Thoughts In Free Time
Thursday, 23 March 2017
C++ GMP Wrapper
Recently beeing busy  writing, in C++, wrapper for GMP Library. Code here  you are invited to improve it!:)
Wednesday, 8 March 2017
Scala Trees
Recently, I've been playing with functional programming, ADT to be precise, and this is the outcome  trees implementation in Scala.
Data are created in common way as a Scala trait:
All looks Okay, lets create some trees and functions.
No problem, but at least for integer binary trees, we can right now do better: create a function to join elements to the tree:
This is the standard way to add elements to a binary tree, we can use it in a loop and don't need to worry about memory  we use persistent data structure.
Two sample functions operate on a tree:
They do what their names promise: count nodes and maximum depth of the trees. Finally map over the tree:
Data are created in common way as a Scala trait:
sealed trait Tree[+A] case object EmptyTree extends Tree[Nothing] case class Node[A](value: A, left: Tree[A], right: Tree[A]) extends Tree[A]
All looks Okay, lets create some trees and functions.
val tree = Node(10, Node(5, EmptyTree, EmptyTree), Node(15, EmptyTree, EmptyTree))
No problem, but at least for integer binary trees, we can right now do better: create a function to join elements to the tree:
def addToTree(elem: Int, tree: Tree[Int]): Tree[Int] = { if (isTreeEmpty(tree)) Node(elem, EmptyTree, EmptyTree) else if (elem == loadTree(tree)) tree else if (elem < loadTree(tree)) Node(loadTree(tree), addToTree(elem, leftBranch(tree)), rightBranch(tree)) else Node(loadTree(tree), leftBranch(tree), addToTree(elem, rightBranch(tree))) }
This is the standard way to add elements to a binary tree, we can use it in a loop and don't need to worry about memory  we use persistent data structure.
Two sample functions operate on a tree:
def length[A](tree: Tree2[A]): Int = tree match { case EmptyTree => 0 case Node(x, left, right) => length2(left) + 1 + length2(right) }
def maxDepth[A](tree: Tree2[A]): Int = tree match { case EmptyTree => 0 case Node(x, left, right) => var lmax = maxDepth2(left) var rmax = maxDepth2(right) if (lmax > rmax) lmax + 1 else rmax + 1 }
They do what their names promise: count nodes and maximum depth of the trees. Finally map over the tree:
def treeMap[A, B](tree: Tree2[A])(f: A => B): Tree2[B] = tree match { case EmptyTree => EmptyTree case Node(x, left, right) => Node(f(x), tree2Map(left)(f), tree2Map(right)(f)) }We can, in a similar way, all the needed function like foldl, traversing a tree and more. Code, including not implemented here functions helped to design joining element to tree, as usually on github. Till the next time!
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:
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!
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(V1)""" 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.
Above is about 4 to 5 times faster than math.Pow from library.
Average:
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!
Integer mean without casing an overflow and exponenation by binary decomposition.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  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 } 
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:
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.
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, n1) 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:
After tokenization, we may have:
Now create a, to be searched, documents list.
For really big corporas, this (doc_list) should be a function fetched and prepared document one by one to create the inverted list structure.
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:
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:
This is the parser part:
Finally, checking how it works, query is:
Making a tree:
As seen parser does the job, and asking it:
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!
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('[azAz]', 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:
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:
But if we add the word 'atcon' to the dictionary, algorithm incorrectly segments to:
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!
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"

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!
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