Tree traversal preorder, postorder: applies to any kind of tree inorder: applies only to binary trees void BST::Traverse (BinaryNode *node) { if (node == NULL) // return an appropriate value return; else // PREORDER: compute/use node here Traverse (node->left); // INORDER: compute/use node here Traverse (node->right); // POSTORDER: compute/use node here } Oct 31, 2001 CSE 373, Autumn 2001

Tree traversal, example int NumLeaves (BNode<Comp> *tree) { if (tree == NULL) return 0; else int sum = 0; sum = NumLeaves(tree->left) + NumLeaves(tree->right); if ((tree->left == NULL) && (tree->right == NULL)) sum += 1; } return sum; Oct 31, 2001 CSE 373, Autumn 2001

Tree traversal, example 2 D B F A C E G preorder: ?? inorder: ?? postorder: ?? level order (use a queue): ?? Oct 31, 2001 CSE 373, Autumn 2001

Open Addressing Examine cells in the order h0(x), h1(x), h2(x), … where hi(k)=(hash(k) + f(i)) mod TableSize Linear Probing f(i) = i After searching spot hash(k) in the array, look in hash(k) + 1, hash(k) + 2, etc. example: insert 38, 19, 8, 109, 10 Oct 31, 2001 CSE 373, Autumn 2001

Linear Probing example 8 1 109 2 10 3 4 5 6 7 38 9 19 38 19 8 109 10 primary clustering Oct 31, 2001 CSE 373, Autumn 2001

Quadratic Probing f(i) = i2 After searching spot hash(k), look in the 1st, 4th, 9th, etc. spots after hash(k). Less likely to encounter primary clustering. … n-1 Oct 31, 2001 CSE 373, Autumn 2001

Double Hashing f(i) = i · hash2(x) Items that hash to the same location with hash(x) won’t have the same probe for hash2(x). Oct 31, 2001 CSE 373, Autumn 2001

Analysis of find Defn: The load, , of a hash table is the ratio:  no. of elements  table size Separate chaining unsuccessful:  (avg. length of a list at hash(k)) successful: 1 + (/2) (one node, plus half the avg. length of a list (not including the item)). Oct 31, 2001 CSE 373, Autumn 2001

Open addressing Linear probing unsuccessful:  successful:  (Analysis from Knuth) Oct 31, 2001 CSE 373, Autumn 2001

Random Collision Resolution Model Goal: see how collisions occur under ideal conditions. On an insertion, each probe is independent of previous probes. On a successful find, the exact probe sequence used in insertion is used. unsuccessful: 1/(1 - ) (1- ) fraction of the cells are empty. successful: (ln means loge ) Oct 31, 2001 CSE 373, Autumn 2001

Comparison Double hashing – performance approximated by the random collision resolution model. linear probing random collision  unsucc. successful 0.1 1.1 0.3 1.5 1.4 1.2 0.5 2.5 2.0 0.7 6.1 3.3 2.2 1.7 0.9 50.5 10.0 5.5 2.6 Oct 31, 2001 CSE 373, Autumn 2001

Rehashing Idea: When the table gets too full, create a bigger table and hash all the items from the original table into the new table. When to rehash? half full ( = 0.5) when an insertion fails some other threshold Cost of rehashing Oct 31, 2001 CSE 373, Autumn 2001