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 memoization completely — what it is, how it transforms exponential recursion to polynomial time, implementing memo dictionaries in Python, using lru_cache and functools.cache, multi-dimensional memoization, when memoization is better than tabulation, and common pitfalls.
What Is Memoization?
Memoization is the technique of storing the results of expensive function calls and returning the cached result when the same inputs occur again.
The word comes from "memo" (note) — you write down the answer so you don't have to recompute it.
Memoization is the natural way to implement dynamic programming: start with a recursive solution (which is often obvious), then add a cache to prevent recomputation.
Adding Memoization — Step by Step
Example
def fibonacci_memo(n, memo={}):
"""
Memoized Fibonacci — O(n) time, O(n) space.
Each value computed exactly once.
"""
# Check cache first
if n in memo:
return memo[n]
# Base cases
if n <= 1:
return n
# Compute and cache
memo[n] = fibonacci_memo(n-1, memo) + fibonacci_memo(n-2, memo)
return memo[n]
print(fibonacci_memo(50)) # 12586269025 — instant!
print(fibonacci_memo(1000)) # 43466557686... — still instant!
What changed: One if n in memo: return memo[n] check and one memo[n] = ... assignment. That's it. The algorithm went from O(2^n) to O(n).
How it works:
- First call:
fib(5)→ callsfib(4)andfib(3) fib(4)→ callsfib(3)(not computed yet) andfib(2)(not computed yet)- When
fib(3)completes, it's stored in memo - When
fib(5)then needsfib(3)directly, it's already cached → O(1) lookup
Python's Built-in Memoization
Python provides memoization through functools:
@lru_cache — Least Recently Used Cache
python example
from functools import lru_cache
@lru_cache(maxsize=None) # maxsize=None means unlimited cache
def fibonacci(n):
if n <= 1: return n
return fibonacci(n-1) + fibonacci(n-2)
print(fibonacci(100)) # 354224848179261915075
print(fibonacci.cache_info()) # CacheInfo(hits=98, misses=101, maxsize=None, currsize=101)@cache — Simpler Version (Python 3.9+)
python example
from functools import cache
@cache
def fibonacci(n):
if n <= 1: return n
return fibonacci(n-1) + fibonacci(n-2)@cache is equivalent to @lru_cache(maxsize=None) but slightly faster.
Multi-Dimensional Memoization
For problems with multiple parameters:
Example
from functools import lru_cache
@lru_cache(maxsize=None)
def knapsack(items, capacity, n):
"""
0/1 Knapsack: items is a tuple of (weight, value) pairs.
State: (n items considered, remaining capacity)
"""
if n == 0 or capacity == 0:
return 0
weight, value = items[n-1]
if weight > capacity:
# Can't include this item
return knapsack(items, capacity, n-1)
else:
# Max of: exclude item n | include item n
return max(
knapsack(items, capacity, n-1),
knapsack(items, capacity - weight, n-1) + value
)
items = ((2, 6), (2, 10), (3, 12)) # (weight, value) pairs
print(knapsack(items, 5, len(items))) # 22
Note: lru_cache requires all arguments to be hashable. Use tuples (not lists) for collections.
Memoization with Dictionary (Manual)
When lru_cache isn't available or when you need more control:
Example
def longest_common_subsequence(s1, s2):
"""LCS using manual memoization dictionary."""
memo = {}
def dp(i, j):
if (i, j) in memo: return memo[(i, j)]
if i == 0 or j == 0:
return 0
if s1[i-1] == s2[j-1]:
result = dp(i-1, j-1) + 1
else:
result = max(dp(i-1, j), dp(i, j-1))
memo[(i, j)] = result
return result
return dp(len(s1), len(s2))
print(longest_common_subsequence("ABCBDAB", "BDCAB")) # 4
Classic Memoization Problems
1. Coin Change (Minimum Coins)
Example
from functools import lru_cache
def coin_change(coins, amount):
"""
Minimum number of coins to make amount.
State: remaining amount
Recurrence: dp(amount) = 1 + min(dp(amount - coin) for each coin)
"""
@lru_cache(maxsize=None)
def dp(remaining):
if remaining == 0: return 0
if remaining < 0: return float('inf')
return 1 + min(dp(remaining - coin) for coin in coins)
result = dp(amount)
return result if result != float('inf') else -1
print(coin_change([1, 5, 6, 9], 11)) # 2 (5+6)
print(coin_change([2], 3)) # -1 (impossible)
2. Unique Paths in Grid
python example
from functools import lru_cache
def unique_paths(m, n):
"""
Count paths from (0,0) to (m-1,n-1) moving only right or down.
State: (row, col)
Recurrence: dp(r,c) = dp(r-1,c) + dp(r,c-1)
"""
@lru_cache(maxsize=None)
def dp(r, c):
if r == 0 or c == 0: return 1 # Only one path along edges
return dp(r-1, c) + dp(r, c-1)
return dp(m-1, n-1)
print(unique_paths(3, 7)) # 28
print(unique_paths(3, 3)) # 63. Word Break
Example
from functools import lru_cache
def word_break(s, word_dict):
"""
Can s be segmented using words from word_dict?
State: starting index in s
"""
words = frozenset(word_dict)
@lru_cache(maxsize=None)
def dp(start):
if start == len(s): return True # Successfully segmented
for end in range(start + 1, len(s) + 1):
if s[start:end] in words and dp(end):
return True
return False
return dp(0)
print(word_break("leetcode", ["leet","code"])) # True
print(word_break("applepenapple", ["apple","pen"])) # True
print(word_break("catsandog", ["cats","dog","and"])) # False
When Memoization Is Better Than Tabulation
Prefer Memoization when
✅ Not all states are needed (sparse state space)
Example: dp(100, 50) might never call dp(1, 99) — why compute it?
✅ Recursion is the natural way to think about the problem
✅ The state space is irregular (tree problems, graph problems)
✅ Prototyping quickly — memoization is faster to code
Prefer Tabulation when
✅ All states need to be computed anyway
✅ No recursion overhead (important for Python speed)
✅ Stack overflow is a concern (deep recursion)
✅ Space optimization needed (slide over table rows)
Common Memoization Pitfalls
Pitfall 1 — Mutable Default Argument
python example
# WRONG: memo={} is shared across all calls!
def fib(n, memo={}):
...
# CORRECT: Use None as default, create inside
def fib(n, memo=None):
if memo is None: memo = {}
...
# OR use @lru_cache which handles this correctlyPitfall 2 — Unhashable Arguments
python example
# WRONG: Lists are not hashable
@lru_cache(maxsize=None)
def dp(arr, target): # arr is a list — TypeError!
...
# CORRECT: Convert to tuple
def solve(arr, target):
return dp(tuple(arr), target)
@lru_cache(maxsize=None)
def dp(arr, target): # arr is now a tuple — hashable ✓
...Pitfall 3 — Not Clearing Cache Between Test Cases
python example
# If you reuse a memoized function across multiple test cases:
fibonacci.cache_clear() # Clear cache between independent problemsComplexity Analysis with Memoization
| States: 1 per value of n | n states |
| Work per state | O(1) (just two recursive calls) |
| Total | O(n) |
| States | m × n (one per (i,j) pair) |
| Work per state | O(1) |
| Total | O(m × n) |
| States | n items × W capacity values = n × W states |
| Work per state | O(1) |
| Total | O(n × W) |
| States | n values (number of nodes) |
| Work per state | O(n) (iterate over root choices) |
| Total | O(n²) |
Summary
Memoization transforms exponential recursion into polynomial DP:
- Start with recursive solution — implement the natural recursion
- Add cache — check if result is already computed before recursing
- Store result — before returning, save to cache
In Python: Use @lru_cache(maxsize=None) or @cache (Python 3.9+) for the cleanest code.
Key insight: Memoization only helps when the same subproblem (same arguments) is called multiple times. This is precisely the "overlapping subproblems" property that makes DP applicable.
*Next Lesson: Tabulation*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Memoization — Top-Down Dynamic Programming with Caching.
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, dynamic, programming, memoization
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