Dynamic Arrays

What is a dynamic array?
- All of the properties of a static arrays
- But don't need to declare the size.
  - It will grow and shrink as needed.
The c++ vector class is an example of this.
In general the array will have
- A size $s$, or the "number of elements" stored in the array.
  - This assumes the elements are stored at locations 0 to $s-1$
- A capacity $c, \backepsilon 0 \le s \le c$
Operations
- The [] still works, but only for access and modification
- size returns the size of the array
- capacity returns the capacity of the array.
- Generally there is an append, push or push_back which inserts an element on the end of the array.
  - ```
  push_back(ItemT e)
    if size ≥ capacity
        capacity ← capacty * 2
        allocate newA 
        copy data from A to newA
        delete A
        let A = newA
    A[size] = e
    size ← size + 1 
```

The timing of push back is interesting

Item Added	Size/Capacity (at start)	Need new array?	Copy Ops Needed	Average Copy Ops
1	0/0	Y	0	0
2	1/1	Y	1 item	1/2
3	2/2	Y	2 items	(2+1)/3 = 3/3
4	3/4	N	0	(2+1)/4 = 3/4
5	4/4	Y	4	(4+2+1)/5 = 7/5
6	5/8	N	0	(4+2+1)/6 = 7/6
7	6/8	N	0	(4+2+1)/7 = 7/7
8	7/8	N	0	(4+2+1)/8 = 7/8
9	8/8	Y	8	(8+4+2+1)/9 = 15/9
10	9/16	N	0	(8+4+2+1)/10 = 15/10
11	10/16	N	0	(8+4+2+1)/11 = 15/11
12	11/16	N	0	(8+4+2+1)/12 = 15/12
13	12/16	N	0	(8+4+2+1)/13 = 15/13
14	13/16	N	0	(8+4+2+1)/14 = 15/14
15	14/16	N	0	(8+4+2+1)/15 = 15/15
16	15/16	N	0	(8+4+2+1)/16 = 15/16

In general
- $T(copy) = 1 + 2 + 4 + 8 + ... + 2^k$
  - Where $k = \lceil \log_2(n)\rceil$
- $T(copy) = 2^{k+1}-1$
- But we can perform up to $2^{k+1}$ insertions.
- So the average price for copy is $\frac{2^{k+1}-1}{2^{k+1}}$
- This is O(1)
- So in general, if we double the array every time we need more space, copy for insert averages out to be a constant time operation.
Since the other part of insert (lines 6 and 7) is constant, the entire insert, when averaged, is constant.