You can learn React, Django, Kubernetes and the latest tools, but programming logic and data structures remain the foundation. Understanding them helps you reason about problems instead of relying on a framework to supply every answer. I have seen developers with years of experience that cannot solve a basic algorithm problem in an interview. I have also seen juniors who, with solid foundations, outperform seniors in problem solving. The difference is not in how many frameworks you know, but in how well you think. This article is the guide I would have liked to have when I started. We go from zero to advanced structures, with practical exercises and a clear route. --- ## Why this matters more than any framework Let me be direct: technologies change, foundations do not. React could be replaced in 5 years. Python could stop being the favorite. But binary trees, recursion and algorithmic complexity will remain relevant in 50 years. They are the universal programming language. ### Concrete benefits of mastering these fundamentals 1. Technical interviews: Most top companies (and more and more medium-sized companies) evaluate data structures and algorithms. It does not matter if you apply for frontend, backend or DevOps. 2. Problem Solving: When you face a complex bug or design a system, your ability to decompose problems and choose the right structure makes the difference. 3. Efficient code: Understanding Big O allows you to write code that scales. The difference between O(n) and O(n2) can be the difference between an app that works and one that collapses. 4. Technical communication: Being able to talk about trade-offs between structures positions you as a senior, even if you have little experience. 5. Learning new languages: When you understand the underlying logic, learning a new language is just a matter of syntax. --- ## Part 1: Programming Logic Before data structures, you need to think as a programmer. This means breaking down problems into steps, understanding control flows and developing the "muscle" of logical thinking. ### 1.1 Algorithmic Thinking An algorithm is simply a sequence of steps to solve a problem. Before writing code, you should be able to explain the solution in Spanish. Mental exercise: Before coding any problem, write it in pseudocode or in your natural language. If you cannot explain it in words, you will not be able to explain it to the computer. #### The 4 steps of algorithmic thinking 1. Understanding the problem: What is the input? What is the expected output? What are the restrictions? 2. Decompose: Divide the big problem into smaller subproblems 3. Finding patterns: Have I seen something similar before? Is there repetition? 4. Abstract: Can I generalize the solution for similar cases? Practical example: Problem: Determine if a year is leap. Step 1 - Understand: - Input: A whole number (year) - Output: True or false - Rules: It is biased if it is divisible by 4, EXCEPT those divisible by 100, UNLESS they are also divisible by 400. Step 2 - Decompose: - Verify divisibility by 400 - Verify divisibility by 100 - Verify divisibility by 4 - Combining the conditions Step 3 - Pattern: - It is a problem of nested conditions - There are exceptions to the rules Step 4 - Pseudocode: ``` SI año es divisible por 400: RETORNAR verdadero SI año es divisible por 100: RETORNAR falso SI año es divisible por 4: RETORNAR verdadero RETORNAR falso ``` ### 1.2 Variables and Types of Data The variables are boxes where you store information. The types of data define what kind of information can go in each box. #### Fundamental primitive types |Type|Description|Example| |------|-------------|---------| |Integer (int)|Numbers without decimals|42, -17, 0| |Float|Numbers with decimals|3.14, -0.001| |string|Text|"Hello world"| |Boolean (bool)|True or false|true, false| |Null (null/None)|Lack of value|null, None| #### Good practices with variables 1. Descriptive names: `precioTotal` is better than `pt` or `x` 2. CamelCase or snake▁Sécase: Be consistent (`precioTotal` or `precio_total`) 3. Constants in MAYUSCULAS: `MAX_INTENTOS = 3` 4. Avoid magic numbers: Instead of `if intentos > 3`, use `if intentos > MAX_INTENTOS` ### 1.3 Operators Operators manipulate data. Mastering them is essential. #### Arithmetic operators ``` + Suma - Resta * Multiplicación / División % Módulo (resto de división) ** Potencia // División entera (en Python) ``` #### Comparison operators ``` == Igual a != Diferente de > Mayor que < Menor que >= Mayor o igual <= Menor o igual ``` #### Logic operators ``` AND (&&) - Verdadero si AMBOS son verdaderos OR (||) - Verdadero si AL MENOS UNO es verdadero NOT (!) - Invierte el valor booleano ``` Table of important truth: |A|B|AND B|A OR B|NOT A| |---|---|---------|--------|-------| |V|V|V|V|F| |V|F|F|V|F| |F|V|F|V|V| |F|F|F|F|V| ### 1.4 Control Structures #### Conditional (if/else) Conditionals allow your program to make decisions. ```python # Estructura básica if condicion: # código si es verdadero elif otra_condicion: # código si la primera es falsa pero esta es verdadera else: # código si todas son falsas ``` Complete example - System of qualifications: ```python def obtener_calificacion(nota): if nota >= 90: return "A - Excelente" elif nota >= 80: return "B - Muy bien" elif nota >= 70: return "C - Bien" elif nota >= 60: return "D - Suficiente" else: return "F - Reprobado" # Pruebas print(obtener_calificacion(95)) # A - Excelente print(obtener_calificacion(73)) # C - Bien print(obtener_calificacion(45)) # F - Reprobado ``` #### Loops Loops repeat code until a condition is fulfilled (or unfulfilled). Loop for - When you know how many times iterate: ```python # Iterar sobre un rango for i in range(5): print(i) # 0, 1, 2, 3, 4 # Iterar sobre una lista frutas = ["manzana", "banana", "naranja"] for fruta in frutas: print(fruta) # Con índice for indice, fruta in enumerate(frutas): print(f"{indice}: {fruta}") ``` Loop while - When you don't know how many iterations you need: ```python # Ejemplo: encontrar el primer número divisible por 7 mayor a 100 numero = 101 while numero % 7 != 0: numero += 1 print(f"Encontrado: {numero}") # 105 ``` Loop controls: ```python # break - Sale del bucle completamente for i in range(100): if i == 5: break print(i) # 0, 1, 2, 3, 4 # continue - Salta a la siguiente iteración for i in range(5): if i == 2: continue print(i) # 0, 1, 3, 4 ``` ### 1.5 Function Functions encapsulate reusable logic. They are fundamental to write clean code. #### Anatomy of a function ```python def nombre_funcion(parametro1, parametro2="valor_default"): """ Docstring: describe qué hace la función Args: parametro1: descripción parametro2: descripción (opcional) Returns: descripción de lo que retorna """ # Cuerpo de la función resultado = parametro1 + parametro2 return resultado ``` #### Principles for good functions 1. A function, a responsibility: If your function does many things, divide it 2. Verbal names: `calcular_impuesto()`, `validar_email()`, `obtener_usuario()` 3. Limited parameters: More than 3-4 parameters is a sign that something is wrong 4. Avoids side effects: A function ideally receives inputs and returns outputs, without modifying external state. 5. Early return: Valid at the beginning and return errors soon Example - Well-designed function: ```python def calcular_descuento(precio, porcentaje_descuento, es_cliente_premium=False): """ Calcula el precio final después de aplicar descuento. Args: precio: Precio original del producto porcentaje_descuento: Descuento a aplicar (0-100) es_cliente_premium: Si es premium, obtiene 5% adicional Returns: Precio final con descuento aplicado """ # Validaciones tempranas if precio < 0: raise ValueError("El precio no puede ser negativo") if not 0 <= porcentaje_descuento <= 100: raise ValueError("El descuento debe estar entre 0 y 100") # Cálculo descuento_total = porcentaje_descuento if es_cliente_premium: descuento_total += 5 descuento_total = min(descuento_total, 100) # Máximo 100% precio_final = precio * (1 - descuento_total / 100) return round(precio_final, 2) ``` ### 1.6 Recurrence Recursion is when a function calls itself. It is elegant for certain problems, but requires care. #### Components of any recursive function 1. Base case: Condition that stops the recursion 2. Recursive case: The function is called with a smaller problem 3. Progress towards the base case: Each call must approach the base case Classical example - Factorial: ```python def factorial(n): # Caso base if n <= 1: return 1 # Caso recursivo return n * factorial(n - 1) # factorial(5) = 5 * factorial(4) # = 5 * 4 * factorial(3) # = 5 * 4 * 3 * factorial(2) # = 5 * 4 * 3 * 2 * factorial(1) # = 5 * 4 * 3 * 2 * 1 # = 120 ``` Example - Fibonacci: ```python # Versión recursiva simple (ineficiente) def fibonacci_recursivo(n): if n <= 1: return n return fibonacci_recursivo(n - 1) + fibonacci_recursivo(n - 2) # Versión con memoización (eficiente) def fibonacci_memo(n, memo={}): if n in memo: return memo[n] if n <= 1: return n memo[n] = fibonacci_memo(n - 1, memo) + fibonacci_memo(n - 2, memo) return memo[n] ``` #### When to use recursion Use it when: - The problem has natural recursive structure (trees, graphs) - The iterative solution would be very complex - You are dividing a problem into smaller identical subproblems Avoid it when: - A simple iterative solution exists - The depth of recursion can be very large (stack overflow) - Performance is critical and you cannot memorize. --- ## Part 2: Fundamental Data Structures Now that you think as a programmer, you need to know where and how to save your data. The correct structure can make your 1000x code faster. ### 2.1 Arrays (Arrays/Lists) The array is the most basic and most used structure. It is an ordered collection of elements. #### Characteristics - Index access: O(1) - instantaneous - Search: O(n) - everything must be covered in the worst case - Insertion at the end: O(1) amortized - Insertion at the beginning/mean: O(n) - elements must be moved #### Common operations ```python # Creación lista = [1, 2, 3, 4, 5] lista_vacia = [] # Acceso primer_elemento = lista[0] # 1 ultimo_elemento = lista[-1] # 5 sublista = lista[1:3] # [2, 3] # Modificación lista[0] = 10 # [10, 2, 3, 4, 5] lista.append(6) # [10, 2, 3, 4, 5, 6] lista.insert(0, 0) # [0, 10, 2, 3, 4, 5, 6] lista.extend([7, 8]) # [0, 10, 2, 3, 4, 5, 6, 7, 8] # Eliminación lista.pop() # Elimina y retorna el último lista.pop(0) # Elimina y retorna el primero lista.remove(10) # Elimina la primera ocurrencia de 10 # Búsqueda if 3 in lista: indice = lista.index(3) # Ordenamiento lista.sort() # Ordena in-place lista_ordenada = sorted(lista) # Retorna nueva lista ordenada # Longitud tamaño = len(lista) ``` #### Important techniques with arrays Two Pointers: Useful for problems involving pairs or search in ordered arrays. ```python def dos_suma_ordenada(nums, target): """ Encuentra dos números que sumen target en array ordenado. """ izquierda = 0 derecha = len(nums) - 1 while izquierda < derecha: suma_actual = nums[izquierda] + nums[derecha] if suma_actual == target: return [izquierda, derecha] elif suma_actual < target: izquierda += 1 else: derecha -= 1 return None ``` Sliding Window: For contiguous subarrays problems. ```python def max_suma_subarray(nums, k): """ Encuentra la suma máxima de un subarray de tamaño k. """ if len(nums) < k: return None # Suma de la primera ventana suma_ventana = sum(nums[:k]) max_suma = suma_ventana # Deslizar la ventana for i in range(k, len(nums)): suma_ventana += nums[i] - nums[i - k] max_suma = max(max_suma, suma_ventana) return max_suma ``` ### 2.2 Strings The strings are character arrays with special operations. #### Essential operations ```python texto = "Hola Mundo" # Acceso print(texto[0]) # 'H' print(texto[-1]) # 'o' print(texto[0:4]) # 'Hola' # Inmutabilidad - los strings no se modifican in-place # texto[0] = 'h' # ¡Error! texto = 'h' + texto[1:] # Crear nuevo string # Métodos útiles texto.lower() # 'hola mundo' texto.upper() # 'HOLA MUNDO' texto.strip() # Elimina espacios al inicio/fin texto.split(' ') # ['Hola', 'Mundo'] ' '.join(['Hola', 'Mundo']) # 'Hola Mundo' texto.replace('Mundo', 'Python') # 'Hola Python' texto.find('Mundo') # 5 (índice donde empieza) texto.count('o') # 2 # Verificaciones texto.startswith('Hola') # True texto.endswith('Mundo') # True texto.isalpha() # False (tiene espacio) texto.isdigit() # False 'mundo' in texto.lower() # True ``` #### Classical problem - Palindromo ```python def es_palindromo(texto): """ Verifica si un texto es palíndromo (ignora espacios y mayúsculas). """ # Limpiar texto limpio = ''.join(c.lower() for c in texto if c.isalnum()) # Comparar con su reverso return limpio == limpio[::-1] # Alternativa con dos punteros def es_palindromo_punteros(texto): limpio = ''.join(c.lower() for c in texto if c.isalnum()) izq, der = 0, len(limpio) - 1 while izq < der: if limpio[izq] != limpio[der]: return False izq += 1 der -= 1 return True ``` ### 2.3 Hash Tables (Dictionaries/Maps) The most versatile structure. It stores key-value pairs with O(1) access. #### Characteristics - Access by key: O(1) average - Insertion: O(1) average - Elimination: O(1) average - Unordered: Keys have no guaranteed order #### Fundamental operations ```python # Creación diccionario = {} diccionario = {'nombre': 'Juan', 'edad': 25} diccionario = dict(nombre='Juan', edad=25) # Acceso nombre = diccionario['nombre'] # 'Juan' nombre = diccionario.get('nombre') # 'Juan' nombre = diccionario.get('apellido', 'No existe') # 'No existe' # Modificación diccionario['edad'] = 26 # Actualiza diccionario['ciudad'] = 'Madrid' # Agrega nuevo # Eliminación del diccionario['ciudad'] valor = diccionario.pop('edad') # Elimina y retorna # Verificación if 'nombre' in diccionario: print("Existe") # Iteración for clave in diccionario: print(clave) for clave, valor in diccionario.items(): print(f"{clave}: {valor}") for valor in diccionario.values(): print(valor) # Métodos útiles claves = list(diccionario.keys()) valores = list(diccionario.values()) items = list(diccionario.items()) ``` #### Common applications Count frequencies: ```python def contar_caracteres(texto): frecuencias = {} for char in texto: frecuencias[char] = frecuencias.get(char, 0) + 1 return frecuencias # Con Counter (más pythónico) from collections import Counter frecuencias = Counter("hola mundo") # {'o': 2, 'h': 1, ...} ``` Two Sum (classic interview problem): ```python def two_sum(nums, target): """ Encuentra dos índices cuya suma sea target. """ vistos = {} # valor -> índice for i, num in enumerate(nums): complemento = target - num if complemento in vistos: return [vistos[complemento], i] vistos[num] = i return None ``` Group anagrams: ```python def agrupar_anagramas(palabras): """ Agrupa palabras que son anagramas entre sí. """ grupos = {} for palabra in palabras: clave = ''.join(sorted(palabra)) if clave not in grupos: grupos[clave] = [] grupos[clave].append(palabra) return list(grupos.values()) # Ejemplo palabras = ["eat", "tea", "tan", "ate", "nat", "bat"] print(agrupar_anagramas(palabras)) # [['eat', 'tea', 'ate'], ['tan', 'nat'], ['bat']] ``` ### 2.4 Sets A set is like a dictionary but only with keys (without values). Unique elements, without order. #### Characteristics - Unique elements: No duplicates - Verification of membership: O(1) - Insertion/Elimination: O(1) average - Without indexes: You cannot access by position #### Operations ```python # Creación conjunto = {1, 2, 3} conjunto = set([1, 2, 2, 3]) # {1, 2, 3} - duplicados eliminados conjunto_vacio = set() # {} crea diccionario, no set # Modificación conjunto.add(4) conjunto.remove(4) # Error si no existe conjunto.discard(4) # No error si no existe conjunto.pop() # Elimina elemento aleatorio # Verificación if 2 in conjunto: print("Existe") # Operaciones de conjuntos a = {1, 2, 3} b = {2, 3, 4} union = a | b # {1, 2, 3, 4} interseccion = a & b # {2, 3} diferencia = a - b # {1} dif_simetrica = a ^ b # {1, 4} # Verificaciones a.issubset(b) # ¿a es subconjunto de b? a.issuperset(b) # ¿a es superconjunto de b? a.isdisjoint(b) # ¿No tienen elementos en común? ``` #### Application - Find duplicates ```python def tiene_duplicados(lista): return len(lista) != len(set(lista)) def encontrar_duplicados(lista): vistos = set() duplicados = set() for elemento in lista: if elemento in vistos: duplicados.add(elemento) vistos.add(elemento) return list(duplicados) ``` ### 2.5 Stacks LIFO: Last In, First Out. The last one that enters is the first one that leaves. #### Fundamental operations ```python # En Python, usamos lista como stack stack = [] # Push - agregar al tope stack.append(1) stack.append(2) stack.append(3) # Pop - remover del tope elemento = stack.pop() # 3 # Peek - ver el tope sin remover tope = stack[-1] # 2 # Verificar si está vacío esta_vacio = len(stack) == 0 ``` #### Classical applications Check balanced parentheses: ```python def parentesis_validos(s): """ Verifica si los paréntesis/brackets están balanceados. """ stack = [] pares = {')': '(', ']': '[', '}': '{'} for char in s: if char in '([{': stack.append(char) elif char in ')]}': if not stack or stack[-1] != pares[char]: return False stack.pop() return len(stack) == 0 # Ejemplos print(parentesis_validos("()[]{}")) # True print(parentesis_validos("([)]")) # False print(parentesis_validos("{[]}")) # True ``` Evaluate reverse Polish notation: ```python def evaluar_rpn(tokens): """ Evalúa expresión en notación polaca inversa. Ejemplo: ["2", "1", "+", "3", "*"] = (2 + 1) * 3 = 9 """ stack = [] operadores = { '+': lambda a, b: a + b, '-': lambda a, b: a - b, '*': lambda a, b: a * b, '/': lambda a, b: int(a / b) # División entera hacia cero } for token in tokens: if token in operadores: b = stack.pop() a = stack.pop() resultado = operadores[token](a, b) stack.append(resultado) else: stack.append(int(token)) return stack[0] ``` ### 2.6 Queues (Colas) FIFO: First In, First Out. The first one that enters is the first one that leaves. #### Implementation ```python from collections import deque # Crear cola cola = deque() # Enqueue - agregar al final cola.append(1) cola.append(2) cola.append(3) # Dequeue - remover del frente elemento = cola.popleft() # 1 # Peek - ver el frente sin remover frente = cola[0] # 2 # Verificar si está vacía esta_vacia = len(cola) == 0 ``` #### Application - BFS (Wide search) ```python def bfs_matriz(matriz, inicio, fin): """ Encuentra el camino más corto en una matriz. """ filas, cols = len(matriz), len(matriz[0]) cola = deque([(inicio, 0)]) # (posición, distancia) visitados = {inicio} direcciones = [(0, 1), (0, -1), (1, 0), (-1, 0)] while cola: (fila, col), distancia = cola.popleft() if (fila, col) == fin: return distancia for df, dc in direcciones: nf, nc = fila + df, col + dc if (0 <= nf < filas and 0 <= nc < cols and (nf, nc) not in visitados and matriz[nf][nc] != 1): # 1 = obstáculo visitados.add((nf, nc)) cola.append(((nf, nc), distancia + 1)) return -1 # No hay camino ``` ### 2.7 Linked Lists A sequence of nodes where each node points to the next. #### Basic implementation ```python class Nodo: def __init__(self, valor): self.valor = valor self.siguiente = None class ListaEnlazada: def __init__(self): self.cabeza = None def agregar_al_final(self, valor): nuevo = Nodo(valor) if not self.cabeza: self.cabeza = nuevo return actual = self.cabeza while actual.siguiente: actual = actual.siguiente actual.siguiente = nuevo def agregar_al_inicio(self, valor): nuevo = Nodo(valor) nuevo.siguiente = self.cabeza self.cabeza = nuevo def eliminar(self, valor): if not self.cabeza: return if self.cabeza.valor == valor: self.cabeza = self.cabeza.siguiente return actual = self.cabeza while actual.siguiente: if actual.siguiente.valor == valor: actual.siguiente = actual.siguiente.siguiente return actual = actual.siguiente def buscar(self, valor): actual = self.cabeza while actual: if actual.valor == valor: return True actual = actual.siguiente return False def imprimir(self): actual = self.cabeza elementos = [] while actual: elementos.append(str(actual.valor)) actual = actual.siguiente print(" -> ".join(elementos)) ``` #### Classical problem - Detect cycle ```python def tiene_ciclo(cabeza): """ Detecta si una lista enlazada tiene ciclo usando Floyd's algorithm. """ if not cabeza or not cabeza.siguiente: return False lento = cabeza rapido = cabeza while rapido and rapido.siguiente: lento = lento.siguiente rapido = rapido.siguiente.siguiente if lento == rapido: return True return False ``` Invest linked list: ```python def invertir_lista(cabeza): """ Invierte una lista enlazada in-place. """ previo = None actual = cabeza while actual: siguiente = actual.siguiente actual.siguiente = previo previo = actual actual = siguiente return previo ``` --- ## Part 3: Advanced Data Structures These structures are less common in day-to-day code, but essential to solve complex problems and interviews in top companies. ### 3.1 Binary Trees Hierarchical structure where each node has maximum two children. #### Implementation ```python class NodoArbol: def __init__(self, valor): self.valor = valor self.izquierdo = None self.derecho = None class ArbolBinario: def __init__(self): self.raiz = None def insertar(self, valor): if not self.raiz: self.raiz = NodoArbol(valor) else: self._insertar_recursivo(self.raiz, valor) def _insertar_recursivo(self, nodo, valor): if valor < nodo.valor: if nodo.izquierdo: self._insertar_recursivo(nodo.izquierdo, valor) else: nodo.izquierdo = NodoArbol(valor) else: if nodo.derecho: self._insertar_recursivo(nodo.derecho, valor) else: nodo.derecho = NodoArbol(valor) ``` #### Fundamental routes ```python def inorder(nodo, resultado=None): """Izquierdo -> Raíz -> Derecho""" if resultado is None: resultado = [] if nodo: inorder(nodo.izquierdo, resultado) resultado.append(nodo.valor) inorder(nodo.derecho, resultado) return resultado def preorder(nodo, resultado=None): """Raíz -> Izquierdo -> Derecho""" if resultado is None: resultado = [] if nodo: resultado.append(nodo.valor) preorder(nodo.izquierdo, resultado) preorder(nodo.derecho, resultado) return resultado def postorder(nodo, resultado=None): """Izquierdo -> Derecho -> Raíz""" if resultado is None: resultado = [] if nodo: postorder(nodo.izquierdo, resultado) postorder(nodo.derecho, resultado) resultado.append(nodo.valor) return resultado def level_order(raiz): """Recorrido por niveles (BFS)""" if not raiz: return [] resultado = [] cola = deque([raiz]) while cola: nivel = [] for _ in range(len(cola)): nodo = cola.popleft() nivel.append(nodo.valor) if nodo.izquierdo: cola.append(nodo.izquierdo) if nodo.derecho: cola.append(nodo.derecho) resultado.append(nivel) return resultado ``` #### Common tree problems Tree height: ```python def altura(nodo): if not nodo: return 0 return 1 + max(altura(nodo.izquierdo), altura(nodo.derecho)) ``` Verify if it is valid BST: ```python def es_bst_valido(nodo, minimo=float('-inf'), maximo=float('inf')): if not nodo: return True if nodo.valor <= minimo or nodo.valor >= maximo: return False return (es_bst_valido(nodo.izquierdo, minimo, nodo.valor) and es_bst_valido(nodo.derecho, nodo.valor, maximo)) ``` Lower Common Ancestor (LCA): ```python def lca(raiz, p, q): """ Encuentra el ancestro común más bajo de dos nodos. """ if not raiz or raiz == p or raiz == q: return raiz izq = lca(raiz.izquierdo, p, q) der = lca(raiz.derecho, p, q) if izq and der: return raiz return izq or der ``` ### 3.2 Heaps (Mounds) Special binary tree that maintains the minimum (or maximum) always at the root. #### Use in Python ```python import heapq # Min heap (por defecto en Python) heap = [] heapq.heappush(heap, 3) heapq.heappush(heap, 1) heapq.heappush(heap, 4) minimo = heapq.heappop(heap) # 1 # Ver mínimo sin remover minimo = heap[0] # Crear heap desde lista numeros = [3, 1, 4, 1, 5, 9, 2, 6] heapq.heapify(numeros) # O(n) # Max heap (truco: negar valores) max_heap = [] heapq.heappush(max_heap, -3) heapq.heappush(max_heap, -1) maximo = -heapq.heappop(max_heap) # 3 # K elementos más grandes k_mayores = heapq.nlargest(3, numeros) k_menores = heapq.nsmallest(3, numeros) ``` #### Application - Median data stream ```python class MedianFinder: def __init__(self): self.pequeños = [] # Max heap (mitad menor) self.grandes = [] # Min heap (mitad mayor) def agregar(self, num): # Agregar a max heap (negamos para simular) heapq.heappush(self.pequeños, -num) # Mover el mayor de pequeños a grandes heapq.heappush(self.grandes, -heapq.heappop(self.pequeños)) # Balancear: pequeños puede tener 1 más que grandes if len(self.grandes) > len(self.pequeños): heapq.heappush(self.pequeños, -heapq.heappop(self.grandes)) def mediana(self): if len(self.pequeños) > len(self.grandes): return -self.pequeños[0] return (-self.pequeños[0] + self.grandes[0]) / 2 ``` ### 3.3 Graphs Set of nodes connected by edges. They can be directed or not directed. #### Representations ```python # Lista de adyacencia (más común y eficiente) grafo = { 'A': ['B', 'C'], 'B': ['A', 'D', 'E'], 'C': ['A', 'F'], 'D': ['B'], 'E': ['B', 'F'], 'F': ['C', 'E'] } # Con pesos grafo_pesos = { 'A': [('B', 4), ('C', 2)], 'B': [('A', 4), ('D', 3)], 'C': [('A', 2), ('D', 1)], 'D': [('B', 3), ('C', 1)] } # Matriz de adyacencia (útil para grafos densos) # A B C D # A [[0, 1, 1, 0], # B [1, 0, 0, 1], # C [1, 0, 0, 1], # D [0, 1, 1, 0]] ``` #### Fundamental algorithms DFS (Deep search): ```python def dfs(grafo, inicio, visitados=None): if visitados is None: visitados = set() visitados.add(inicio) print(inicio) # Procesar nodo for vecino in grafo[inicio]: if vecino not in visitados: dfs(grafo, vecino, visitados) return visitados # Versión iterativa con stack def dfs_iterativo(grafo, inicio): visitados = set() stack = [inicio] while stack: nodo = stack.pop() if nodo not in visitados: visitados.add(nodo) print(nodo) for vecino in grafo[nodo]: if vecino not in visitados: stack.append(vecino) return visitados ``` BFS (Width Search): ```python def bfs(grafo, inicio): visitados = set([inicio]) cola = deque([inicio]) while cola: nodo = cola.popleft() print(nodo) for vecino in grafo[nodo]: if vecino not in visitados: visitados.add(vecino) cola.append(vecino) return visitados ``` Dijkstra (shortest path): ```python def dijkstra(grafo, inicio): """ Encuentra el camino más corto desde inicio a todos los nodos. grafo: diccionario con listas de tuplas (vecino, peso) """ distancias = {nodo: float('inf') for nodo in grafo} distancias[inicio] = 0 heap = [(0, inicio)] # (distancia, nodo) visitados = set() while heap: dist_actual, nodo = heapq.heappop(heap) if nodo in visitados: continue visitados.add(nodo) for vecino, peso in grafo[nodo]: if vecino not in visitados: nueva_dist = dist_actual + peso if nueva_dist < distancias[vecino]: distancias[vecino] = nueva_dist heapq.heappush(heap, (nueva_dist, vecino)) return distancias ``` Detect cycle in directed graph: ```python def tiene_ciclo_dirigido(grafo): """ Detecta ciclo usando coloración: 0 = blanco (no visitado), 1 = gris (en proceso), 2 = negro (terminado) """ colores = {nodo: 0 for nodo in grafo} def dfs(nodo): colores[nodo] = 1 # Gris for vecino in grafo.get(nodo, []): if colores[vecino] == 1: # Encontramos un gris -> ciclo return True if colores[vecino] == 0 and dfs(vecino): return True colores[nodo] = 2 # Negro return False for nodo in grafo: if colores[nodo] == 0: if dfs(nodo): return True return False ``` ### 3.4 Tries (trees of prefixes) Optimized structure to store and search strings by prefixes. ```python class TrieNode: def __init__(self): self.hijos = {} self.es_fin_palabra = False class Trie: def __init__(self): self.raiz = TrieNode() def insertar(self, palabra): nodo = self.raiz for char in palabra: if char not in nodo.hijos: nodo.hijos[char] = TrieNode() nodo = nodo.hijos[char] nodo.es_fin_palabra = True def buscar(self, palabra): nodo = self._buscar_prefijo(palabra) return nodo is not None and nodo.es_fin_palabra def empieza_con(self, prefijo): return self._buscar_prefijo(prefijo) is not None def _buscar_prefijo(self, prefijo): nodo = self.raiz for char in prefijo: if char not in nodo.hijos: return None nodo = nodo.hijos[char] return nodo def autocomplete(self, prefijo): """Retorna todas las palabras que empiezan con el prefijo.""" nodo = self._buscar_prefijo(prefijo) if not nodo: return [] resultados = [] self._dfs_palabras(nodo, prefijo, resultados) return resultados def _dfs_palabras(self, nodo, prefijo_actual, resultados): if nodo.es_fin_palabra: resultados.append(prefijo_actual) for char, hijo in nodo.hijos.items(): self._dfs_palabras(hijo, prefijo_actual + char, resultados) ``` --- ## Part 4: Algorithmic Complexity (Big O) Understanding Big O is what separates someone who "codes" from someone who understands what they do. ### 4.1 What is Big O? Big O describes how an algorithm scales when the input grows. It does not measure exact time, but how time/space grows with larger inputs. ### 4.2 Common complexities |Notation|Name|Example| |----------|--------|---------| |O(1)|Constant|Index array access| |O(log n)|Logarithmic|Binary search| |O(n)|Linear|Go through array| |O(n log n)|Linearithmic|Merge sort, quick sort| |O(n2)|Quadratic|Two nested loops| |O(2n)|Exponential|Recursive Fibonacci without memo| |O(n!)|Factorial|Permutations| ### 4.3 How to calculate Big O Basic rules: 1. Ignorance constants: O(2n) O O(n) 2. Ignores minor terms: O(n2 + n) = O(n2) 3. Nested loops multiply: Loop inside loop O O(n m m) 4. Sequential loops sum: Loop followed by loop lo O(n + m) = O(n) Practical examples: ```python # O(1) - Constante def obtener_primero(lista): return lista[0] # O(n) - Lineal def sumar_todos(lista): total = 0 for num in lista: # n iteraciones total += num # O(1) por iteración return total # O(n²) - Cuadrática def encontrar_pares(lista): pares = [] for i in range(len(lista)): # n iteraciones for j in range(len(lista)): # n iteraciones por cada i pares.append((lista[i], lista[j])) return pares # O(log n) - Logarítmica def busqueda_binaria(lista, objetivo): izq, der = 0, len(lista) - 1 while izq <= der: # Se divide a la mitad cada vez mid = (izq + der) // 2 if lista[mid] == objetivo: return mid elif lista[mid] < objetivo: izq = mid + 1 else: der = mid - 1 return -1 # O(n log n) - Linearítmica def merge_sort(lista): if len(lista) <= 1: return lista mid = len(lista) // 2 izq = merge_sort(lista[:mid]) # log n niveles de recursión der = merge_sort(lista[mid:]) return merge(izq, der) # O(n) por nivel def merge(izq, der): resultado = [] i = j = 0 while i < len(izq) and j < len(der): if izq[i] <= der[j]: resultado.append(izq[i]) i += 1 else: resultado.append(der[j]) j += 1 resultado.extend(izq[i:]) resultado.extend(der[j:]) return resultado ``` ### 4.4 Spatial complexity Not only time matters, but also memory. ```python # O(1) espacio - Solo variables simples def suma(a, b): return a + b # O(n) espacio - Lista que crece con input def duplicar_lista(lista): return [x * 2 for x in lista] # O(n) espacio por recursión (stack) def factorial(n): if n <= 1: return 1 return n * factorial(n - 1) # n llamadas en el stack ``` --- ## Part 5: Recommended Study Plan ### Weeks 1-2: Fundamentals of logic - Variables, types, operators - Conditional and loops - Functions - Practice: 30 easy problems in LeetCode ### Weeks 3-4: Arrays and Strings - Basic operations - Two pointers - Sliding window - Practice: 40 array/string problems ### Weeks 5-6: Hash Tables and Sets - When using each one - Frequency patterns - Practice: 30 problems with hashmaps ### Weeks 7-8: Stacks, Queues, Linked Lists - Implementations from scratch - Classical problems - Practice: 30 problems ### Weeks 9-10: Recursion and trees - Recursive thinking - Binary trees and BST - Routes - Practice: 40 tree problems ### Weeks 11-12: Graphs and advanced algorithms - DFS, BFS - Basic Dijkstra - Cycle detection - Practice: 30 graph problems ### Continuous: Maintenance - 1-2 daily problems - Participate in contests - Review solutions from others --- ## Recommended Resources ### Practice platforms 1. LeetCode - The standard for interviews 2. HackerRank - Good initial tutorials 3. Codewars - Fun Katas 4. NeetCode - Pattern-Organized Problems ### Books 1. "Grokking Algorithms" - Perfect for beginners 2. Cracking the Coding Interview 3. "Introduction to Algorithms" (CLRS) - The Academic Bible ### Videos 1. NeetCode on YouTube - Clear explanations 2. CS Dojo - Well explained foundations 3. MIT OpenCourseWare - If you want to deepen --- ## Common mistakes to avoid 1. skip the fundamentals: Don't go to graphs without mastering arrays 2. Memorize without understanding: Understand why, not just the code 3. Not practicing enough: Theory without practice is useless 4. Surrender very soon: The first problems are frustrating, it is normal 5. Compare yourself with others: Your rhythm is valid 6. Not implementing from scratch: Use built-in structures after implementing them yourself --- ## Frequently asked questions Do I need to know advanced mathematics for this? No. You need logical thinking, not advanced calculation. Big O uses some notation, but 90% is reasoning about how your code behaves, not solving integrals. In what language do I practice logic and data structures? In which you want: the concepts are universal. Here we use Python for its readability; if you don't know it, it starts with my [Python tutorial for beginners](/en/python-tutorial-for-beginners-with-a-project) and then returns to this guide. Does this help me for technical interviews? A lot. Most technical interviews evaluate exactly this. And if you doubt which stack to accompany, look at the most demanded [languages](/en/programming-languages-and-starter-projects) and [frameworks](/en/popular-frameworks-and-learning-paths) in 2026. ## Conclusion Mastering programming logic and data structures is not something that happens in a week. It is a process that requires consistency and deliberate practice. But it is the most profitable investment you can make in your career as a developer. You don't need to be a math genius or have studied computer science. You need curiosity, persistence and a clear plan. This article gives you the plan, the rest depends on you. It begins today. One problem at a time. In 3 months you will be surprised how much progress you have made. And remember: every senior you admire went through exactly the same process. The difference is that they started earlier. Your time to start is now.

Tag: Algorithms

Programming Logic and Data Structures: A Complete Guide