sandbox/notes/chapter1.txt

# Chapter 1

## Efficiency

Concerns:
1. Number of operations
2. Processor speeds
3. Storage space

## Interfaces

* Interface / abstract data type

### Queue interface

* `add(x)` (aka `queue`): add `x` to the queue
* `remove()` (aka `dequeue`): remove the next value from queue and return it

* Normal queue: the first element inserted is removed first
* Priority queue: elements are inserted with a priority, and the smallest
  element is removed. This function is usually called `deleteMin`.
* LIFO queue: a stack; add and remove are called `push` and `pop`.
* Deque: generalisation of these
  * `addFirst(x)`
  * `removeFirst(x)`
  * `addLast(x)`
  * `removeLast(x)`
* Stack: addFirst, removeFirst
* Queue: addLast, removeFirst

### List interface

The List interface subsumes the Queue interface. A list is just a sequence of
values, and a Queue becomes a special case of it.

Interface:

* size()
* get(i): get i'th element
* set(i, x): set the i'th element to x
* add(i, x): insert x at position i
* remove(i): remove the i'th element

### USet (unordered sets)

USets are a collection of unique items in no particular order; this mimics a
mathematical set.

Interface:

* `size()`: returns the number of elements in the set
* `add(x)`: add x to the set if it doesn't already exist
* `remove(x)`: remove x from the set if it doesn't already exist
* `find(y)`: membership test

Note that y and x may be distinct objects, and only need to satisfy an
equality test. For example, a dictionary or hashmap is created using a tuple
`(key, value)`; `find` compares on `key` and two objects are considered equal
if their keys match.

### SSet (sorted set)

A USet where order matters. Its interface only changes in the `find` function:

* `find(x)`: find the smallest y s.t. y >= x. thereby returning a useful value
  even if x isn't in the set. AKA successor search.

Difference between USet and SSet: sorting requires more steps (run time) and
complexity. A USet should be used unless an SSet is explicitly required.

## Mathematical background

(See notebook).

## The model of computation

Proper analysis requires a mathematical model of computation. The model in the
book is on a w-bit word-RAM model.

* we can access cells of memory, each of which stores a w-bit word
* basic operations (arithmetic and logical) take constant time
* cells can be read or written in constant time
* the memory manager allows allocating a block of k cells of memory in O(k)
  time
* size constraint: w >= log(n) where n is the number of elements stored in a
  data structure
* data structures use a generic type T such that T occupies one word

## Correctness, time complexity, and space complexity

Three factors for analysing a data structure:

* correctness: data structure must implement the interface
* time complexity: run times of operations on the data structure should
  be as small as possible
* space complexity: the storage space used by a data structure should be
  as small as possible

Run times come in three flavours:

1. Worst-case: an operation never takes longer than this
2. Amortized: if a data structure has an amortized run time of f(n), then
   a sequence of m operations takes at most m f(n) time.
3. Expected: the actual run time is a random variable, and the expected
   value of this run time is at most f(n).

## Exercises

1. See src/ch01ex01.cc --- note that the last three exercises were skipped for
   time.

2. A Dyck word is a sequence of +1’s and -1’s with the property that the
   sum of any prefix of the sequence is never negative. For example,
   +1,−1,+1,−1 is a Dyck word, but +1,−1,−1,+1 is not a Dyck word since the
   prefix +1 − 1 − 1 < 0. Describe any relationship between Dyck words and
   Stack push(x) and pop() operations.

   A +1 corresponds to a push, and a -1 corresponds to a pop. At any point,
   the stack must not overflow.

3. A matched string is a sequence of {, }, (, ), [, and ] characters that are
   properly matched. For example, “{{()[]}}” is a matched string, but this
   “{{()]}” is not, since the second { is matched with a ]. Show how to use a
   stack so that, given a string of length n, you can determine if it is a
   matched string in O(n) time.

   The program should push each opening character onto a stack. When a closing
   character is encountered, the top of the stack should be the matching
   opening character. See src/ch01ex03.cc.