When you’re preparing for coding interviews or trying to deepen your understanding of algorithmic thinking, it’s easy to get lost in hundreds of problems. But behind the chaos, there’s a consistent mental model that makes everything easier:

All algorithms are just different ways to traverse a space of states or possibilities.

Whether it’s a list, a tree, or an abstract state space — you’re either going deep (DFS), broad (BFS), jumping (binary search), or reusing paths (DP).

🚦 Why Think in Terms of Traversal?

Imagine solving a puzzle — a maze, a Rubik’s cube, or even planning a trip. You’re essentially moving from a start state toward a goal, trying different paths, avoiding dead ends, and learning from past moves.

That’s what algorithms do:

  • Recursion is like exploring a maze by diving in until you hit a wall, then backtracking.
  • DP is remembering dead ends so you never go back there again.
  • BFS is checking all routes layer by layer.
  • Binary Search is teleporting halfway each time in a structured environment.

Benefits of Traversal Thinking:

  • ✅ You can break any problem into smaller moves or decisions
  • ✅ You focus on state + transition instead of syntax
  • ✅ You spot when to cache, prune, or reverse

This post explores this mindset through Fibonacci as a warmup, then suggests better general problems at the end.

🧠 1. DFS (Recursion) — Brute Force State Traversal

def fib(n):
    if n <= 1:
        return n
    return fib(n - 1) + fib(n - 2)

Each call explores two sub-branches → full binary tree of calls. This is depth-first search over the state space fib(n) → fib(n-1), fib(n-2).

🧱 Time: O(2^n) — exponential explosion.

🧠 2. Memoization — Caching Repeated Traversals

memo = {}
def fib(n):
    if n in memo:
        return memo[n]
    if n <= 1:
        return n
    memo[n] = fib(n - 1) + fib(n - 2)
    return memo[n]

Still DFS — but now you remember visited states, just like a visited set in graph traversal.

🧱 Time: O(n), Space: O(n)

🔁 3. Bottom-Up DP — Traversal by Building Up

def fib(n):
    if n <= 1:
        return n
    dp = [0] * (n + 1)
    dp[1] = 1
    for i in range(2, n + 1):
        dp[i] = dp[i - 1] + dp[i - 2]
    return dp[n]

No recursion. Just iterate through all reachable states from 0 → n. Still traversal — but more efficient.

🧵 4. Space-Optimized DP — Traversal with Constant Memory

def fib(n):
    if n <= 1:
        return n
    a, b = 0, 1
    for _ in range(2, n + 1):
        a, b = b, a + b
    return b

Only care about last 2 states → no full table needed.

🔄 5. Stack-Based Simulation — DFS Without Recursion

def fib(n):
    if n <= 1:
        return n
    stack = [(n, False)]
    memo = {}
    
    while stack:
        cur, visited = stack.pop()
        if cur in memo:
            continue
        if cur <= 1:
            memo[cur] = cur
        elif not visited:
            stack.append((cur, True))
            stack.append((cur - 1, False))
            stack.append((cur - 2, False))
        else:
            memo[cur] = memo[cur - 1] + memo[cur - 2]

    return memo[n]

Simulate DFS using explicit stack → useful when recursion is limited or stack overflows.

🌐 6. BFS-style Traversal (Theoretical for Fib)

from collections import deque

def fib(n):
    if n <= 1:
        return n
    q = deque([(0, 0), (1, 1)])
    while q:
        i, val = q.popleft()
        if i == n:
            return val
        q.append((i + 1, val + (q[0][1] if q else 0)))

Less efficient, but demonstrates level-order traversal thinking.

📌 Summary: One Problem, Many Traversals

Pattern Method Notes
DFS Recursion Simple but slow
DFS + Cache Memoization Top-down DP
Linear Traversal Bottom-up DP Fast and easy
Minimal Traversal 2-var DP O(1) space
Explicit DFS Stack simulation No recursion
Breadth-first (BFS) Queue-based simulation Rare for Fib

👉 These are not separate ideas — just different ways to walk through the state space.

🌟 Bonus: Better General Problems for Traversal Patterns

While Fibonacci is minimal and great for learning, these problems scale the idea better:

  • 🧗‍♂️ Climbing Stairs – each step can go +1 or +2
  • 🧮 Coin Change – how many ways to reach a target amount
  • 🎯 Target Sum – pick + or - on each number to hit target
  • 🎲 Dice Combinations – from 0 to n, ways to roll
  • 🧩 Word Ladder / Graph Paths – classic BFS

All of these can be solved by:

  • DFS brute force
  • DFS + memo
  • Bottom-up DP
  • BFS
  • Explicit stack/queue

Once you adopt traversal thinking, all these problems become different flavors of the same pattern. Practice them deeply — not widely.

Mastering algorithms is not about solving more problems. It’s about seeing one problem in many different ways.