---
title: 'Sliding Window'
description: 'Two pointers plus incremental state between them.'
---

## The idea

Two pointers that move the **same direction**, carrying state between them.

Think of a caterpillar:
- The head crawls forward and eats (expand right).
- When the rule breaks, the tail pulls in until the rule holds again (contract left).

The window is always a **contiguous** slice. You never rebuild it — you update the carried state as edges move. That is what makes it O(n).

**The recipe:**
1. Expand right. Add the new element to your state.
2. Rule broken? Contract left until it holds.
3. Record the best window. Repeat.

## The picture

```mermaid
flowchart TD
    A["a b c a b c<br/>[a]           window = a"] --> B["[a b]         expand → ab"]
    B --> C["[a b c]       expand → abc, best = 3"]
    C --> D["a [b c a]     'a' repeats → contract, then expand"]
    D --> E["a b [c a b]   keep sliding, best stays 3"]

    style C fill:#2e7d32,color:#fff
```

## Longest Substring Without Repeating (LC 3)

State = a set of chars in the window. Repeat found? Shrink from the left until it is gone.

```python
def length_of_longest_substring(s: str) -> int:
    window = set()
    left = 0
    best = 0

    for right, c in enumerate(s):
        while c in window:            # rule broken
            window.remove(s[left])    # contract left
            left += 1
        window.add(c)                 # expand right
        best = max(best, right - left + 1)
    return best
```

## Best Time to Buy and Sell (LC 121) — the hidden window

Looks like a stock problem. It is a window problem.
Left = cheapest buy so far. Right = today. Carry one number: the min price.

```python
def max_profit(prices: list[int]) -> int:
    min_price = prices[0]
    best = 0

    for p in prices[1:]:
        best = max(best, p - min_price)
        min_price = min(min_price, p)
    return best
```

## Character Replacement (LC 424) — window with a budget

Rule: window is valid if `window size − count of top letter ≤ k`.
That many replacements fix the window. Carry a frequency map.

```python
from collections import defaultdict

def character_replacement(s: str, k: int) -> int:
    count = defaultdict(int)
    left = 0
    best = 0
    top = 0                                  # highest letter count seen

    for right, c in enumerate(s):
        count[c] += 1
        top = max(top, count[c])
        if (right - left + 1) - top > k:     # over budget
            count[s[left]] -= 1              # contract exactly one step
            left += 1
        best = max(best, right - left + 1)
    return best
```

## When the window fails

Sliding window needs a one-way rule: **growing can only hurt, shrinking can only help.**
Negative numbers break this — a bigger window can flip from bad to good.
That is when you reach for prefix sum + hashmap (LC 560) instead.
Say this decision out loud in the interview.

## Complexity

| | Time | Space |
|---|---|---|
| All patterns above | O(n) — each index enters and leaves once | O(1) or O(alphabet) |

## Close-out ritual

Before you submit, say out loud:
1. Time and space complexity — note each element enters and leaves the window at most once.
2. One edge case trace: empty string, all same char, or k larger than the string.

## Plan problems

**A-set:** LC 121 · 3 · 424 · 567 · 76
**B-set:** LC 209 · 1004 · 643

<YouTube id="QGNAVBn1_bc" title="Sliding Window" />