CPSC 212, Test #2B, October 23, 2006

How to explain why O(n log n) is the best performance possible for an algorithm that sorts by comparisons.
- Draw for your friend a decision tree for n=3 and n=4 values.
- Explain that the height of the decision tree represents the minimum number of comparisons necessary to reach a leaf, and a leaf contains a sorted list.
- This decision tree has n! leaves, one for each possible sorted order for n values.
- The tree is approximately balanced and height of this tree, therefore, is log(n!).
- Stirling's formula shows that log(n!) is O(n log n)
- Therefore, the minimum number of comparisons required by an algorithm that sorts by comparisons is O(n log n).

B-Tree
Leaf 1: 4 5 7 Leaf 2: 15 16 19 23 Leaf 3: 24 25 27 Leaf 4: 29 34 36 43 Leaf 5: 61 72 98 Node 1: (parent of Leaf 1 and Leaf 2) keys: 15 Node 2: (parent of Leaf 3, Leaf 4, and Leaf 5) keys: 29 61 Node 3: (root and parent of Node 1 and Node 2) keys: 24

Red-Black Tree
A red-black tree is a binary search tree with the following red-black properties:

each node is either red or black
the root is black
each leaf (NULL) is black
if a node is red, its children are black
every simple path from root to leaf has the same number of black nodes.

Hash Table
i: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 a: 1 20 2 39 21 78 58 40 77 59

Leftist minHeap
root : 15 children of 15: 19 16 children of 19: 27 23 children of 16: 24 children of 27: 61 children of 23: 34 children of 24: 29

Binomial maxQueue
Bo: root : 6 B3: root : 27 children of 27: 24 7 19 children of 7: 5 children of 24: 23 15 children of 23: 4

RadixSort
After Pass 1 After Pass 2 1792 8710 4205 4865 2190 8710 2412 1792 2412 2456 2412 2412 8710 2412 1453 2412 1453 2456 4567 4865 4865 1297 4205 4567 1453 2456 2190 2190 4567 1792 4205 1297 1297

QuickSort
pivot = 41 i: 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 a: 19 29 39 25 28 41 91 95 63 46 43 57 77 63 72 63 69 48 86

Why InsertionSort?
For short arrays, i.e., arrays whose length is less than or equal to CUTOFF, InsertionSort will execute faster than QuickSort.
Short Answers

True
False
Separate chaining, Linear Probing, Quadratic Probing, Double Hashing
HeapSort, MergeSort, QuickSort
ShellSort