COEN 352 Data Structures and Algorithms Stuart Thiel Advanced Trees COEN 352 Data Structures and Algorithms Tries Balanced Trees Stuart Thiel Concordia University Department of Electrical Computer Engineering Summer, 2014 1/12 COEN 352 Data Outline Structures and Algorithms Stuart Thiel Advanced Trees Advanced Trees Tries Tries Balanced Trees Balanced Trees 2/12 COEN 352 Data Tries Structures and Algorithms Stuart Thiel I pronounced 'try' Advanced Trees I object space decomposition: BST Tries I key space decomposition: Trie Balanced Trees I data usually just in leaves I branching factor based on alphabet I Huffman Coding Tree: binary trie 3/12 COEN 352 Data Searching for Words Structures and Algorithms Stuart Thiel I depth bound by word length Advanced Trees I clumping on common prefix: meh Tries I Unlike Huffman Coding Tree, need terminating Balanced Trees character I show page 432 for example 4/12 COEN 352 Data PAT Tries Structures and Algorithms Stuart Thiel I compressing leaves worked above Advanced Trees I we can also compress internal nodes Tries I Practical Algorithm To Retrieve Information Coded In Balanced Trees Alphanumeric I aka PATRICIA aka PAT trie I nodes point to the next relevant part of the input I full comparison at leaves I show page 433 5/12 COEN 352 Data Balanced Trees Structures and Algorithms Stuart Thiel I BSTs can become unbalanced Advanced Trees I makes search expensive Tries I adjust access, improve performance Balanced Trees I ensuring complete binary tree too expensive I relax requirement, good enough 6/12 COEN 352 Data AVL Tree vs. Splay Tree Structures and Algorithms Stuart Thiel I AVL constrain difference between depth of subtrees Advanced Trees I SPLAY improves balance on each access Tries I both have advantages Balanced Trees 7/12 COEN 352 Data AVL Tree Structures and Algorithms Stuart Thiel I named for its inventors Adelson-Velskii and Landis) Advanced Trees I left and right subtrees differ in depth by at most 1 Tries I ALL NODES Balanced Trees I max depth is O (logn) I search is therefore O (logn) I updates can be done in time proportional to search 8/12 COEN 352 Data AVL Tree Balance Structures and Algorithms Stuart Thiel I Each node knows its level of unbalance Advanced Trees I −1, 0 or 1 Tries I tree starts balanced according to this Balanced Trees I changes propagate up to root I when can we stop early? I propagation stops logically I show page 435 9/12 COEN 352 Data AVL Tree Rotations Structures and Algorithms Stuart Thiel I what are these changes? Advanced Trees I we call them rotations Tries I AVL needs a single rotation or double rotation Balanced Trees I rotation about node with bad balance I if sign of imbalance is the same, single I if sign of imbalance differs, double I double rotation starts by making sign of imbalance the same I rotate on node with awkward balance 10/12 COEN 352 Data AVL Tree Deletion Structures and Algorithms Stuart Thiel I addition looks easy. it is! Advanced Trees I deletion is only slightly trickier Tries I can we optimize? Balanced Trees I if balance is 1 grab smallest in right subtree? I if balance is −1 grab biggest in left subtree? I at least it cannot hurt I work up from secondary deletion 11/12 COEN 352 Data AVL Tree Examples Structures and Algorithms Stuart Thiel I Lets try some trees on the chalkboard! Advanced Trees Tries Balanced Trees 12/12.