## Monday, 20 February 2017

### Graphs in Python

In one of the post from series Python Data Structures I promised, that graphs come to the family, so here they are.
Design, I decided to adjacency list implementation, as underlying data structures there are: a list of vertices and list of lists (list of adjacent vertices for any given). I could have used a symbol table (more operation), but decided to keep things as simple as possible (graph algorithms are complicted on its own). A sample of code:

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90``` ```# Simple graph API in Python, implementation uses adjacent lists. # look also here: # Classes: Graph, Depth_first_search, Depth_first_paths # Usage: # Creating new graph: gr1 = Graph(v) - creates new # graph with no edges and v vertices; # Search object: gr2 = Depth_first_search(graph, vertex) - # creates search object, # gr2.marked_vertex(vertex) - returns true if # given vertex is reachable from source(above) # Path object: gr3 = Depth_first_paths(graph, vertex)- # creates a new path object, # gr3.has_path(vertex) - thee same as above # gr3.path_to(vertex) - returns path from source vertex (to the given) from collections import deque class Graph: """Class graph, creates a graph - described by integers - number of vertices - V : v0, v1, ..., v(V-1)""" def __init__(self, v_in): """constructor - takes number of vertices and creates a graph with no edges (E = 0) and an empty adjacent lists of vertices""" self.V = v_in self.E = 0 self.adj = [] for i in range(v_in): self.adj.append([]) def V(self): """returns number of vertices""" return self.V def E(self): """returns number of edges""" return self.E def add_edge(self, v, w): """adds an edge to the graph, takes two integers (two vertices) and creates an edge v,w - by modifying appropriate adjacent lists """ self.adj[v].append(w) self.adj[w].append(v) self.E += 1 def adj_list(self, v): """Takes an integer - a graph vertex and returns the adjacency lists of it""" return self.adj[v] def __str__(self): """to string method, prints the graph""" s = str(self.V) + " vertices, " + str(self.E) + " edges\n" for v in range(self.V): s += str(v) + ": " for w in self.adj[v]: s += str(w) + " " s += "\n" return s class Depth_first_search: """class depth forst search, creates an object, constructor takes graph and a vertex""" def __init__(self, gr_obj, v_obj): self.marked = [False] * gr_obj.V self.cnt = 0 self.__dfs(gr_obj, v_obj) def __dfs(self, gr, v): """private depth first search, proceed recursively, mutates marked - marks the all possible to reach from given (v) vertices; also mutates cnt - number of visited vert""" self.marked[v] = True self.cnt += 1 for w in gr.adj_list(v): if self.marked[w] == False: self.__dfs(gr, w) def marked_vertex(self, w): """Takes an integer - a graph vertex and returns True if it's reachable from vertex v (source)""" return self.marked[w] def count(self): """returns number of visited verticles (from given in the constructor vertex)""" return self.cnt ```

Why I did it in the java style? (Additional classes to given graph opertions) Primarly to avoid set and mutate global variables (in that case cnt in class Depth_first_search, I don't know how to solve it using function no object :/), also there is lots of graph tasks and operatons, and, imo, this approach is more clear (Did I really write this?:)). There are  going to be updates on this definitely! Code obviously, also on github. Thanks, till the next time!

## Thursday, 9 February 2017

### Bit Hacks in Go

Some time ago I wrote about little bit tweddlings in C, I, recently, have tried something similar in GO. It's not easy, for example bit shifts (>>) works only for unsigned integers and other differences. For now I reproduced and tested two things (both work on unsigned ints):
Integer mean without casing an overflow and exponenation by binary decomposition.

```
1
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func iexp(x uint32, n uint32) uint32 {
if (n == 0) {
return 1
}
var p, y uint32
y = 1;
p = x;
for {
if (n & 1 != 0) {y = p*y}
n >>= 1;
if (n == 0) {return y}
p *= p;
}
return 0
}

```
Above is about 4 to 5 times faster than math.Pow from library.
Average:

 ```1 2 3``` ```func average(x uint32, y uint32) uint32 { return (x & y) + ((x ^ y) >> 1) } ```

This time the adventage is not speed, but the fact, that we have the average without overflow (works,  for ex., for:, 2^32 - 1 and 2^32 - 1).
That's it, code also on github, till the next time!

## Tuesday, 7 February 2017

### Python Pollard's rho Algorithm

I've recently was looking for some number theoretic algorithms in Python and Go. While searching, found this  on the first position on duckduckgo and google but it's not really good. Even for example used (1200), gives wrong answer (missing factor 600).
Applying small fix:

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```def pollard_rho(n): s = set() i = 0 xi = randint(0, n-1) y = xi k = 2 while i < 2 * n: i += 1 xi = ((xi^2) - 1)%n d = gcd(y - xi, n) if d != 1 and d != n: s.add(d) if i == k: y = xi k *= 2 return sorted(s) ```

Where gcd is Greater Common Divisor, 2 * n in line 7, makes it work correctly, but it's still slow.
Anybody needs more efficient factorization (and more), I would recommend this library.

## Friday, 3 February 2017

### Python Inverted Index Algorithm

As a continuation of interesting algorithms, today another text processing tool: Inverted Index.
How it works, briefly. Let's say, that we have a list of documents, maybe large: thousands books, articles or so and we want effectively query this list, looking for information. It's done in the way, that the corpora is saved on disk (or we can use database) and we create a special data structure to do search and then retrieve matched documents (the database part is not the part of this article, I hope I will implement it soon in this project).
I used Python's dicts and sets, the best would be  go through this:
I have 2 tests documents:

 ```1 2 3 4 5``` ```d1 = ['i', 'did', 'enact', 'julius', 'caesar', 'i', 'was', 'kill' , 'i', 'the', 'capitol', 'brutus','me'] d1 = set(d1) d2 = ['i', 'so', 'let', 'it', 'be', 'with' , 'caesar', 'the', 'noble', 'brutus', 'hath', 'told', 'you', 'caesar', 'was', 'ambitious'] d2 = set(d2) ```

After tokenization, we may have:

```d1 = ['i', 'did', 'enact', 'julius', 'caesar', 'i', 'was', 'kill' , 'i', 'the', 'capitol',  'brutus','me']
d1 = set(d1)
d2 = ['i', 'so', 'let', 'it', 'be', 'with' , 'caesar', 'the', 'noble', 'brutus', 'hath', 'told', 'you',
'caesar', 'was', 'ambitious']
d2 = set(d2)
```

Now create a, to be searched, documents list.

 ```1 2 3``` ```doc_list = [] doc_list.append(d1) doc_list.append(d2) ```

For really big corporas, this (doc_list) should be a function-  fetched and prepared document one by one to create the inverted list structure.

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20``` ```tokens = d1.union(d2) inv_index = {} for x in tokens: tmp, tmp1 = find(x, doc_list) inv_index[(x, tmp1)] = tmp def find(elem, d_list): l_return = [] i = 0 for x in d_list: if elem in x: l_return.append(i) i += 1 return (l_return, len(l_return)) def find_key(x, d): for y in d.keys(): if x in y: return y return "word not in a texts" ```

Tokens is a set of the all unique words in the all documents, as a keys in the inv_index dict there are tuples: token, frequency: in how many documents the word is present - this is the length of an actual value of the key. Values: as 've said, for every key word it is  a list of documents which contains this word, it's implemented here as a list of integers - zero for the first document, one for the next and so on. The functions find and find_key are just helping to create the dict. Searches, in the documents, are now boolean queries on our dict values, for example:

 `1` ```('cat' AND 'lion') AND NOT 'mouse' ```

In fact I'll do it in slightly different manner, I will parse it and use own, modified Propositional Logic Parser. Here are boolean functions:

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25``` ```def NOT(l1): indexes = l1 ret_list = [] for x in range(len(doc_list)): if x not in indexes: ret_list.append(x) return ret_list def OR(l1, l2): tmp0 = set(l1) tmp1 = set(l2) indexes = tmp0.union(tmp1) ret_list = [] for x in indexes: ret_list.append(x) return ret_list def AND(l1, l2): tmp0 = set(l1) tmp1 = set(l2) indexes = tmp0.intersection(tmp1) ret_list = [] for x in indexes: ret_list.append(x) return ret_list ```

This is the parser part:

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47``` ```from porpositional_logic_eval import * import re def query_tree_evaluate(tree): opers = {'||': OR, '&&': AND, '~': NOT} leftT = tree.getLeftChild() rightT = tree.getRightChild() #pdb.set_trace() if leftT and not rightT: fn = opers[tree.getRootVal()] return fn(query_tree_evaluate(leftT)) elif leftT and rightT: fn = opers[tree.getRootVal()] return fn(query_tree_evaluate(leftT), query_tree_evaluate(rightT)) else: return tree.getRootVal() def build_query_parse_tree(exp): exp_list = exp.replace('(', ' ( ').replace(')', ' ) ').replace('~', ' ~ ').split() e_tree = BinaryTree('') current_tree = e_tree for token in exp_list: if token == '(': current_tree.insertLeft('') current_tree = current_tree.getLeftChild() elif token in ['||','&&', '->', '==', 'XR']: if current_tree.getRootVal() == '~': current_tree.getParent().setRootVal(token) current_tree.insertRight('') current_tree = current_tree.getRightChild() else: current_tree.setRootVal(token) current_tree.insertRight('') current_tree = current_tree.getRightChild() elif token == '~': current_tree.setRootVal('~') current_tree.insertLeft('') current_tree = current_tree.getLeftChild() elif token == ')': current_tree = current_tree.getParent() elif re.search('[a-zA-z]', token): current_tree.setRootVal(inv_index[find_key(token, inv_index)]) current_tree = current_tree.getParent() if current_tree.getRootVal() == '~': current_tree = current_tree.getParent() else: raise ValueError return e_tree ```

Finally, checking how it works, query is:

 `1` ```exp = "((i && caesar) && ~julius)" ```

Making a tree:

 ```1 2 3 4 5 6 7 8 9``` ```tr = build_query_parse_tree(exp) inorder_traversal(tr) # output: [0, 1] && [0, 1] && [0] ~ ```

As seen parser does the job, and asking it:

 ```1 2 3``` ```query_tree_evaluate(tr) # output: [1] ```

Gives as what we expected: the second document (contains 'caesar' and 'i', but not 'julius').     Coplexity, as seen is linear, so it should be pretty quick.
Hopefully updates here soon! Code, as usually on github. Thank you!

## Monday, 23 January 2017

### Text Segmentation (Maximum Matching) in Python

Today another algorithm in the set Algorithms in Python, part one here - maximum matching - it's a text segmentation algorithm - separates word in a  text, with laguages with no clear word separator, like Chinesse. It's simple, that's why works only for short words texts, again, an example is  Chinesse. This is a code:

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```D = ['danny', 'condo', 'a', 'the','to','has', 'been', 'unable', 'go', 'at'] def max_match(sentence, dictionary): if not sentence: return "" for i in range(len(sentence), -1, -1): first_word = sentence[:i] remainder = sentence[i:] if first_word in dictionary: return first_word + " " + max_match(remainder, dictionary) first_word = sentence[0] remainder = sentence[1:] return first_word + max_match(remainder, dictionary) print(max_match('atcondogo', D))```

Let's go through it, dictionary is a simple Python list, algorithm starts at the beginning of a input string and try to match, with the dictionary, the longest word - line 8 (in that case this is a whole string and the reminder is empty), if it succeed returns first word with added space bar separator and recursive call from the remainder; if not, pointer goes one char backwards and  so on in this loop. If no word matches, pointer advances one character (creates one character word), and is recursively applied to the rest of the string - lines 12 - 14.
Line 16 corectrly prints:

 `1` `"at condo go" `
But if we add the word 'atcon' to the dictionary, algorithm incorrectly segments to:

 `1` ```"atcon dogo" ```

Not very usefull to English though. That's it, code of course goes to github, together with: Extended Euclicid, modular exponeneation and the longest palindrome finder. Enjoy!

## Thursday, 19 January 2017

### Algorithms in Python

I've decided to start another repo: Python Algorithms, during my programming works I've collected some amount of code which is stacked in different places as a useful functions, algorithms, even Project Euler Helper - designed to only this website. I'm going to slowly check, test that stuff (my goal is, that this is suppose to be solid and ready to work code) and put it on github. Let's bring some algorithms then!

First edit distance:

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21``` ```def edit_dist_dp(str1, str2, m, n): store = [[0 for x in range(n + 1)] for x in range(m + 1)] for i in range(m + 1): for j in range(n + 1): if i == 0: store[i][j] = j elif j == 0: store[i][j] = i elif str1[i - 1] == str2[j - 1]: store[i][j] = store[i - 1][j - 1] else: store[i][j] = 1 + min(store[i][j - 1], store[i - 1][j], store[i - 1][j - 1]) return store[m][n] ```

This is dynamic programming version, naive, recursive is extremely ineffective. What is does is, according to dp paradigm, first it creates an array to store temporary values (line 2, where m and n are input strings lengths accordingly) and then in a loop starts filling it bottom up, i.e. counting minimum edit distance.
If the first string is empty we have only insert all the characters of the second string (lines 7, 8, #j operations). If the second string is empty, than we insert all the first string characters (line 13, 14, time complexity i).
If the last characters are the same we ignore them and move to remaining string(lines 12, 14).
If the last characters are different, we have to rise distance by at least one plus recursively find minimum between the all possibilities: Insert, remove and replace (lines 17 to 20). Complexity of the whole thing is O(m*n).

Another is simple, but nice algorithm to shuffle elements of a given array.

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```# algorithm for shuffling given subscriptable data structure (Knuth) import random def swap(alist, i, j): """swaps input lists i, j elements""" alist[i], alist[j] = alist[j], alist[i] def shuffle(data): """randomly shuffles element in the input data""" n = len(data) for token in range(n - 1): swap(data, token, random.randrange(token, n)) ```

Maybe it's obvious (in that case sorry for that:)), but I think the reason for that algorithm being published is, that is easy to make a mistake and write the last line like this:

 `1` ```swap(data, token, random.randrange(token, n)) ```

Which is definitely no good:)

Thats it for now, more stuff on github, incuding clasics merge sort, quicksort and binary search; obvioususly more algorithms will come, stay in tuch!

## Friday, 13 January 2017

### Web Page Crawler in Python

Again, here is another jupyter notebook guest post - this time it's simple recursive web crawler and html parser (using n ary trees). Code also here. Cheers!
PS Python Data Structures updated as well