CS521 Final Term Notes Set 2

CS521 \ Notes for the Final Exam · Asymptotic Notations: · Notation · Definition · Limit

Big-O

Small-o

Big-

(non-negative constant) Small-

Big-

(some constant)

Notes:

Some hierarchies: ( is “o”, is “ ”)

Divide and Conquer – recurrences:

Break a problem into sub-problems, solve sub problems and merge solutions. Hanoi-tower: ( ) ( ) ( )

Solving Recurrences:

1. Substitution method:

Have a guess for ( ) Assume correctness for all Substitute -terms in the recurrence by the assumption, using constants, and derive upper / lower bounds (or both). Example:

To get an upper bound: ( ) ( ) – develop and derive an expression that is exactly .

2. Recursion tree:

Build a tree from the levels in the substitution: Level 0: ( ) Level 1: elements, each costs ( )

Level : elements, each costs ( ) Last level: elements , each costs ( )

3. The Master Theorem:

For , ( ) function over non-negative integers and ( ) the recurrence:

1. ( ) ( ) ( ) ( )
2. ( ) ( ) ( ) ( )
3. ( ) ( ) and ( ) ( ) ( ) ( ( ))

Notes for MT: The addition/subtraction denotes polynomial difference in running time between ( ) and . Cases (1) and (2) derive for some relation between . Simplified Master Theorem: For ( ) ( ) ( ) :

1. ( ) ( )
2. ( ) ( )
3. ( ) ( )

Some recurrences:

Binary search: ( ) ( ) ( ) Linear search: ( ) ( ) ( ) ( ) ( ) ( ) (MT case 1) ( ) ( ) ( ) (tree) ( ) ( ) √ ( ) (MT case 3) ( ) ( ) ( ) (bounds) ( ) ( ) ( ) (tree) ( ) ( ) ( ) ( ) ( ) (bounds) ( ) ( ) ( ) (substitution) ( ) ( ) ( ) (bounds) ( ) ( ) ( ) (bounds) ( ) √ (√ ) ( ) ( ) (tree)

Useful math:

Harmonic series: ∑ when

Sorting algorithms:

Insertion Sort:

Each element is pushed as far as possible to the left (sorted) part of the array. Running time: ( ) worst-case (when sorted in reverse), ( ) best-case (when already sorted). Sorts in-place, space complexity: ( ).

Merge-sort:

Divide the sequence into two sequences, apply the algorithm recursively and merge solutions.

Quick Sort:

Partition( ): linear procedure that partitions in-place around , - s.t.: | | and returns the index of

• the pivot.

Preferable over merge-sort, as it sorts in-place. General recurrence: ( ) ( ) ( ) ( ) Worst-case: array already sorted, partition at the beginning: ( ) ( ) ( ) ( ) For any -proportional division we get ( ):

For we get a balanced tree. Expected running time: , ( )- ∑ , - ( | ) Heap: Left filled structure that satisfies partial order (node its children, no particular relation between children). For node : left child at , right at , parent ⌊ ⌋ Height is : the cost of insertion, extract-max, delete. Every level of height has ⌈ ⌉ nodes.

Insert: put node at the end and heapify up (switch with parent( ) until parent( ) or is root. Remove-max: remove root and put last node of the heap in its place. Heapify it down, each level switching with the greater child of the two until satisfies heap properties. Build-heap: for node down to , heapify down. Amortized analysis – each level but lowest will be visited at most ( ) . Total: ( ) ∑ . The sum part is ( ) total: ( ).