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Fibonacci memoization

Speed up recursive Fibonacci with memoization.

Fibonacci memoization

Speed up recursive Fibonacci with memoization: store each n's result in a dictionary.

Compute and store on first call; return the stored value on later calls.

This small change drops fibonacci from Θ(2^n) to Θ(n); a big difference.

Dictionary check

First check if n is in the cache. If yes return directly; else compute and store.

This ensures each subproblem is solved only once.

The cache check is the heart of memoization; it must be implemented correctly.

Practice Fibonacci memo

Write fib_memo(n, memo). Recursive Fibonacci with memoization. Print 55 with print(fib_memo(10, {})).

Starter code

solution.py
def fib_memo(n, memo):
    # Fibonacci memo

Sample solution

solution.py
def fib_memo(n, memo):
    # Fibonacci memo
    if n in memo:
        return memo[n]
    if n <= 1:
        return n
    memo[n] = fib_memo(n - 1, memo) + fib_memo(n - 2, memo)
    return memo[n]
print(fib_memo(10, {}))

Expected output

55

Questions and answers

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

1 Cache check

Question 
if n in ____:
Answer 
if n in memo:

Explanation: memo is right.

2 Fibonacci memoization: match the terms

Term Meaning
memo Cache
dict Dictionary
n Parameter
return Return

3 What is Fibonacci memoization's complexity?

  • Θ(log n)
  • Θ(2^n)
  • Θ(n^2)
  • Θ(n) Correct answer

Explanation: Θ(n).

All lessons in Advanced

1 DP - Memoization

  1. What is memoization?
  2. Fibonacci memoization
  3. Dict-based cache
  4. Summary - Memoization

2 DP - Tabulation

  1. What is tabulation?
  2. Building the table
  3. Fibonacci tabulation
  4. Summary - Tabulation

3 LCS

  1. LCS definition
  2. DP table
  3. Backtracking the table
  4. Summary - LCS

4 Knapsack

  1. Knapsack problem
  2. 0/1 knapsack DP
  3. Decision table
  4. Summary - Knapsack

5 DP project

  1. Project: Overview
  2. Cost table
  3. DP solution
  4. Print the result

6 Matrix chain

  1. Multiplication order
  2. DP table
  3. Split point
  4. Summary - Matrix chain

7 Floyd-Warshall

  1. All pairs
  2. Triple loop
  3. Table update
  4. Summary - Floyd-Warshall

8 Dijkstra

  1. Single source
  2. Priority queue
  3. Relaxation
  4. Summary - Dijkstra

9 Bellman-Ford

  1. Negative weights
  2. Edge relaxation loop
  3. Negative cycle
  4. Summary - Bellman-Ford

10 Kruskal

  1. Minimum spanning tree
  2. Edge sorting
  3. Cycle check
  4. Summary - Kruskal

11 Prim

  1. Prim idea
  2. Priority queue
  3. Visited set
  4. Summary - Prim

12 Topological sort

  1. Dependency order
  2. DFS-based sort
  3. Kahn algorithm
  4. Summary - Topological sort

13 Union-find, trie, segment tree

  1. Union-find
  2. Trie
  3. Segment tree
  4. Summary - Advanced structures

14 Advanced final

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