#+TITLE: Notes & Exercises: The Algorith Design Manual
#+AUTHOR: Joseph Ferano
#+STARTUP: overview
#+OPTIONS: ^:{}

* Chapter 1

** 1.1 Robots

An /algorithm/ is a procedure that takes any of the possible input instances
and transforms it to the desired output. 

The Robot arm problem is presented where it is trying to solder contact points
and visit all points in the shortest path possible.

The first algorithm considered is ~NearestNeighbor~. However this is a naïve, and
the arm hopscotches around

Next we consider ~ClosestPair~, but that too misses in certain instances.

Next is ~OptimalTSP~, which will always give the correct result because it
enumerates all possible combinations and return the one with the shortest
path. For 20 points however, the algorithm grows at a right of O(n!). TSP stands
for Traveling Salesman Problem.

*** TODO Implement NearestNeighbor
*** TODO Implement ClosestPair
*** TODO Implement OptimalTSP for N < 8

** 1.2 Right Jobs
Here we are introduced to the Movie Scheduling Problem where we try to pick the
largest amount of mutually non-overlapping movies an actor can pick to maximize
their time. Algorithms considered are ~EarliestJobFirst~ to start as soon as
possible, and then ~ShortestJobFirst~ to be done with it the quickest, but both
fail to find optimal solutions.

~ExhaustiveScheduling~ grows at a rate of O(2^{n}) which is much better than O(n!) as
in the previous problem. Finally ~OptimalScheduling~ improves efficiency by first
removing candidates that are overlapping such that it doesn't even compare them.

** 1.3 Correctness

It's important to be clear about the steps in pseudocode when designing
algorithms on paper. There are important things to consider about algorithm
correctness;

- Verifiability
- Simplicity
- Think small
- Think exhaustively
- Hunt for the weakness
- Go for a tie
- Seek extremes

Other tecniques include *Induction*, *Recursion*, and *Summations*.

** 1.4 Modeling

Most algorithms are designed to work on rigorously defined abstract
structures. These fundamental structures include;

- Permutations
- Subsets
- Trees
- Graphs
- Points
- Polygons
- Strings

** 1.5-1.6 War Story about Psychics


* Chapter 2

** 2.1 RAM Model of Computation

This is a simpler kind of Big Oh where

- Each simple operation is 1 step
- Loops and Subroutines are composition of simple operations
- Each memory access is one time step

Like flat earth theory, in practice we use it when engineering certain
structures because we don't take into account the curvature of the Earth.

We can already apply the concept of worst, average, and best case to this model.

** 2.2 Big Oh

The previous model often requires concrete implementations to actually measure
correctly, so instead Big Oh gives us a better, simpler framework for discussing
the relative performance between algorithms. It ignores factors that don't
impact how algorithms scale.

** 2.3 Growth Rates and Dominance Relations

These are the functions that occur in algorithm analyses;

- *Constant O(1)*
    Hashtable look up, array look up, consing a list
- *Logarithmic O(log n)*
    Binary Search
- *Linear O(n)*
    Iterating over a list
- *Superlinear O(n log n)*
    Quicksort and Mergesort
- *Quadratic* O(n^{2})
    Insertion Sort and Selection Sort
- *Cubic* O(n^{3})
    Some dynamic programming problems
- *Exponential* O(C^{n}^{}) *c for any constant c > 1*
    Enumerate all subsets
- *Factorial O(n!)*
    Generating all permutations or orderings

*Notes*:
- O(n!) algorithms become useless for anything n >= 20
- O(2^{n}) algorithms become impractical for anything n > 40
- O(n^{2}^{}) algorithms start deteriorating after n > 10,000, a million is hopeless
- O(n^{2}^{}) and O(n log n) Are fine up to 1 billion

** 2.4 Working with Big Oh

Apparently you can do arithmetic on the Big Oh functions

** 2.5 Efficiency

*** Selection Sort
**** C

#+begin_src C :includes stdio.h
void print_nums(int *nums, int length) {
    for (int i = 0; i < length; i++) {
        printf("%d,", nums[i]);
    }
    printf("\n");
}

void selection_sort(int *nums, int length) {
    int i, j;
    int min_idx;
    for (i = 0; i < length; i++) {
        print_nums(nums, length);
        min_idx = i;
        for (j = i+1; j < length; j++) {
            if (nums[j] < nums[min_idx]) {
                min_idx = j;
            }
        }
        int temp = nums[min_idx];
        nums[min_idx] = nums[i];
        nums[i] = temp;
    }
}

int nums[9] = { 2, 4, 9, 1, 3, 8, 5, 7, 6 };
selection_sort(nums, 9);
#+end_src

#+RESULTS:
| 2 | 4 | 9 | 1 | 3 | 8 | 5 | 7 | 6 |   |
| 1 | 4 | 9 | 2 | 3 | 8 | 5 | 7 | 6 |   |
| 1 | 2 | 9 | 4 | 3 | 8 | 5 | 7 | 6 |   |
| 1 | 2 | 3 | 4 | 9 | 8 | 5 | 7 | 6 |   |
| 1 | 2 | 3 | 4 | 9 | 8 | 5 | 7 | 6 |   |
| 1 | 2 | 3 | 4 | 5 | 8 | 9 | 7 | 6 |   |
| 1 | 2 | 3 | 4 | 5 | 6 | 9 | 7 | 8 |   |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 9 | 8 |   |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |   |


*** Insertion Sort
**** C

#+begin_src C :includes stdio.h
void insertion_sort(int *nums, int len) {
    int i, j;
    for (i = 1; i < len; i++) {
        j = i;
        while (nums[j] < nums[j -1] && j > 0) {
            int temp = nums[j];
            nums[j] = nums[j - 1];
            nums[j - 1] = temp;
            j--;
        }
    }
}

int nums[8] = {1,4,5,2,8,3,7,9};
insertion_sort(nums, 8);
for (int i = 0; i < 8; i++) {
    printf("%d", nums[i]);
}
#+end_src

#+RESULTS:
: 12345789


*** TODO String Pattern Matching
*** TODO Matrix Multiplication

** 2.6 Logarithms

Logarithms are the inverse of exponents. Binary search is great for sorted
lists. There are applications related to fast exponentiation, binary trees,
harmonic numbers, and criminal sentencing.

** 2.7 Properties of Logarithms

Common bases for logarithms include 2, /e/, and 10. The base of the logarithm has
no real impact on the growth rate; log_{2} and log_{3} are roughly equivalent.

** 2.8 War Story Pyramids

Cool story bro

** 2.9 Advanced Analysis

Some advanced stuff
- *Inverse Ackerman's Function*
  Union-Find data structure
- *log log n*
  Binary search on a sorted array of only log n items
- *log n / log log n*
- log^{2} n
- \sqrt{,}n

There are also limits and dominance relations

* Chapter 3

** 3.1 Contiguous vs Linked Data Structures

Advantages of Arrays
- Constant-time access given the index
- Space efficiency
- Memory locality

Downsides is that they don't grow but dynamic arrays fix this by allocating a
new bigger array when needed.

Advantages of Linked Structures
- No overflow, can keep growing
- Insertions/Deletions are simpler
- A collection of pointers are lighter than contiguous data

However, pointers require extra space for storing pointer fields

** 3.2 Stacks and Queues

*** Stacks
/(PUSH, /POP/) LIFO, useful in executing recursive algorithms.

*** Queues
(/ENQUEUE/, /DEQUEUE/) FIFO, useful for breadth-first searches in graphs.

** 3.3 Dictionaries

Not just hashtables but anything that can provide access to data by
content. Some dictionaries implement trees instead of hashing. Both contiguous
and linked structures can be used with tradeoffs between them.

** 3.4 Binary Search Trees

BSTs have a parent and two child nodes; left and right. They support insertion,
deletion, traversal. Interestingly, Min and Max can be calculated by seeking the
leftmost and rightmost node respectively, provided the tree is balanced. BSTs
can have good performance for most cases so long as they remain balanced. O(h)
refers to the time being the height of the BST.

** 3.5 Priority Queues

They allow new elements to enter a system at arbitrary intervals.

** 3.6 War Story

Rather than storing all of the vertices of a mesh, you can share them between
the different triangles, but connecting all vertices requires visiting each
vertice once, a Hamiltonian path, but that's NP-Complete. Using a greedy
heuristic where it tries to always grab the best possible thing first. Then
using a priority queue, they were able to reduce the running time by several
orders of magnitude compared to the naïve approach.

** 3.7 Hasing and Strings

Take a map to a big int, use modulo to spin around, and if /m/ is a large prime
you'll get fairly uniform distribution. The two main ways to solve collisions
are /Chaining/ and /Open Addressing/. Chaining is where each bucket has a linked
list and collisions are appended. Open addressing looks for adjacent empty
buckets.

Hashing is also useful when dealing strings, in particular, substring pattern
matching. Overlaying pattern /p/ over every position in text /t/ would result in
O(m*n). With hashing, you can hash the slices of /t/ and compare them to /p/, and
get slower growth. This is called the *Rabin-Karp algorithm*. While
false-positives may occur, a good hashing function would avoid this.

Hashing is so important Yahoo! employs them extensively.

** 3.8 Specialized Data Structures

These include;

- String
  Characters in an array
- Geometric
  Collection of points and regions/polygons
- Graph
  Using adjacency matrices
- Set
  Dicionaries and bit vectors


** 3.9 War Story

They were trying to implement sequencing by hybridization (SBH), but ran into
issues when they used a BST. Then they tried a hashtable, then a trie. Finally
what worked was a compressed suffix tree.


** Exercises

*** 3.42

Reverse the words in a sentence—that is, “My name is Chris” becomes “Chris is
name My.” Optimize for time and space.

#+begin_src C :includes stdlib.h stdio.h string.h
void reverse_word(char *string, int length) {
    for (int i = 0; i < length / 2; i++) {
        char temp = string[i];
        string[i] = string[length - 1 - i];
        string[length - 1 - i] = temp;
    }
}

void reverse_words(char *string, int length) {
    printf("Before: %s\n", string);
    reverse_word(string, length);
    printf("After: %s\n", string);
    int start = 0;
    for (int i = 0; i < length; i++) {
        if (string[i] == ' ' || i == length - 1) {
            if (i == length - 1) i++;
            reverse_word(&string[start], i - start);
            start = i + 1;
        }
    }
}

char str[] = "My name is Chris";
reverse_words(str, strlen(str));
printf("Final: %s\n", str);
#+end_src

#+RESULTS:
| Before: | My    | name | is   | Chris |
| After:  | sirhC | si   | eman | yM    |
| Final:  | Chris | is   | name | My    |

* Chapter 4

** 4.1 Applications of Sorting

Apparently sorting is a big deal. There are a lot of problems that can be solved by using a sorted
list, for example; closest pair, searching, frequency distribution, convex hulls, etc;

The fastest sorting algorithms are ~n log n~, here is how it scales compared to an algorithm with
quadratic growth. Even crazier is that this table divides it by 4 so it's not that ridiculous.

| n       | n^{2}/4          | n lg n    |
|---------+---------------+-----------|
| 10      | 25            | 33        |
| 100     | 2,500         | 664       |
| 1,000   | 250,000       | 9,965     |
| 10,000  | 25,000,000    | 132,877   |
| 100,000 | 2,500,000,000 | 1,660,960 |

Whenever you have an algorithmic problem, don't be afraid to use sorting, because if it results in
the ability to then do a linear scan, at worst is becomes ~2n log n~, which in the end is just ~n log n~

** 4.2 Pragmatics of Sorting

There are some important things to consider when sorting; the order, keys and their data, equality,
and non-numerical data. For resolving these, we would use /comparison functions/. Here is the
signature of the C function for quicksort

#+begin_src C :include stdlib.h
void qsort(void *base, size_t nitems, size_t size, int (*compar)(const void *, const void*))
#+end_src

It takes a comparison function that might look like this;

#+begin_src C :include stdlib.h
int compare(int *i, int *j) {
    if (*i > *j) return 1;
    if (*i < *j) return -1;
    return 0;
}
#+end_src


** 4.3 Heapsort

An O(n^{2}) sorting algorithm like Selection Sort can be made faster by using the right data structure,
in this case either a heap or a balanced binary tree. The first initial construction will take
one O(n), but subsequent operations within the loop will now take O(log n) rather than O(n), giving
a final complexity of O(n log n).


*** Heaps

They can either be min or max heaps and the root node will dominate its children in min-ness or
max-ness. It's also different in that it can be implemented in an array and still be reasonably
conservative with it's space complexity. However it should be noted that searching isn't efficient
because the nodes aren't guaranteed to be ordered, only the relationship between the parent/child.

Here's the full heap implementation;

#+begin_src C :includes stdio.h stdlib.h
#define PQ_SIZE 256
#define item_type int

struct priority_queue {
    item_type *q;
    int len;
};

struct priority_queue *pq_create() {
    item_type *q = calloc(PQ_SIZE, sizeof(item_type));
    struct priority_queue *pq = malloc(sizeof(struct priority_queue));
    pq->q = q;
    pq->len = 0;
}

int pq_parent(int n) {
    if (n == 1) return -1;
    else return ((int) n/2);
}

int pq_young_child(int n) {
    return 2 * n;
}

// I mean technically we shouldn't need to provide the parent, since we can
// just call that ourselves
void pq_swap(struct priority_queue *pq, int n, int parent) {
    item_type item = pq->q[n];
    pq->q[n] = pq->q[parent];
    pq->q[parent] = item;
}

void pq_bubble_up(struct priority_queue *pq, int n) {
    int pidx = pq_parent(n);
    if (pidx == -1) {
        return;
    }
    item_type parent = pq->q[pidx];
    item_type node = pq->q[n];

    if (parent > node) {
        pq_swap(pq, n, pidx);
        pq_bubble_up(pq, pidx);
    }
}

void pq_bubble_down(struct priority_queue *pq, int n) {
    int cidx = pq_young_child(n);
    if (cidx > pq->len) {
        return;
    }
    item_type child = pq->q[cidx];
    item_type node = pq->q[n];
    int min_idx = n;

    if (cidx <= pq->len && node > child) {
        min_idx = cidx;
    }
    if (cidx + 1 <= pq->len && pq->q[min_idx] > pq->q[cidx + 1]) {
        min_idx = cidx + 1;
    }

    if (node > child) {
        pq_swap(pq, n, min_idx);
        pq_bubble_down(pq, min_idx);
    }
}

void pq_insert(struct priority_queue *pq, item_type x) {
    if (pq->len >= PQ_SIZE) {
        printf("Error: Priority Queue Overflow");
        return;
    }
    pq->q[++pq->len] = x;
    pq_bubble_up(pq, pq->len);
}

item_type pq_pop_top(struct priority_queue *pq) {
    if (pq->len == 0) {
        printf("Error: No elements in Priority Qeueu");
        return -1;
    }
    item_type top = pq->q[1];
    pq->q[1] = pq->q[pq->len--];
    pq_bubble_down(pq, 1);
    return top;
}

struct priority_queue* pq = pq_create();

#define ELEMENTS 8
item_type arr[ELEMENTS];

for (int i = 0; i < ELEMENTS; i++) {
    arr[i] = i + 1;
}

void reverse(item_type *arr) {
    for (int i = 0; i < ELEMENTS / 2; i++) {
        item_type temp = arr[i];
        arr[i] = arr[ELEMENTS - i - 1];
        arr[ELEMENTS - i - 1] = temp;
    }
}

void shuffle(item_type *arr) {
    for (int i = 0; i < ELEMENTS - 1; i++) {
        float r = (float)rand() / RAND_MAX;
        int idx = (int)(ELEMENTS * r);
        item_type temp = arr[idx];
        arr[idx] = arr[i];
        arr[i] = temp;
    }
}

void print_elems(item_type *arr) {
    for (int i = 0; i < ELEMENTS; i++) {
        printf("%d,", arr[i]);
        if (i == ELEMENTS - 1) {
            printf("\n");
        }
    }
}

void print_pq(struct priority_queue *pq) {
    for (int i = 1; i <= pq->len; i++) {
        printf("%d,", pq->q[i]);
        if (i == pq->len) {
            printf("\n");
        }
    }
}

reverse(arr);
for (int i = 0; i < ELEMENTS; i++) {
    pq_insert(pq, arr[i]);
}

print_pq(pq);

pq_pop_top(pq);

print_pq(pq);
#+end_src


** 4.4 War Story

Apparently calculating airline tickets is hard

** 4.5 Mergesort

Uses Divide-and-Conquer, recursively partitioning elements into two groups. It
takes O(n log n), however the space complexity is linear, because in the
following code, we have to allocate memory to construct a new sorted array.
Doing so in place doesn't work because you're effectively destroying the
previous sorting. However, when working with linked lists, no extra allocations
are required since you can just rearrange what the pointers point to.

#+begin_src C :includes stdio.h stdlib.h
int *merge_sort(int *array, int start, int len) {
    int *sorted = malloc(sizeof(int) * len);
    if (len <= 1) {
        sorted[0] = array[start];
        return sorted;
    }
    int half = (len + 1) / 2;
    int *sorted_l = merge_sort(array, start, half);
    int *sorted_r = merge_sort(array, start + half, len / 2);
    int size_r = len / 2;
    int size_l = half;
    int i = 0, ir = 0, il = 0;
    for (; i < len; i++) {
        if ((il >= size_l && ir < size_r) || (ir < size_r && sorted_l[il] > sorted_r[ir])) {
            sorted[i] = sorted_r[ir++];
        } else if (il < size_l) {
            sorted[i] = sorted_l[il++];
        }
    }
    free(sorted_l);
    free(sorted_r);
    return sorted;
}

#define AL 10
int array[AL] = { 'y','z','x','n','k','m','d','a','b','c' };
/* int array[AL] = { 9,8,6,7,4,5,2,3,1,0 }; */
/* int array[AL] = { 0,1,2,3,4,5,6,7,8,9 }; */

for (int i = 0; i < AL; i++) {
    printf("%c-", array[i]);
} printf("\n");

int *sorted = merge_sort(array, 0, AL);

for (int i = 0; i < AL; i++) {
    printf("%c-", sorted[i]);
}
#+end_src

#+RESULTS:
| y-z-x-n-k-m-d-a-b-c- |
| a-b-c-d-k-m-n-x-y-z- |

** 4.6 Quicksort

Quicksort depends on a randomly selected pivot in order to get O(n log n) in the
average case. This is because if you select the same index for the pivot each
time, there will always exist an arrangement of elements in an array that will
be the worst case and result in O(n^{2}).

The moral of the story is that randomization is helpful for improving
algorithms involved in sampling, hashing, and searching. The  nuts and bolts
problem is a good example; if you have /n/ different sized bolts and /n/
matching nuts, how long would it take to match them all. Well if you pick a bolt
randomly and make two piles based on whether they are smaller or larger, then
you effectively ran a quicksort and were able to get it done in O(n log n) time,
rather than having to test each bolt with each nut.

One important note about Quicksort and why it's preferred over Mergesort is
because apparently in real world benchmarks, it outperforms it 2-3x. This is
likely due to the extra space requirements of mergesort. In particular if you
have to allocate memory on the heap

#+begin_src C :includes stdio.h stdlib.h time.h
void swap(int *i, int *j) {
    int temp = *i;
    ,*i = *j;
    ,*j = temp;
}

void skiena_quicksort(int A[], int p, int q) {
    if(p >= q) return;

    swap(&A[p + (rand() % (q - p + 1))], &A[q]);   // PIVOT = A[q]
    int i = p - 1;
    for(int j = p; j <= q; j++) {
        if(A[j] <= A[q]) {
            swap(&A[++i], &A[j]);
        }
    }

    skiena_quicksort(A, p, i - 1);
    skiena_quicksort(A, i + 1, q);
}

void quicksort(int *array, int start, int end) {
    int len = end - start + 1;
    if (len <= 1) {
        return;
    }
    int r = rand() % len;
    int pivot = array[start + r];
    int mid = start;
    int s = mid;
    int h = end;
    while (mid <= h) {
        if (array[mid] < pivot) {
            swap(&array[mid++], &array[s++]);
        } else if (array[mid] > pivot) {
            swap(&array[mid], &array[h--]);
        } else {
            mid++;
            quicksort(array, start, s-1);
            quicksort(array, s+1, end);
            return;
        }
    }
    quicksort(array, start, s-1);
    quicksort(array, s+1, end);
}

#define LEN 10
/* int array[LEN] = {5,4,3,2,1}; */
int array[LEN] = {5,4,3,2,3,2,1,1,8,6};

for (int i = 0; i < LEN; i++) {
    printf("%i", array[i]);
} printf("\n");

/* skiena_quicksort(array, 0, LEN - 1); */
quicksort(array, 0, LEN - 1);

for (int i = 0; i < LEN; i++) {
    printf("%i", array[i]);
}
#+end_src

Usually in higher level languages like Python and Haskell, you would allocate
new arrays to hold the new sorted elements. However, it becomes more challenging
when doing it in place. This algorithm requires 3 pointers to keep track of the
mid point, the iterator, and the high, then finish once mid passes h. 

** 4.7 Distribution Sort: Bucketing

Two other sorting algorithms function similarly by subdividing the sorting
groups; bucketsort and distribution sort. The example given is that of a
phonebook. However, these are more heuristic and don't guarantee good
performance if the distribution of the data is not fairly uniform.

** 4.8 War Story

Apparently serving as an expert witness is quite interesting. That and hardware
is the platform.

** 4.9 Binary Search and Related Algorithms

If anyone ever asks you to play 20 questions where you have to guess a word,
just use binary search and before 20 tries you will have narroed down the
correct word. Counting occurences can have pretty good time if you want to have
constant space growth; just sort the array then count the repeat ocurrences in a
contiguous sequence. There's also a one-sided binary search in case you're
dealing with lazy sequences, where you search first by A[1], then A[2], A[4],
A[8], A[16], and so forth. You can also use these sorts of bisections on square
root problems, whatever those are.

** 4.10 Divide-and-Conquer

Algorithms that use this technique, Mergesort being the classic example, have
other important applications such as Fourier transforms and Strassen's matrix
multiplication algorithm. What's important is understanding recurrence
relations.

*** Recurrence Relations

It is an equation that is defined in terms of itself, so I guess recursive.
Fibonacci is a good example F_{n} = F_{n-1} + F_{n-2}...

* Chapter 5