tailieunhanh - Báo cáo toán học: " ON THE NUMBER OF DESCENDANTS AND ASCENDANTS IN RANDOM SEARCH TREES"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: ON THE NUMBER OF DESCENDANTS AND ASCENDANTS IN RANDOM SEARCH TREES. | ON THE NUMBER OF DESCENDANTS AND ASCENDANTS IN RANDOM SEARCH TREES Cgnradg Martinez Alois Panhglzer Departament de Llenguatges i Sistemes Informatics Polytechnical University of Catalonia Pau Gargallo 5 E-08028 Barcelona Spain. email www http conrado Institut fur Algebra und Diskrete Mathematik Technical University of Vienna Wiedner Hauptstrasse 8-10 A-1040 Vienna Austria. email Helmut Prgdinger Institut fuur Algebra und Diskrete Mathematik Technical University of Vienna Wiedner Hauptstrasse 8-10 A-1040 Vienna Austria. email www http theoinf Submitted January 7 1997 Accepted March 26 1998. Abstract. The number of descendants of a node in a binary search tree BST is the size of the subtree having this node as a root the number of ascendants is the number of nodes on the path connecting this node with the root. Using a purely combinatorial approach generating functions and differential equations we are able to extend previous results. For the number of descendants we get explicit formulae for all moments for the number of ascendants which is harder we get the variance. A natural extension of binary search trees occurs when performing local reorganisations. Poblete and Munro have already analyzed some aspects of these locally balanced binary search trees LBSTs . Here we relate these structures with the performance of median-of-three Quicksort. We get as new results the variances for ascendants and descendants in this setting. If the rank of the node itself is picked at random grand averages the corresponding parameters only depend on the size n. In this instance we get all the moments for the descendants BST and LBST as well as the probabilities. For ascendants LBST we get the variance and in principle the higher moments as well as the normal limiting distribution. The emphasis is on explicit formulae and these are sometimes quite