1. 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

2. 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

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

4. 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

5. Linear Probing example8 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

6. 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

7. 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

8. 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

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

10. 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

11. 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

12. 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