, Sahih Ibn-e-Habban, Sahih Ibn-e-Khuzeema. Sunan: Sunan Abu Dawud, Sunan al-Tirmidhi, Sunan al-Nasai, Sunan Ibn Majah, and Sunan al-Kubra. Masaneed: Musnad Ahmad ibn Hanbal, Musnad Abu Ya`la al-Mawsili. How many people have reported them and what is the standard of the weakness and soundness of those narrators have also been discussed in this article. Besides these books of Ahadith, these Ahadith have been searched in other books of Ahadith also. The status of these Qura has been explained in the light of the commentary of Muhadithin. Whether Ahadith critics have declared them thiqa or weak or have declared them as average sadooq. The most important thing is that there is no one weak reporter in these imam qura. Two out of three imam qura are ranked as thiqa and one sadooq. And among the narrators of these qura one is thiqa, one sadooq, and nobody are weak reporters. There is silence about the remaining four reporters of these qura. The reason is that there is no hadith reported from them. Because of all this their religious and scholarly authenticity could be determined. The narrations of these thalathah (ثلاثہ) Qura are confined to reporting the Holy Quran but they have also reported about every part of fiqh and they have been utilized and refered to  

Combinatorial Generation Algorithms for Subgraphs

The field of computer science has greatly benefitted from the advances in combina- torial algorithms in the last few decades. This is because the advent of high speed computers has made it possible to generate lists of combinatorial objects in a practi- cal amount of time. In areas such as genome science and data mining the problems are often vaguely defined, and researchers have to look for meaningful information in huge datasets. In this thesis, efficient generation algorithms for subgraphs such as bicliques and paths in cliques are developed. Cliques and bicliques are used to model various real-world problems encountered in bio-informatics, data mining and networks. We consider two variations of bicliques: pseudo-bicliques and c-isolated bicliques. Pseudo- bicliques relax the rigid connectivity requirement of bicliques to cater for missing and noisy data. On the other hand, the c-isolated bicliques enforce a restriction on the external connectivity of the vertices in a biclique to model cohesive communities. This thesis presents an algorithm based on reverse search to list all pseudo-bicliques in a graph G. The algorithm takes linear time on average to generate each pseudo- biclique. On the other hand, our generation algorithm for c-isolated bicliques exploit underlying properties of an isolated biclique to trim the input graph. Furthermore, the algorithm deploys the vertex cover enumeration algorithm based on fixed point tractability (FTP) and lists all isolated bicliques in linear time, in the case where c is constant. The performance of the proposed algorithms is evaluated on random graphs and real-world problems. The results are quite promising and confirm our theoretical findings. In this research work, we also explore another combinatorial object called a clique. A constant amortized time algorithm is proposed to generate all spanning paths and all paths in a clique in minimal change order (an ordering in which successive elements differ in a small way)