Graph Partitioning is a well-known problem in the literature. In this paper, path Partitioning of trees in which the given tree is partitioned into edge-disjoint paths is considered. A linear time algorithm is given for computing a path Partitioning of minimum height.

A thorough recognition of the nature and duties of the genes is based upon having adequate information about the proteins. However, the proteomic projects follow a slow trend, therefore, solving the protein-related problems has become as one of the most important challenges in bio-informatics. Consequently, the presence of tools which can enhance the structural recognition, classification, and interpretation of proteins will be advantageous. Statistical methods are among the tools to help solve bio-informatics problems. These methods may be used to help predict the third structures of proteins, study proteins collectively, as well as extract new interactions among the protein collections. One of the very efficient and useful methods in the collective study of protein subsets is the cluster analysis. In the present study, the recognized protein sequences related to esophagus, stomach, and colon cancers are analyzed through Partitioning, non-Partitioning, and fuzzy clustering methods. Needleman-Wunsch global alignment algorithm was used to determine pair-wise similarities. The evaluations have shown that the clusters obtained through using the AGNES method have produced more powerful structures, yet, it can be said that the PAM clustering method, compared to other ones, has produced the best results in predicting ability of the 3D structure of the unknown protein sequences.

Based on rapid upsurge in the demand and usage of electronic media devices such as tablets, smart phones, laptops, personal computers, etc. and its different display specifications including the size and shapes, image retargeting became one of the key components of communication technology and internet. The existing techniques in image resizing cannot save the most valuable information of images on display devices with different resolutions. Seam carving is a standard technique for content-aware resizing of images and videos with negligible distortion. However, seam carving resize high-resolution videos and high quality images with high computational complexity; this limits its real-time applications. In this paper, we present a novel approach to reduce seam carving process time. In the proposed technique, the image was split into three equal parts: upper-middle-lower (or right-middle-left) using horizontal or vertical strips. The middle strip was analyzed by original seam carving technique. For other strips (upper-lower), the seam was obtained employing Dijkstra fixed start point technique. In our proposed technique, unlimited Dijkstra depth search was replaced with a limited depth search. It enhances the computational efficiency of Dijkstra technique for the upper and lower strips. The experimental results showed much better computational efficiency than the current enhanced seam carving techniques. These results indicate that computational complexity is superior, while still maintaining the output quality of the original seam carving method.