Welcome to Last Module 10 of our comprehensive Python course, designed to transform you from a beginner to an advanced Python programmer!
In Module 9, we explored web application development with Flask and Django, building dynamic websites and APIs. Now, we dive into algorithms and problem solving with Python, covering over 20 core algorithms across searching, sorting, recursion, number theory, dynamic programming, and graph algorithms. These algorithms are the foundation of efficient programming, used in applications like search engines, data processing, and route optimization.
This blog is beginner-friendly yet detailed enough for intermediate and advanced learners, offering real-world scenarios, multiple code examples, pros and cons, best practices, and alternatives. Whether you're optimizing search functionality, sorting datasets, or solving complex graph problems, this guide will equip you with the skills to tackle algorithmic challenges. Let’s dive in!
Table of Contents
1. Searching AlgorithmsLinear SearchLinear search checks each element in a list until the target is found or the list ends.Code:Binary SearchBinary search works on sorted lists, dividing the search space in half each step.Code:Pros, Cons, and AlternativesPros:This example demonstrates searching for products, comparing linear and binary search.
2. Sorting AlgorithmsBubble SortBubble sort repeatedly swaps adjacent elements if they are in the wrong order.Code:Insertion SortInsertion sort builds a sorted array by inserting elements in their correct position.Code:Selection SortSelection sort selects the smallest element and places it at the beginning.Code:Merge SortMerge sort divides the list into halves, sorts them, and merges them back.Code:Quick SortQuick sort picks a pivot, partitions the list, and recursively sorts sublists.Code:Heap SortHeap sort uses a max-heap to sort elements.Code:Counting SortCounting sort counts occurrences of elements to sort non-negative integers.Code:Radix SortRadix sort sorts numbers by processing digits place by place.Code:Pros, Cons, and AlternativesPros:This example demonstrates sorting customer orders, comparing bubble and merge sort.
3. Recursion-Based AlgorithmsFibonacci NumbersFibonacci numbers sum the two preceding numbers: 0, 1, 1, 2, 3, 5, ...Code:FactorialFactorial multiplies a number by all positive integers below it.Code:Tower of HanoiMove disks between pegs, following specific rules.Code:Pros, Cons, and AlternativesPros:This example uses recursion with memoization for financial calculations.
4. Number Theory AlgorithmsPrime NumbersCheck if a number is prime by testing divisibility.Code:GCD & LCMGreatest Common Divisor (GCD) uses Euclid’s algorithm; LCM uses GCD.Code:Sieve of EratosthenesGenerate all primes up to a number.Code:Pros, Cons, and AlternativesPros:This example uses the Sieve of Eratosthenes for cryptographic applications.
5. Dynamic ProgrammingLongest Common Subsequence (LCS)Find the longest subsequence common to two strings.Code:Knapsack ProblemMaximize value within a weight constraint.Code:Coin Change ProblemFind the minimum number of coins to make a target amount.Code:Pros, Cons, and AlternativesPros:This example optimizes delivery item selection based on value and weight constraints.
6. Graph AlgorithmsBreadth-First Search (BFS)BFS explores nodes level by level, using a queue.Code:Depth-First Search (DFS)DFS explores as far as possible along each branch.Code:Dijkstra’s AlgorithmFinds the shortest path in a weighted graph.Code:Minimum Spanning Tree (Kruskal/Prim)Finds a tree with minimum total edge weight.Kruskal’s Code:Pros, Cons, and AlternativesPros:This example uses BFS to find all friends of a user in a social network.
7. ConclusionCongratulations on mastering Module 10! You’ve learned over 20 core algorithms, including searching, sorting, recursion, number theory, dynamic programming, and graph algorithms. These skills enable you to solve complex problems like inventory searches, order sorting, financial calculations, and network analysis.
Table of Contents
- Searching Algorithms
- Linear Search
- Binary Search
- Pros, Cons, and Alternatives
- Best Practices
- Example: Finding Products in an Inventory
- Sorting Algorithms
- Bubble Sort
- Insertion Sort
- Selection Sort
- Merge Sort
- Quick Sort
- Heap Sort
- Counting Sort
- Radix Sort
- Pros, Cons, and Alternatives
- Best Practices
- Example: Sorting Customer Orders
- Recursion-Based Algorithms
- Fibonacci Numbers
- Factorial
- Tower of Hanoi
- Pros, Cons, and Alternatives
- Best Practices
- Example: Calculating Loan Interest Recursively
- Number Theory Algorithms
- Prime Numbers
- GCD & LCM
- Sieve of Eratosthenes
- Pros, Cons, and Alternatives
- Best Practices
- Example: Generating Encryption Keys
- Dynamic Programming
- Longest Common Subsequence
- Knapsack Problem
- Coin Change Problem
- Pros, Cons, and Alternatives
- Best Practices
- Example: Optimizing Delivery Routes
- Graph Algorithms
- Breadth-First Search (BFS)
- Depth-First Search (DFS)
- Dijkstra’s Algorithm
- Minimum Spanning Tree (Kruskal/Prim)
- Pros, Cons, and Alternatives
- Best Practices
- Example: Social Network Analysis
- Conclusion & Next Steps
1. Searching AlgorithmsLinear SearchLinear search checks each element in a list until the target is found or the list ends.Code:
python
def linear_search(arr, target):
for i, item in enumerate(arr):
if item == target:
return i
return -1
# Example
inventory = ["apple", "banana", "orange"]
print(linear_search(inventory, "banana")) # Output: 1
print(linear_search(inventory, "grape")) # Output: -1python
def binary_search(arr, target):
left, right = 0, len(arr) - 1
while left <= right:
mid = (left + right) // 2
if arr[mid] == target:
return mid
elif arr[mid] < target:
left = mid + 1
else:
right = mid - 1
return -1
# Example
sorted_inventory = ["apple", "banana", "orange"]
print(binary_search(sorted_inventory, "banana")) # Output: 1- Linear Search: Simple, works on unsorted data.
- Binary Search: Fast (O(log n)) for sorted data.
- Linear Search: Slow (O(n)) for large lists.
- Binary Search: Requires sorted data, not adaptive for frequent updates.
- Hash Tables: O(1) lookup with dictionaries.
- Ternary Search: For unimodal functions.
- Jump Search: For sorted lists with fewer comparisons.
- Use linear search for small or unsorted lists.
- Use binary search for large, sorted lists.
- Ensure input validation to handle empty lists.
- Sort data before binary search if needed.
python
def search_product(products, target, method="linear"):
if method == "linear":
return linear_search(products, target)
elif method == "binary":
sorted_products = sorted(products)
return binary_search(sorted_products, target)
return -1
# Test
products = ["laptop", "phone", "tablet", "mouse"]
print(search_product(products, "phone", "linear")) # Output: 1
print(search_product(products, "phone", "binary")) # Output: 2 (sorted: laptop, mouse, phone, tablet)2. Sorting AlgorithmsBubble SortBubble sort repeatedly swaps adjacent elements if they are in the wrong order.Code:
python
def bubble_sort(arr):
n = len(arr)
for i in range(n):
for j in range(0, n - i - 1):
if arr[j] > arr[j + 1]:
arr[j], arr[j + 1] = arr[j + 1], arr[j]
return arr
# Example
print(bubble_sort([64, 34, 25, 12, 22])) # Output: [12, 22, 25, 34, 64]python
def insertion_sort(arr):
for i in range(1, len(arr)):
key = arr[i]
j = i - 1
while j >= 0 and arr[j] > key:
arr[j + 1] = arr[j]
j -= 1
arr[j + 1] = key
return arr
# Example
print(insertion_sort([64, 34, 25, 12, 22])) # Output: [12, 22, 25, 34, 64]python
def selection_sort(arr):
for i in range(len(arr)):
min_idx = i
for j in range(i + 1, len(arr)):
if arr[j] < arr[min_idx]:
min_idx = j
arr[i], arr[min_idx] = arr[min_idx], arr[i]
return arr
# Example
print(selection_sort([64, 34, 25, 12, 22])) # Output: [12, 22, 25, 34, 64]python
def merge_sort(arr):
if len(arr) <= 1:
return arr
mid = len(arr) // 2
left = merge_sort(arr[:mid])
right = merge_sort(arr[mid:])
return merge(left, right)
def merge(left, right):
result = []
i = j = 0
while i < len(left) and j < len(right):
if left[i] <= right[j]:
result.append(left[i])
i += 1
else:
result.append(right[j])
j += 1
result.extend(left[i:])
result.extend(right[j:])
return result
# Example
print(merge_sort([64, 34, 25, 12, 22])) # Output: [12, 22, 25, 34, 64]python
def quick_sort(arr, low, high):
if low < high:
pi = partition(arr, low, high)
quick_sort(arr, low, pi - 1)
quick_sort(arr, pi + 1, high)
return arr
def partition(arr, low, high):
pivot = arr[high]
i = low - 1
for j in range(low, high):
if arr[j] <= pivot:
i += 1
arr[i], arr[j] = arr[j], arr[i]
arr[i + 1], arr[high] = arr[high], arr[i + 1]
return i + 1
# Example
arr = [64, 34, 25, 12, 22]
print(quick_sort(arr, 0, len(arr) - 1)) # Output: [12, 22, 25, 34, 64]python
def heap_sort(arr):
def heapify(arr, n, i):
largest = i
left = 2 * i + 1
right = 2 * i + 2
if left < n and arr[left] > arr[largest]:
largest = left
if right < n and arr[right] > arr[largest]:
largest = right
if largest != i:
arr[i], arr[largest] = arr[largest], arr[i]
heapify(arr, n, largest)
n = len(arr)
for i in range(n // 2 - 1, -1, -1):
heapify(arr, n, i)
for i in range(n - 1, 0, -1):
arr[0], arr[i] = arr[i], arr[0]
heapify(arr, i, 0)
return arr
# Example
print(heap_sort([64, 34, 25, 12, 22])) # Output: [12, 22, 25, 34, 64]python
def counting_sort(arr):
max_val = max(arr)
count = [0] * (max_val + 1)
for num in arr:
count[num] += 1
result = []
for i, freq in enumerate(count):
result.extend([i] * freq)
return result
# Example
print(counting_sort([4, 2, 2, 8, 3])) # Output: [2, 2, 3, 4, 8]python
def counting_sort_for_radix(arr, exp):
n = len(arr)
output = [0] * n
count = [0] * 10
for i in range(n):
index = arr[i] // exp
count[index % 10] += 1
for i in range(1, 10):
count[i] += count[i - 1]
i = n - 1
while i >= 0:
index = arr[i] // exp
output[count[index % 10] - 1] = arr[i]
count[index % 10] -= 1
i -= 1
return output
def radix_sort(arr):
max_val = max(arr)
exp = 1
while max_val // exp > 0:
arr = counting_sort_for_radix(arr, exp)
exp *= 10
return arr
# Example
print(radix_sort([170, 45, 75, 90, 802])) # Output: [45, 75, 90, 170, 802]- Bubble/Insertion/Selection: Simple, good for small lists.
- Merge/Quick/Heap: Efficient for large datasets (O(n log n)).
- Counting/Radix: Fast for specific data (O(n)).
- Bubble/Insertion/Selection: Slow (O(n²)) for large lists.
- Merge: Requires extra memory.
- Quick: Worst-case O(n²) with poor pivot choice.
- Counting/Radix: Limited to non-negative integers.
- Python’s sorted(): Uses Timsort, hybrid of merge and insertion sort.
- NumPy sort: For numerical arrays.
- External Sorting: For massive datasets.
- Use sorted() for general-purpose sorting.
- Choose algorithms based on data size and type.
- Avoid bubble sort in production due to inefficiency.
- Test sorting stability for complex objects.
python
def sort_orders(orders, algorithm="bubble"):
if algorithm == "bubble":
return bubble_sort(orders)
elif algorithm == "merge":
return merge_sort(orders)
return orders
# Test
orders = [100, 50, 200, 75]
print(sort_orders(orders, "bubble")) # Output: [50, 75, 100, 200]
print(sort_orders(orders, "merge")) # Output: [50, 75, 100, 200]3. Recursion-Based AlgorithmsFibonacci NumbersFibonacci numbers sum the two preceding numbers: 0, 1, 1, 2, 3, 5, ...Code:
python
def fibonacci(n):
if n <= 1:
return n
return fibonacci(n - 1) + fibonacci(n - 2)
# Example
print(fibonacci(6)) # Output: 8python
def factorial(n):
if n == 0:
return 1
return n * factorial(n - 1)
# Example
print(factorial(5)) # Output: 120python
def tower_of_hanoi(n, source, auxiliary, target):
if n > 0:
tower_of_hanoi(n - 1, source, target, auxiliary)
print(f"Move disk {n} from {source} to {target}")
tower_of_hanoi(n - 1, auxiliary, source, target)
# Example
tower_of_hanoi(3, 'A', 'B', 'C')- Elegant solutions for recursive problems.
- Simplifies complex problems like tree traversals.
- High memory usage for deep recursion.
- Risk of stack overflow for large inputs.
- Slower than iterative solutions in some cases.
- Iterative Solutions: For memory efficiency.
- Memoization: To cache recursive results.
- Tail Recursion: Not optimized in Python.
- Use recursion for naturally recursive problems.
- Implement memoization for performance (e.g., @lru_cache).
- Limit recursion depth for large inputs.
- Test base cases thoroughly.
python
from functools import lru_cache
@lru_cache(maxsize=None)
def compound_interest(principal, rate, time):
if time == 0:
return principal
return compound_interest(principal * (1 + rate), rate, time - 1)
# Test
print(compound_interest(1000, 0.05, 3)) # Output: 1157.6254. Number Theory AlgorithmsPrime NumbersCheck if a number is prime by testing divisibility.Code:
python
def is_prime(n):
if n < 2:
return False
for i in range(2, int(n ** 0.5) + 1):
if n % i == 0:
return False
return True
# Example
print(is_prime(17)) # Output: Truepython
def gcd(a, b):
while b:
a, b = b, a % b
return a
def lcm(a, b):
return abs(a * b) // gcd(a, b)
# Example
print(gcd(48, 18)) # Output: 6
print(lcm(48, 18)) # Output: 144python
def sieve(n):
primes = [True] * (n + 1)
primes[0] = primes[1] = False
for i in range(2, int(n ** 0.5) + 1):
if primes[i]:
for j in range(i * i, n + 1, i):
primes[j] = False
return [i for i in range(n + 1) if primes[i]]
# Example
print(sieve(30)) # Output: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]- Efficient for cryptographic and mathematical applications.
- Sieve is fast for generating multiple primes.
- Sieve requires significant memory for large n.
- GCD/LCM are specialized algorithms.
- Prime checking is slow for very large numbers.
- Miller-Rabin: For probabilistic primality testing.
- Math Libraries: Like sympy for advanced number theory.
- Brute Force: For small inputs, but inefficient.
- Use optimized algorithms (e.g., Euclid’s for GCD).
- Cache results for repeated calculations.
- Validate inputs for negative numbers.
- Use Sieve for generating primes up to a limit.
python
def generate_primes(limit):
return sieve(limit)
# Test
primes = generate_primes(100)
print(primes[:5]) # Output: [2, 3, 5, 7, 11]5. Dynamic ProgrammingLongest Common Subsequence (LCS)Find the longest subsequence common to two strings.Code:
python
def lcs(str1, str2):
m, n = len(str1), len(str2)
dp = [[0] * (n + 1) for _ in range(m + 1)]
for i in range(1, m + 1):
for j in range(1, n + 1):
if str1[i - 1] == str2[j - 1]:
dp[i][j] = dp[i - 1][j - 1] + 1
else:
dp[i][j] = max(dp[i - 1][j], dp[i][j - 1])
return dp[m][n]
# Example
print(lcs("ABCD", "ACDF")) # Output: 3 (ACD)python
def knapsack(values, weights, capacity):
n = len(values)
dp = [[0] * (capacity + 1) for _ in range(n + 1)]
for i in range(1, n + 1):
for w in range(capacity + 1):
if weights[i - 1] <= w:
dp[i][w] = max(dp[i - 1][w], dp[i - 1][w - weights[i - 1]] + values[i - 1])
else:
dp[i][w] = dp[i - 1][w]
return dp[n][capacity]
# Example
values = [60, 100, 120]
weights = [10, 20, 30]
capacity = 50
print(knapsack(values, weights, capacity)) # Output: 220python
def coin_change(coins, amount):
dp = [float('inf')] * (amount + 1)
dp[0] = 0
for i in range(1, amount + 1):
for coin in coins:
if coin <= i:
dp[i] = min(dp[i], dp[i - coin] + 1)
return dp[amount] if dp[amount] != float('inf') else -1
# Example
print(coin_change([1, 5, 10], 11)) # Output: 2 (10 + 1)- Solves complex problems efficiently by breaking them down.
- Avoids redundant calculations with memoization.
- Requires significant memory for DP tables.
- Can be complex to design and debug.
- Not always intuitive for beginners.
- Greedy Algorithms: For simpler problems, but may not be optimal.
- Brute Force: For small inputs, but inefficient.
- Divide and Conquer: Without memoization, less efficient.
- Use bottom-up DP for memory efficiency.
- Initialize DP tables with appropriate base cases.
- Test edge cases (e.g., empty inputs, zero capacity).
- Optimize space with rolling arrays when possible.
python
def optimize_delivery(items, capacity):
values = [item['value'] for item in items]
weights = [item['weight'] for item in items]
return knapsack(values, weights, capacity)
# Test
items = [{'value': 60, 'weight': 10}, {'value': 100, 'weight': 20}, {'value': 120, 'weight': 30}]
print(optimize_delivery(items, 50)) # Output: 2206. Graph AlgorithmsBreadth-First Search (BFS)BFS explores nodes level by level, using a queue.Code:
python
from collections import deque
def bfs(graph, start):
visited = set()
queue = deque([start])
visited.add(start)
while queue:
vertex = queue.popleft()
print(vertex, end=' ')
for neighbor in graph[vertex]:
if neighbor not in visited:
visited.add(neighbor)
queue.append(neighbor)
# Example
graph = {0: [1, 2], 1: [0, 3, 4], 2: [0], 3: [1], 4: [1]}
bfs(graph, 0) # Output: 0 1 2 3 4python
def dfs(graph, vertex, visited=None):
if visited is None:
visited = set()
visited.add(vertex)
print(vertex, end=' ')
for neighbor in graph[vertex]:
if neighbor not in visited:
dfs(graph, neighbor, visited)
# Example
dfs(graph, 0) # Output: 0 1 3 4 2python
import heapq
def dijkstra(graph, start):
distances = {node: float('inf') for node in graph}
distances[start] = 0
pq = [(0, start)]
while pq:
current_dist, current = heapq.heappop(pq)
if current_dist > distances[current]:
continue
for neighbor, weight in graph[current]:
distance = current_dist + weight
if distance < distances[neighbor]:
distances[neighbor] = distance
heapq.heappush(pq, (distance, neighbor))
return distances
# Example
graph = {0: [(1, 4), (2, 8)], 1: [(0, 4), (3, 5)], 2: [(0, 8), (3, 2)], 3: [(1, 5), (2, 2)]}
print(dijkstra(graph, 0)) # Output: {0: 0, 1: 4, 2: 7, 3: 9}python
def kruskal(graph, vertices):
parent = {v: v for v in vertices}
rank = {v: 0 for v in vertices}
def find(v):
if parent[v] != v:
parent[v] = find(parent[v])
return parent[v]
def union(u, v):
root1, root2 = find(u), find(v)
if root1 != root2:
if rank[root1] < rank[root2]:
parent[root1] = root2
elif rank[root1] > rank[root2]:
parent[root2] = root1
else:
parent[root2] = root1
rank[root1] += 1
edges = []
for u in graph:
for v, weight in graph[u]:
edges.append((weight, u, v))
edges.sort()
mst = []
for weight, u, v in edges:
if find(u) != find(v):
union(u, v)
mst.append((u, v, weight))
return mst
# Example
graph = {0: [(1, 4), (2, 8)], 1: [(0, 4), (3, 5)], 2: [(0, 8), (3, 2)], 3: [(1, 5), (2, 2)]}
vertices = [0, 1, 2, 3]
print(kruskal(graph, vertices)) # Output: [(2, 3, 2), (0, 1, 4), (1, 3, 5)]- BFS/DFS: Simple, versatile for graph traversal.
- Dijkstra’s: Optimal for shortest paths in weighted graphs.
- Kruskal/Prim: Efficient for network optimization.
- BFS/DFS: Not optimal for weighted graphs.
- Dijkstra’s: Doesn’t handle negative weights.
- Kruskal/Prim: Complex for dense graphs.
- A*: For heuristic-based shortest paths.
- Bellman-Ford: For graphs with negative weights.
- Floyd-Warshall: For all-pairs shortest paths.
- Use adjacency lists for sparse graphs.
- Validate graph inputs for cycles or disconnected nodes.
- Optimize Dijkstra’s with a priority queue.
- Test edge cases (e.g., empty graphs, single nodes).
python
def find_friends(graph, user):
visited = set()
queue = deque([user])
visited.add(user)
friends = []
while queue:
person = queue.popleft()
friends.append(person)
for friend in graph[person]:
if friend not in visited:
visited.add(friend)
queue.append(friend)
return friends[1:] # Exclude the user
# Test
network = {'Alice': ['Bob', 'Charlie'], 'Bob': ['Alice', 'David'], 'Charlie': ['Alice'], 'David': ['Bob']}
print(find_friends(network, 'Alice')) # Output: ['Bob', 'Charlie', 'David']7. ConclusionCongratulations on mastering Module 10! You’ve learned over 20 core algorithms, including searching, sorting, recursion, number theory, dynamic programming, and graph algorithms. These skills enable you to solve complex problems like inventory searches, order sorting, financial calculations, and network analysis.
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