Merge pull request #13 from NicolaBernini/combinatorics_20190422_1032_1

Combinatorics 20190422 1032 1

combinatorics/generation/permutations/permutations1.cpp

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+#include <iostream>
+#include <vector>
+using namespace std;
+
+// remove_item :: [Int] -> Int -> [Int]
+vector<int> remove_item (const vector<int>& a, const unsigned int id)
+{
+	vector<int> res; 
+	for(unsigned int i=0; i<a.size(); ++i) if(i!=id) res.push_back(a[i]); 
+	return res;
+}
+
+
+// add_item :: [Int] -> Int -> [Int]
+vector<int> add_item(const vector<int>& a, const int b)
+{
+	vector<int> res=a; 
+	res.push_back(b); 
+	return res;
+}
+
+vector< vector<int> > merge(const vector< vector<int> >& a, const vector< vector<int> >& b)
+{
+	vector< vector<int> > res=a; 
+	for(const auto& e : b) res.push_back(e); 
+	return res;
+}
+
+
+// perm_comp :: [Int] -> [Int] -> [[Int]]
+// Rel 1:N : at each layer, choose N parallel paths 
+// Design 
+// Complexity needs to be O(N!) = O(N*(N-1)...) so Recursive 
+// State / Args 
+// - Partial Items List 
+// - Remaining Items List 
+vector< vector<int> > perm_comp(const vector<int>& a, const vector<int>& b)
+{
+
+	if(b.size()==0) return { a }; 
+
+	vector< vector<int> > res; 
+	for(unsigned int i=0; i<b.size(); ++i) res=merge(res, perm_comp(add_item(a, b[i]), remove_item(b,i))); 
+	return res; 
+}
+
+
+vector< vector<int> > permutations(const vector<int>& v)
+{	
+	return perm_comp({}, v); 
+}
+
+int main() {
+	// your code goes here
+
+	auto res = permutations({1,2,3,4,5}); 
+	cout << "Sol Num = " << res.size() << endl; 
+	for(const auto& a : res) 
+	{
+		for(const auto& b : a) cout << to_string(b) << " "; 
+		cout << endl; 
+	}
+	return 0;
+}
+
+
+

combinatorics/generation/permutations/readme.md

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+
+# Overview 
+
+Some Permutations Computation related material 
+
+# Basic Permutation 
+
+## Problem Statement
+
+> Given a vector of int, compute all the possible vectors resulting from the permutation of the elements 
+
+## Solutions 
+
+Analysis 
+
+- Equivalent to drawing elements from a container 
+- Recursive + Iterative 
+  - Recursion i --> Recursion i+1 = Remove an element 
+  - Iterations inside a Recursion = Run one recursion for each of the remaining elements 
+
+
+Implementations 
+
+- [CPP](permutations1.cpp)
+
+
+
+
+
+Work in progress 
+

combinatorics/generation/readme.md

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+
+# Overview 
+
+The Generation related problems and solutions 
+
+
+
+
+
+Work in progress 
+

combinatorics/readme.md

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+
+# Overview 
+
+This section is related to Combinatorics Problems in Discrete Math hence including the following categories 
+
+- Generation 
+  - Examples 
+    - Compute all the possible strings resulting from the permutation of a given set of chars 
+    - Compute all the possible substrings for a given string 
+    - Compute all the possible submatrixes for a given matrix 
+
+- Enumaration / Counting 
+  - Examples 
+    - Count all the possible paths connecting 2 given points on a given graph 
+    - Count the number of prime numbers in a given interval 
+
+
+- Search 
+  - Examples 
+    - Search for a given element in a sorted array 
+    - Search a pair of numbers in a given array which sum up to a certain given value 
+
+
+Work in progress