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#+AUTHOR: Joseph Ferano
#+STARTUP: overview
#+OPTIONS: ^:{}
* Chapter 1 - Introduction
** 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 - Algorithm Analyses
** 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
**** OCaml
Really reinventing the wheel on this one...
#+begin_src ocaml
let max_element = function
| [] -> invalid_arg "empty list"
| x::xs ->
let rec f acc = function
| [] -> acc
| x::xs -> f (if x > acc then x else acc) xs
in f x xs
let remove item list =
let rec f acc item = function
| [] -> List.rev acc
| x::xs -> if item = x then (List.rev acc) @ xs else f (x::acc) item xs
in f [] item list
let selection_sort list =
let rec f acc = function
| [] -> acc
| xs ->
let m = max xs
in f (m::acc) (remove m xs)
in f [] list
#+end_src
**** Python
#+begin_src python
def selection_sort(arr):
sorted_list = []
for i in range(len(arr)):
max = arr[0]
for count, value in enumerate(arr):
if value > max:
max = value
sorted_list.append(max)
arr.remove(max)
return sorted_list
selection_sort([2,1,5,3,4])
#+end_src
**** 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
*** 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 - Data Structures
** 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.
#+begin_src C :includes stdlib.h stdio.h stdbool.h
typedef struct Node {
struct Node* next;
/* void *data; */
int data;
} Node;
typedef struct Stack {
Node* top;
int length;
} Stack;
/* void stack_create(void* data) { */
Stack *stack_create(int data) {
Stack *stack = malloc(sizeof(Stack));
Node *node = malloc(sizeof(Node));
node->data = data;
node->next = NULL;
stack->top = node;
stack->length = 1;
return stack;
}
int stack_pop(Stack *stack) {
Node *top = stack->top;
Node *next = top->next;
if (stack->length > 0 && next != NULL) {
stack->top = next;
stack->length--;
int value = top->data;
free(top);
return value;
} else {
// A better API design would be to return a bool and the int as a pointer
// Although once we switch away from int and use void pointers, might not be needed
return -1;
}
}
/* void stack_push(Stack *stack, void *data) { */
void stack_push(Stack *stack, int data) {
Node *node = malloc(sizeof(Node));
Node *top = stack->top;
node->data = data;
node->next = top;
stack->top = node;
stack->length++;
}
/* void* stack_peak(Stack *stack) { */
int stack_peak(Stack *stack) {
return stack->top->data;
}
void stack_print(Stack *stack) {
Node *current = stack->top;
int i = 0;
while (current != NULL) {
printf("Stack at %d: %d\n", ++i, current->data);
current = current->next;
}
printf("------------------\n");
}
#+end_src
*** Queues
(/ENQUEUE/, /DEQUEUE/) FIFO, useful for breadth-first searches in graphs.
#+name: queue
#+begin_src C :includes stdio.h stdlib.h
#define QBUFSIZE 64
#define T int
struct queue {
T buf[QBUFSIZE];
int start;
int end;
int size;
};
struct queue *q_create() {
struct queue *q = calloc(1, sizeof(struct queue));
q->start = 0;
q->end = 0;
}
void q_enqueue(struct queue *q, T item) {
if (q->size == QBUFSIZE) {
printf("Queue Overflow");
exit(1);
}
q->buf[q->end] = item;
q->end = ++q->end % QBUFSIZE;
q->size++;
}
T q_dequeue(struct queue *q) {
if (q->size == 0) {
printf("Queue empty");
exit(1);
}
T item = q->buf[q->start++];
q->size--;
return item;
}
T q_peek(struct queue *q) {
if (q->size == 0) {
printf("Queue empty");
exit(1);
}
return q->buf[q->start];
}
bool q_empty(struct queue *q) {
return q->size <= 0;
}
void q_print(struct queue *q) {
printf("Qeueu_Elements: ");
for (int i = 0; i < q->size; i++) {
printf("%i-", q->buf[(i + q->start) % QBUFSIZE]);
}
printf("\n");
}
// struct queue *q = q_create();
// q_enqueue(q, 1);
// q_enqueue(q, 2);
// q_enqueue(q, 3);
// q_enqueue(q, 4);
// q_dequeue(q);
// q_dequeue(q);
// q_enqueue(q, 5);
// q_enqueue(q, 6);
// q_print(q);
#+end_src
** 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 - Sorting and Searching
** 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 item_type int
struct priority_queue {
// TODO: Why did I make this a pointer?
item_type *q;
int len;
int capacity;
};
struct priority_queue *pq_create(int capacity) {
item_type *q = calloc(capacity, sizeof(item_type));
struct priority_queue *pq = malloc(sizeof(struct priority_queue));
pq->q = q;
pq->len = 0;
pq->capacity = capacity;
}
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->capacity) {
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 the price of 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.
*** Python
#+begin_src python
import random
def quicksort(arr):
if len(arr) < 2:
return arr
elif len(arr) == 2:
if arr[0] > arr[1]:
temp = arr[1]
arr[1] = arr[0]
arr[0] = temp
return arr
else:
# Pick a random pivot
index = random.randrange(0, len(arr))
pivot = arr.pop(index)
left = [x for x in arr if x <= pivot]
right = [x for x in arr if x > pivot]
return quicksort(left) + [pivot] + quicksort(right)
#+end_src
** 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.
Here's a simple binary search guessing game;
#+begin_src ocaml
let binary_search items target =
let rec f low high =
match (high - low) / 2 + low with
| mid when target = items.(mid) -> Some items.(mid)
| mid when target < items.(mid) -> f low mid
| mid when target > items.(mid) -> f mid high
| _ -> None
in f 0 (Array.length items);;
binary_search [|1;2;3;4;5|] 3;;
#+end_src
** 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 - Graph Traversal
** 5.1 Graphs
Graphs are G = (E,V), so a set of edges and a set of vertices. Many real world
systems can be modeled as graphs, such as road/city networks. Many algorithmic
problems become much simpler when modeled with graphs. There are several flavors
of graphs;
- Directed vs Undirected
- Weighted vs Unweighted
- Simple vs Non-simple
- Spares vs Dense
- Cyclic vs Acyclic
- Embedded vs Topological
- Implicted vs Explicit
- Labeled vs Unlabeled
Social networks provide an interesting way to analyze each one of these
considerations.
** 5.2 Data Structures for Graphs
Which data structure we use will impact the time and space complexity of certain
operations.
- Adjecency Matrix
You can use an /n/ x /m/ matrix, where each /(i,j)/ index answers whether an edge
exists. However, with sparse graphs, there might be a lot of wasted space.
- Adjencency Lists
Use linked lists instead, however it is harder to verify if a certain edge
exists. This can be mitigated by collecting them in a BFS of DFS.
#+name: graph
#+begin_src C :includes stdio.h stdlib.h stdbool.h
#define MAXV 1000
struct edgenode {
int y;
int weight;
struct edgenode *next;
};
struct graph {
struct edgenode *edges[MAXV];
int degree[MAXV];
int nvertices;
int nedges;
bool directed;
};
void initialize_graph(struct graph *g, bool directed) {
g->nvertices = 0;
g->nedges = 0;
g->directed = directed;
for (int i = 0; i < MAXV; i++) g->degree[i] = 0;
for (int i = 0; i < MAXV; i++) g->edges[i] = NULL;
}
// TODO: We have to read whether the graph is directed or not and
// insert edges twice
void insert_edge(struct graph *g, int x, int y) {
struct edgenode *p;
p = malloc(sizeof(struct edgenode));
p->weight = 0;
p->y = y;
p->next = g->edges[x];
g->edges[x] = p;
g->degree[x]++;
g->nedges++;
}
void print_graph(struct graph *g) {
int i;
struct edgenode *p;
for (i = 0; i < g->nvertices; i++) {
printf("V%d", i+1);
p = g->edges[i];
while (p != NULL) {
printf("->%d", p->y);
p = p->next;
}
printf("\n");
}
}
struct graph *g = malloc(sizeof(struct graph));
initialize_graph(g, true);
g->nvertices = 3;
insert_edge(g, 0, 1);
insert_edge(g, 0, 2);
insert_edge(g, 0, 3);
insert_edge(g, 1, 1);
insert_edge(g, 1, 3);
insert_edge(g, 2, 1);
print_graph(g);
#+end_src
** 5.3 War Story
Apparently, computers were really slow before. That and it's better to keep
asymptotics in mind even if it's about the same.
** 5.4 War Story
Apparently loading data itself can be really slow.
** 5.5 Traversing a Graph
When traversing a graph, it's useful to keep track of the state of each vertex,
whether the vertex has been /undiscovered/, /discovered/, or /processed/.
** 5.6 Breadth-First Search (BFS)
This is a C implementation of BFS, however at the moment it doesn't really do
much. The important thing here is that a BFS uses a queue to visit every node in
the graph.
#+begin_src C :includes stdio.h stdlib.h stdbool.h :noweb yes
<<queue>>
<<graph>>
bool processed[QBUFSIZE];
bool discovered[QBUFSIZE];
int parent[QBUFSIZE];
void initialize_search(struct graph *g) {
for (int i = 0; i < g->nvertices; i++) {
processed[i] = discovered[i] = false;
parent[i] = -1;
}
}
void process_edge(T vertex, T edge) {}
void bfs(struct graph *g, int start) {
struct queue *q = q_create();
initialize_search(g);
q_enqueue(q, start);
discovered[start] = true;
while (!q_empty(q)) {
T v = q_dequeue(q);
processed[v] = true;
struct edgenode *p = g->edges[v];
while (p != NULL) {
T y = p->y;
if ((processed[y] == false) || g->directed) {
process_edge(v, y);
}
if (discovered[y] == false) {
q_enqueue(q, y);
discovered[y] = true;
parent[y] = v;
}
p = p->next;
}
}
free(q);
}
#+end_src
** 5.7 Applications of BFS
The one project we worked on, the "Six Degrees of Bacon", which is just a six
degrees of separation problem. However, one interesting point to the "six
degrees of separation" between all humans is that it assumes that all human
social networks are "connected", meaning that there may be vertices on the graph
that are not connected to the others and form their own little cluster. BFS
works well for this, since it's basically a shortest path problem.
Other applications include solving a Rubiks cube so in theory you should now
know enough to be able to create a solver, because each legal move up until the
solved move should be connected on the graph.
The vertex-coloring problem assigns each vertex a label or color such that no
edge links any two vertices of the same label/color, ideally using as few colors
as possible. A graph is /bipartite/ if it can be colored without conflicts with
just two colors.
** 5.8 Depth-first Search (DFS)
While BFS uses a queue, DFS uses a stack, but since recursion models a stack
already, we don't need an extra data structure. Here is the C implementation
mostly just copied from the book.
#+begin_src C :includes stdio.h stdlib.h stdbool.h :noweb yes
<<graph>>
void dfs(struct graph *g, int v) {
edgenode *p;
int y;
// No idea where this symbol comes from
// They seem to be global
if (finished) return;
discovered[v] = true;
// And these two as well
entry_time[v] = ++time;
p = g->edges[v];
while (p != NULL) {
y = p->y;
if (discovered[y] == false) {
parent[y] == v;
process_edge(v, y);
dfs(g, y);
} else if (!processed[y] || g->directed) {
process_edge(v, y);
if (finished) return;
p = p->next;
}
time = time + 1;
exit_time[v] = time;
processed[v] = true;
}
}
#+end_src
This uses a technique called Backtracking which still hasn't been explored, but
the idea is that it goes further down until it cannot process anything further,
then goes all the way back up to where it can start branching out again.
** 5.9 Applications of DFS
DFS is useful for finding articulation vertices, which are basically weak points
where removal will cause other vertices to become disconnected. It's also useful
for finding cycles.
** 5.10 DFS on Directed Graphs
The important operation here is topological sorting which is useful with
Directed Acyclic Graphs (DAGs). We can also check if a DAG is strongly
connected, meaning that we won't run into any dead ends since we cannot
backtrack. However, these graphs can be partitioned.
* Chapter 6 - Weighted Graph Algorithms
** 6.1 Minimum Spanning Trees
Weighted graphs open up a whole new universe of algorithms which tend to be a
bit more helpful in modeling real world stuff like roads. A minimum spanning
tree would have all vertices connected but uses the smallest weights for all its
edges.
Prim's Algorithm can be used to construct a spanning tree but because it is a
greedy algorithm. Kruskal's algorithm is a more efficient way to find MSTs with
the use of a /union-find/. This data structure is a /set partition/, which finds
disjointed subsets. These can also be used to solve other interesting problems;
- Maximum Spanning Trees
- Minimum Product Spanning Trees
- Minimum Bottleneck Spanning Tree
However, Steiner Tree and Low-degree Spanning Tree apparently cannot be solved
with the previous two algorithms.
** 6.2 War Story
Minimum Spanning Trees and its corresponding algorithms can help solve Traveling
Salesman Problems
** 6.3 Shortest Paths
In an unweighted graph, BFS will find the shortest path between two nodes. For
weighted graphs the shortest path might have many more edges. Dijstra's
algorithm can help us find the shortest path in a weighted path. It runs in
O(n^{2}) time. One caveat is that it does not work with negative weighted edges.
Another problem in this space is the all-pairs shortest path, and for this we
would use the O(n^{3}) Floyd-Warshall algorithm which uses an adjacency matrix
instead since we needed to construct one anyway to track all the possible pairs.
It's a useful algorithm for figuring out /transitive closure/ which is a fancy way
of asking if some vertices are reachable from a node.
*** TODO Dijstra's
#+begin_src C :includes stdio.h stdlib.h
#+end_src
** 6.4 War Story
Kids are probably too young to know what the author is even talking about with these
phone codes.
** 6.5 Network Flows and Bipartite Matching
Bipartite matching is where no two edges share a vertex. There are also residual
flow graphs which are useful for network bandwidth optimization. Flow algorithms
generally solve problems related to edge and vertex connectivity. Edmonds and
Karp runs at O(n^{3}).
*** TODO Network Flow
** 6.6 Design Graphs, Not Algorithms
Modeling problems as graphs is useful for a variety of applications; pathfinding
in games, DNA sequencing, smallest set of non-overlapping rectangles in a plain,
filename shortening, line segmentation for optical character-recognition, and
the list goes on.
* Chapter 7 - Combinatorial Search and Heuristic Methods
** 7.1 Backtracking
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