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State tree

Branch solutions into a tree and walk it.

State tree

Backtracking walks solutions as a tree. Each node is a partial solution, each branch a choice.

Leaves are full solutions; internal nodes are intermediate states and pruning happens here.

The state tree visualizes the solution space; branches are cut early via pruning.

Permutation example

All permutations of an array are built by picking an unused element at each level of the state tree.

For n elements there are n! leaves; each gives a different ordering.

Generating permutations is one of the best known applications of backtracking.

Practice Generate permutations

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

Starter code

solution.py
def perms(arr):
    # Generate permutations

Sample solution

solution.py
def perms(arr):
    # Generate permutations
    result = []
    def backtrack(path, used):
        if len(path) == len(arr):
            result.append(list(path))
            return
        for i in range(len(arr)):
            if not used[i]:
                used[i] = True
                path.append(arr[i])
                backtrack(path, used)
                path.pop()
                used[i] = False
    backtrack([], [False] * len(arr))
    return result
print(perms([1, 2]))

Expected output

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

Questions and answers

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

1 Add choice

Question 
path.____(arr[i])
Answer 
path.append(arr[i])

Explanation: append is right.

2 State tree: match the terms

Term Meaning
tree Tree
node Partial solution
leaf Full solution
branch Choice

3 What does a leaf represent in the state tree?

  • Full solution Correct answer
  • Partial solution
  • Start
  • Pruning

Explanation: Full solution.

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