DSA Notes
HubAlgorithm vs Data Structure — Key Differences, Definitions, and How They Work TogetherApplications of DSA — How Data Structures and Algorithms Power Real-World SoftwareTime and Space Complexity — A Complete BeginnerWhat is DSA? — A Complete BeginnerWhy Learn DSA? — Career Benefits, Real-World Impact, and Long-Term ValueAsymptotic Analysis — Complete Framework with Recurrence Relations and Master TheoremBest, Average, and Worst Case Analysis — A Practical Guide for Every AlgorithmBig O Notation — Complete Guide with Formal Definition, Rules, and ExamplesBig Omega Notation — Lower Bounds, Best Case, and Formal DefinitionBig Theta (Θ) Notation — Tight Bounds and Exact Growth RatesSpace Complexity — Memory Analysis, Auxiliary Space, and Time-Space TradeoffsTime Complexity Examples — 25 Worked Problems from Basic to AdvancedArray Interview Problems — 15 Classic Problems with Complete SolutionsArray Operations — Every Operation Explained with Code, Complexity, and Edge CasesArray Rotation — Algorithms, Techniques, and Interview ProblemsArray Searching — Linear Search, Binary Search, and Advanced TechniquesArrays — The Complete Introduction to the Most Fundamental Data StructureDynamic Arrays — How Python Lists and ArrayLists Work InternallyMultidimensional Arrays — 2D Arrays, Matrices, and Memory LayoutSliding Window Technique — Complete Guide with 10 Classic ProblemsAnagram Problems — Detection, Grouping, and Sliding Window VariationsPalindrome Problems — Check, Longest, Count, and Expand Around CenterPattern Searching — KMP, Rabin-Karp, and Boyer-Moore AlgorithmsString Interview Problems — 10 Classic Problems with Complete SolutionsString Matching — Longest Common Subsequence, Edit Distance, and Z-AlgorithmString Operations — Complete Reference with Complexity and ImplementationStrings — Complete Introduction to String Data Structure and OperationsCircular Linked List — Structure, Operations, and ApplicationsDoubly Linked List — Complete Implementation with Bidirectional OperationsLinked List Interview Problems — 12 Classic Problems with Complete SolutionsLinked Lists — A Complete Introduction with Memory Model and TypesLinked List Operations — Every Technique with Code and ComplexityReverse Linked List — All Approaches from Basic to AdvancedSingly Linked List — Complete Implementation with All OperationsBalanced Parentheses — Stack-Based Validation with All VariantsInfix, Prefix, and Postfix Expressions — Stack-Based Conversion and EvaluationStack Implementation — Three Complete Implementations with Edge CasesStack Interview Problems — 10 Classic Problems with Complete SolutionsStack — Complete Introduction to LIFO Data StructureStack Using Arrays — Memory Layout, Capacity Management, and PerformanceStack Using Linked List — Implementation, Analysis, and When to Use ItCircular Queue — The Ring Buffer Explained with ImplementationDeque — Double-Ended Queue with Complete Implementation and ApplicationsPriority Queue — Complete Guide from Concept to ImplementationQueue Applications — BFS, Level-Order Traversal, 01-BFS, and MoreQueue Implementation — Array, Linked List, and Two-Stack ApproachesQueue Interview Problems — 10 Classic Problems with Complete SolutionsQueue — Complete Introduction to the FIFO Data StructureAVL Tree - Self-Balancing Binary Search TreeBinary Search Tree (BST)Binary Tree - Structure and OperationsFenwick Tree (Binary Indexed Tree)Red-Black Tree - Another Balanced BSTSegment Tree - Range Query Data StructureTree Interview Problems CollectionTree Data Structure FundamentalsTree Traversals - Inorder, Preorder, Postorder, Level OrderHeap — Complete Introduction to the Priority-Based Tree Data StructureHeap Operations — Complete Reference with Complexity AnalysisHeap Problems — 8 Classic Interview Problems with Complete SolutionsMax Heap — Complete Implementation and ApplicationsMin Heap — Complete Implementation with All OperationsPriority Queue Using Heap — Implementation, Applications, and PatternsBellman-Ford Algorithm — Shortest Path with Negative Weights and Cycle DetectionBFS Algorithm — Breadth-First Search with All Applications and VariantsDFS Algorithm — Depth-First Search with Cycle Detection and All ApplicationsDijkstraFloyd-Warshall Algorithm — All-Pairs Shortest Paths with Complete AnalysisGraph Interview Problems — 12 Classic Problems with Complete SolutionsGraph — Complete Introduction to the Most Powerful Data StructureGraph Representation — Adjacency List, Matrix, Edge List with Complete AnalysisKruskalMinimum Spanning Tree — Definition, Properties, and ApplicationsPrimShortest Path Algorithms — Complete Comparison and Selection GuideTopological Sorting — DFS and BFS (KahnUnion-Find (DSU) in Graphs — Connected Components and Dynamic ConnectivitySeparate Chaining — Complete Implementation and AnalysisCollision Handling — Chaining vs Open Addressing Complete ComparisonHash Functions — Design, Properties, and Common ImplementationsHashing — Complete Introduction to Hash Tables and Hash FunctionsHashing Problems — 10 Classic Interview Problems with Complete SolutionsOpen Addressing — Linear Probing, Quadratic Probing, and Double HashingBubble Sort — Complete Analysis with Optimizations and When to Use ItCounting Sort — O(n+k) Sorting That Breaks the Comparison Lower BoundHeap Sort — Guaranteed O(n log n) with O(1) SpaceInsertion Sort — The O(n) Best Case Sort That Powers PythonMerge Sort — Complete Guide to the Guaranteed O(n log n) Stable SortQuick Sort — The Fastest Practical Sorting Algorithm with Complete AnalysisRadix Sort — O(nk) Digit-by-Digit Sorting for Large Integer RangesSelection Sort — Complete Analysis with Swaps Minimization PropertySorting Algorithm Comparison — The Complete Decision GuideSorting — Complete Introduction to Sorting AlgorithmsBinary Search — Complete Guide with All Variants and Proof of CorrectnessInterpolation Search — O(log log n) Search for Uniformly Distributed DataLinear Search — Complete Guide with All Variants and ApplicationsSearch Algorithm Comparison — Choosing the Right Search for Every SituationSearching Interview Problems — 10 Classic Problems with Complete SolutionsTernary Search — Finding Maximum/Minimum of Unimodal FunctionsBacktracking Introduction — The Recursive Exploration with UndoRecursion Basics — The Complete BeginnerRecursion Problems — 12 Classic Problems from Basic to AdvancedRecursion Tree — Visualizing Recursion for Complexity AnalysisRecursion vs Iteration — When to Use Each and How to ConvertTail Recursion — Definition, Optimization, and Converting to IterationDP Interview Problems — 12 Classic Problems with Pattern Recognition GuideDynamic Programming — Complete Introduction with Properties and Problem IdentificationFibonacci DP — From Exponential to Linear with Memoization and Matrix ExponentiationKnapsack Problem — 0/1, Unbounded, and Fractional with Complete AnalysisLongest Common Subsequence — Complete Guide with DP Table and String VariantsLongest Increasing Subsequence — O(n log n) with Patience SortingMatrix Chain Multiplication — Interval DP with Complete AnalysisMemoization — Top-Down Dynamic Programming with CachingTabulation — Bottom-Up Dynamic Programming with Complete Table BuildingActivity Selection Problem — Greedy Scheduling with Proof of CorrectnessFractional Knapsack — Greedy by Value Density with ProofGreedy Interview Problems — 10 Classic Problems with Complete SolutionsGreedy Algorithms — Complete Introduction with Proof TechniquesHuffman Coding — Optimal Prefix-Free Data CompressionJob Sequencing Problem — Greedy Scheduling with DeadlinesBacktracking — The Complete Guide to Recursive Exploration with UndoBacktracking Problems — 12 Classic Interview Problems with Complete SolutionsN-Queens Problem — Complete Solution with Backtracking and OptimizationRat in a Maze — Backtracking Path Finding with All SolutionsSubset Generation — All Subsets, Combinations, and Partitions with BacktrackingSudoku Solver — Backtracking with Constraint PropagationCodeforces Guide — Complete Strategy to Reach 1800+ RatingCommon CP Techniques — Essential Algorithms Every Competitive Programmer Must KnowContest Strategies — How to Perform Your Best in Competitive Programming ContestsCompetitive Programming — Complete Introduction for BeginnersFast Input/Output — Why I/O Speed Matters and Complete Templates for C++ and PythonLeetCode Guide — Complete Strategy to Crack FAANG InterviewsBit Manipulation — Complete Guide with All Tricks and Interview ProblemsBloom Filter — Space-Efficient Probabilistic Membership TestingComputational Geometry — Points, Lines, Convex Hull, and Geometric AlgorithmsDisjoint Set Union (DSU) — Union-Find with Path Compression and Union by RankRandomized Algorithms — Las Vegas, Monte Carlo, and Probabilistic TechniquesSkip List — The Probabilistic Data Structure that Competes with Balanced BSTsSuffix Array — Construction, LCP Array, and String Search ApplicationsTrie Data Structure — Complete Guide with Implementation and ApplicationsAutocomplete System Using TrieLibrary Management System Using DSARoute Finder Using GraphsMini Search Engine ProjectSocial Network Analysis Using GraphsEssential Coding Patterns for InterviewsDSA Cheatsheet: Quick ReferenceEssential DSA Interview QuestionsFAANG DSA Sheet: Company-Specific Patterns and ProblemsProblem Solving Strategies for Coding InterviewsComplexity CheatsheetImportant Algorithms SummaryBest Practice Platforms for DSARecommended Books for DSA
Master the sliding window technique — one of the most powerful array problem-solving patterns. Learn fixed-size and variable-size windows, when to apply this technique, and solve 10 classic interview problems including maximum sum subarray, longest substring without repeats, minimum window substring, and more.
Introduction
The sliding window technique is one of the most powerful patterns for solving array and string problems efficiently. It transforms problems that would naively require O(n²) or O(n³) nested loops into elegant O(n) solutions.
The key insight: instead of recomputing results from scratch for every possible window position, we *slide* the window incrementally — adding one element at the right edge and removing one element at the left edge — and update our result in O(1) per step.
This technique is essential. It appears in a large percentage of medium and hard array/string interview problems.
Type 1 — Fixed-Size Sliding Window
The window has a fixed size k. It slides one position at a time from left to right.
Template:
Example
def fixed_window_template(arr, k):
n = len(arr)
if n < k:
return None # Handle edge case
# Initialize window (first k elements)
window_result = compute(arr[:k]) # Initial window
best = window_result
# Slide the window from position 1 to n-k
for i in range(k, n):
# Add arr[i] (new right element)
# Remove arr[i - k] (old left element that fell off)
window_result = update(window_result, arr[i], arr[i - k])
best = max(best, window_result) # or min, or other operation
return best
Problem 1 — Maximum Sum Subarray of Size K
Example
def max_sum_subarray(arr, k):
"""
Find the maximum sum of any contiguous subarray of exactly k elements.
Time: O(n), Space: O(1)
"""
n = len(arr)
if n < k:
return None
# Compute sum of first window
window_sum = sum(arr[:k]) # O(k)
max_sum = window_sum
# Slide window: add new element, remove old element
for i in range(k, n):
window_sum += arr[i] # Add incoming element
window_sum -= arr[i - k] # Remove outgoing element
max_sum = max(max_sum, window_sum)
return max_sum
print(max_sum_subarray([2, 1, 5, 1, 3, 2], 3)) # 9 (subarray [5,1,3])
print(max_sum_subarray([2, 3, 4, 1, 5], 2)) # 7 (subarray [3,4])
Problem 2 — Average of Subarrays of Size K
Example
def find_averages(arr, k):
"""Find averages of all subarrays of size k."""
n = len(arr)
averages = []
window_sum = sum(arr[:k])
averages.append(window_sum / k)
for i in range(k, n):
window_sum += arr[i]
window_sum -= arr[i - k]
averages.append(window_sum / k)
return averages
print(find_averages([1, 3, 2, 6, -1, 4, 1, 8, 2], 5))
# [2.2, 2.8, 2.4, 3.6, 2.8]
Problem 3 — First Negative in Every Window
Example
from collections import deque
def first_negative_in_window(arr, k):
"""
For each window of size k, find the first negative number.
Use a deque to track negative number positions.
Time: O(n), Space: O(k)
"""
n = len(arr)
result = []
neg_queue = deque() # Stores indices of negatives in current window
for i in range(n):
# Add index if element is negative
if arr[i] < 0:
neg_queue.append(i)
# Remove indices outside current window
while neg_queue and neg_queue[0] < i - k + 1:
neg_queue.popleft()
# Window is complete starting from index k-1
if i >= k - 1:
result.append(arr[neg_queue[0]] if neg_queue else 0)
return result
print(first_negative_in_window([-8, 2, 3, -6, 10], 2)) # [-8, 0, -6, -6]
Type 2 — Variable-Size Sliding Window
The window size is not fixed. It expands to the right and shrinks from the left based on a condition.
Template:
Example
def variable_window_template(arr):
left = 0
result = initial_value
window_state = {} # Could be a hash map, counter, sum, etc.
for right in range(len(arr)):
# Expand window: add arr[right]
window_state = update_add(window_state, arr[right])
# Shrink window while condition is violated
while not is_valid(window_state):
window_state = update_remove(window_state, arr[left])
left += 1
# Update result with current valid window
result = update_result(result, right - left + 1)
return result
Problem 4 — Smallest Subarray with Sum ≥ S
Example
def smallest_subarray_sum(arr, s):
"""
Find the length of the smallest contiguous subarray with sum >= s.
If no such subarray, return 0.
Time: O(n), Space: O(1)
"""
n = len(arr)
min_length = float('inf')
window_sum = 0
left = 0
for right in range(n):
window_sum += arr[right] # Expand window
while window_sum >= s: # Shrink window while sum is valid
min_length = min(min_length, right - left + 1)
window_sum -= arr[left]
left += 1
return min_length if min_length != float('inf') else 0
print(smallest_subarray_sum([2, 1, 5, 2, 3, 2], 7)) # 2 (subarray [5,2])
print(smallest_subarray_sum([2, 1, 5, 2, 8], 7)) # 1 (subarray [8])
Problem 5 — Longest Substring Without Repeating Characters
Example
def length_of_longest_substring(s):
"""
Find the length of the longest substring with all unique characters.
Time: O(n), Space: O(min(m, n)) where m = charset size
"""
char_index = {} # Maps char → its most recent index in window
max_length = 0
left = 0
for right in range(len(s)):
# If char already in window, shrink left past its previous occurrence
if s[right] in char_index and char_index[s[right]] >= left:
left = char_index[s[right]] + 1
char_index[s[right]] = right # Update char's position
max_length = max(max_length, right - left + 1)
return max_length
print(length_of_longest_substring("abcabcbb")) # 3 ("abc")
print(length_of_longest_substring("bbbbb")) # 1 ("b")
print(length_of_longest_substring("pwwkew")) # 3 ("wke")
Trace for "abcabcbb":
| right=0 (a) | window=[a], max=1 |
| right=1 (b) | window=[ab], max=2 |
| right=2 (c) | window=[abc], max=3 |
| right=3 (a): a is in window at idx 0 | left=1, window=[bca], max=3 |
| right=4 (b): b is in window at idx 1 | left=2, window=[cab], max=3 |
| right=5 (c): c is in window at idx 2 | left=3, window=[abc], max=3 |
| right=6 (b): b is in window at idx 4 | left=5, window=[cb], max=3 |
| right=7 (b): b is in window at idx 6 | left=7, window=[b], max=3 |
Problem 6 — Longest Subarray with Sum ≤ K
Example
def longest_subarray_sum_k(arr, k):
"""
Find the longest subarray where sum does not exceed k.
Works for positive numbers.
Time: O(n), Space: O(1)
"""
max_length = 0
window_sum = 0
left = 0
for right in range(len(arr)):
window_sum += arr[right]
while window_sum > k:
window_sum -= arr[left]
left += 1
max_length = max(max_length, right - left + 1)
return max_length
print(longest_subarray_sum_k([1, 2, 1, 0, 1, 1, 0], 4)) # 5
Problem 7 — Longest Subarray with At Most K Distinct Characters
Example
from collections import defaultdict
def longest_k_distinct(s, k):
"""
Longest substring with at most k distinct characters.
Time: O(n), Space: O(k)
"""
char_count = defaultdict(int)
max_length = 0
left = 0
for right in range(len(s)):
char_count[s[right]] += 1 # Add character
# Shrink window if more than k distinct characters
while len(char_count) > k:
char_count[s[left]] -= 1
if char_count[s[left]] == 0:
del char_count[s[left]]
left += 1
max_length = max(max_length, right - left + 1)
return max_length
print(longest_k_distinct("araaci", 2)) # 4 ("araa")
print(longest_k_distinct("cbbebi", 3)) # 5 ("cbbeb")
Type 3 — Two-Pointer Window (Special Case)
Some problems use two pointers where both can move in interesting ways.
Problem 8 — Container With Most Water
Example
def max_area(heights):
"""
Two vertical lines form a container. Find maximum water area.
Two pointers starting at both ends, moving toward each other.
Time: O(n), Space: O(1)
"""
left, right = 0, len(heights) - 1
max_water = 0
while left < right:
# Width = right - left, height = min of two sides
width = right - left
height = min(heights[left], heights[right])
max_water = max(max_water, width * height)
# Move the shorter side inward (moving taller would only decrease width)
if heights[left] < heights[right]:
left += 1
else:
right -= 1
return max_water
print(max_area([1,8,6,2,5,4,8,3,7])) # 49
Problem 9 — Minimum Window Substring (Hard)
Example
from collections import Counter
def min_window_substring(s, t):
"""
Find the smallest substring of s containing all characters of t.
Time: O(|s| + |t|), Space: O(|s| + |t|)
"""
if not t or not s:
return ""
need = Counter(t) # Characters we need and their counts
missing = len(t) # Total characters still needed
best = ""
best_start = 0
left = 0
for right, char in enumerate(s):
if need[char] > 0: # This character is needed
missing -= 1
need[char] -= 1 # We have one more of this character
# Window contains all required characters
if missing == 0:
# Shrink from left as much as possible
while need[s[left]] < 0:
need[s[left]] += 1
left += 1
# Update best window
window = s[left:right + 1]
if not best or len(window) < len(best):
best = window
# Move left to search for next valid window
need[s[left]] += 1
missing += 1
left += 1
return best
print(min_window_substring("ADOBECODEBANC", "ABC")) # "BANC"
print(min_window_substring("a", "a")) # "a"
print(min_window_substring("a", "b")) # ""
Recognizing Sliding Window Problems
A problem is likely solvable with sliding window if it asks about:
| Keyword | Example |
|---|---|
| "contiguous subarray" | Maximum sum subarray of size k |
| "substring" with a constraint | Longest substring without repeating |
| "window" | Moving average |
| "k elements" | Maximum in every window of size k |
| "minimum/maximum subarray" | Minimum window containing all chars |
| "at most k distinct" | Longest substring with k distinct |
Ask yourself:
- Does the problem involve a contiguous sequence?
- Is there a constraint (sum ≤ k, at most k distinct chars, etc.)?
- Can adding one element on the right and removing one on the left update the state in O(1)?
If yes to all three, sliding window applies.
Problem 10 — Maximum Consecutive Ones with K Flips
Example
def longest_ones(nums, k):
"""
You can flip at most k zeros to ones.
Find the length of the longest subarray of 1s.
Time: O(n), Space: O(1)
"""
left = 0
zeros_in_window = 0
max_length = 0
for right in range(len(nums)):
if nums[right] == 0:
zeros_in_window += 1
# Shrink window if we have more than k zeros
while zeros_in_window > k:
if nums[left] == 0:
zeros_in_window -= 1
left += 1
max_length = max(max_length, right - left + 1)
return max_length
print(longest_ones([1,1,1,0,0,0,1,1,1,1,0], 2)) # 6
print(longest_ones([0,0,1,1,0,0,1,1,1,0,1,1,0,0,0,1,1,1,1], 3)) # 10
Summary
| Type | Window Size | Direction | Key Operation |
|---|---|---|---|
| Fixed Window | Constant k | One direction | Add right, remove left[i-k] |
| Variable Window | Varies | One direction | Expand right, shrink left until valid |
| Two-Pointer | Varies | Both directions | Both pointers move based on condition |
Time complexity: O(n) — each element enters and leaves the window at most once
Space complexity: O(1) for simple windows, O(k) if storing window contents
The sliding window pattern eliminates redundant recomputation — the defining characteristic that makes it so powerful. Master this pattern and a large class of array/string problems becomes straightforward.
*Next Lesson: Array Interview Problems — Applying everything learned about arrays to solve classic and hard interview problems.*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Sliding Window Technique — Complete Guide with 10 Classic Problems.
Interview Use
Prepare one clear explanation, one practical example, and one common mistake for this Data Structures & Algorithms topic.
Search Terms
data-structures-algorithms, data structures & algorithms, data, structures, algorithms, arrays, sliding, window
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