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What is backtracking?

Trial-and-error search method.

What is backtracking?

Backtracking searches by trial and error. At an infeasible step it returns and tries another path.

It walks the solution space like a tree; pruning speeds it up and cuts unnecessary branches.

Backtracking is often used for N-queens, sudoku and maze solving; thanks to its flexibility.

Trial and error

Backtracking takes a step, checks if it can proceed. If yes it deepens; if no it returns.

This is implemented naturally with recursion; each call is a step.

Thanks to recursion, backtracking happens automatically; the call stack holds the state.

Practice Generate subsets

Write subsets(arr). Generate all subsets of an array with backtracking. Print [[], [1], [2], [1, 2]] with print(subsets([1, 2])).

Starter code

solution.py
def subsets(arr):
    # Generate subsets

Sample solution

solution.py
def subsets(arr):
    # Generate subsets
    result = []
    def backtrack(start, path):
        result.append(list(path))
        for i in range(start, len(arr)):
            path.append(arr[i])
            backtrack(i + 1, path)
            path.pop()
    backtrack(0, [])
    return result
print(subsets([1, 2]))

Expected output

[[], [1], [2], [1, 2]]

Questions and answers

Review the lesson questions together with their answers. To test yourself, solve the lesson interactively.

1 When does backtracking return?

  • Every step
  • When a step is infeasible Correct answer
  • Never
  • Only at the start

Explanation: When a step is infeasible.

2 Recursion base

Question 
if len(path) == ____:
Answer 
if len(path) == n:

Explanation: n is right.

3 What is backtracking: match the terms

Term Meaning
backtrack Return
try Try
undo Undo
prune Prune

All lessons in Intermediate

1 Merge sort

  1. Divide and conquer
  2. Merge operation
  3. Full merge sort
  4. Summary - Merge sort

2 Quick sort

  1. Pivot selection
  2. Partitioning
  3. Recursive quick sort
  4. Summary - Quick sort

3 Heap sort

  1. What is a heap?
  2. Heapify
  3. Build heap and sort
  4. Summary - Heap sort

4 Counting sort

  1. Counting idea
  2. Count array
  3. Output array
  4. Summary - Counting sort

5 Radix sort

  1. Radix idea
  2. Digit pass
  3. Stable sub-sort
  4. Summary - Radix sort

6 Sorting project

  1. Project: Overview
  2. Score array
  3. Apply merge sort
  4. Print statistics

7 Tree

  1. What is a tree?
  2. Node and edge
  3. Binary tree
  4. Summary - Trees

8 Binary search tree

  1. BST rule
  2. BST insert
  3. BST search
  4. Summary - BST

9 AVL tree

  1. Balanced tree
  2. Rotation
  3. Balance after insert
  4. Summary - AVL tree

10 Heap

  1. Min and max heap
  2. Insert
  3. Extract
  4. Summary - Heap

11 Priority queue

  1. What is a priority queue?
  2. Heap-based implementation
  3. Push and pop
  4. Summary - Priority queue

12 Data structures project

  1. Project: Overview
  2. Undo stack
  3. Pending queue
  4. Print the state

13 Graph basics

  1. What is a graph?
  2. Adjacency list
  3. Adjacency matrix
  4. Summary - Graph

14 Breadth-first search

  1. BFS idea
  2. Using a queue
  3. Marking visited
  4. Summary - BFS

15 Depth-first search

  1. DFS idea
  2. Stack or recursion
  3. Backtracking
  4. Summary - DFS

16 Greedy algorithms

  1. Greedy approach
  2. Coin change
  3. Greedy limits
  4. Summary - Greedy

17 Divide and conquer

  1. Three steps
  2. Recursive structure
  3. Master theorem
  4. Summary - Divide and conquer

18 Backtracking

  1. What is backtracking?
  2. State tree
  3. Pruning
  4. Summary - Backtracking

19 Graph project

  1. Project: Overview
  2. City graph
  3. BFS for reachability
  4. Print the result

20 Intermediate final

  1. Challenge 1
  2. Challenge 2
  3. Challenge 3
  4. Challenge 4