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 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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 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Master suffix arrays — what a suffix array is, how to build it using simple and O(n log n) algorithms, the LCP (Longest Common Prefix) array, how to use suffix arrays for pattern searching in O(m log n), finding the longest repeated substring, and counting distinct substrings.
What Is a Suffix Array?
A suffix array is a sorted array of all suffixes of a string. Given a string S, its suffix array SA stores the starting indices of all suffixes of S in lexicographically sorted order.
| Index 0 | "banana" |
| Index 1 | "anana" |
| Index 2 | "nana" |
| Index 3 | "ana" |
| Index 4 | "na" |
| Index 5 | "a" |
| "a" | index 5 |
| "ana" | index 3 |
| "anana" | index 1 |
| "banana" | index 0 |
| "na" | index 4 |
| "nana" | index 2 |
The suffix array is compact (only indices, not actual strings) and enables O(m log n) pattern matching where a full suffix tree would use O(n) space but be harder to build.
O(n log n) Construction — Prefix Doubling
The efficient approach uses prefix doubling (Manber-Myers algorithm):
Example
def build_suffix_array(s: str) -> list:
"""
O(n log n) suffix array construction using prefix doubling.
At each step, ranks are computed for substrings of length 2^k.
"""
n = len(s)
if n == 0:
return []
# Step 1: Initial rank = ASCII value
sa = list(range(n))
rank = [ord(s[i]) for i in range(n)]
tmp = [0] * n
gap = 1
while gap < n:
# Sort by (rank[i], rank[i+gap])
def sort_key(i):
return (rank[i], rank[i + gap] if i + gap < n else -1)
sa.sort(key=sort_key)
# Update ranks based on new order
tmp[sa[0]] = 0
for j in range(1, n):
tmp[sa[j]] = tmp[sa[j-1]]
if sort_key(sa[j]) != sort_key(sa[j-1]):
tmp[sa[j]] += 1
rank = tmp[:]
if rank[sa[n-1]] == n - 1:
break # All ranks are unique, no need to continue
gap *= 2
return sa
sa = build_suffix_array("banana")
print(sa) # [5, 3, 1, 0, 4, 2]
LCP Array (Longest Common Prefix)
The LCP array stores the length of the longest common prefix between consecutive suffixes in the sorted suffix array.
| SA[0]=5 | "a" |
| SA[1]=3 | "ana" LCP with prev = 1 ("a") |
| SA[2]=1 | "anana" LCP with prev = 3 ("ana") |
| SA[3]=0 | "banana" LCP with prev = 0 |
| SA[4]=4 | "na" LCP with prev = 0 |
| SA[5]=2 | "nana" LCP with prev = 2 ("na") |
Example
def build_lcp_array(s: str, sa: list) -> list:
"""
Kasai's algorithm: O(n) LCP array construction.
Key insight: LCP[rank[i]] >= LCP[rank[i-1]] - 1
"""
n = len(s)
rank = [0] * n
lcp = [0] * n
# Inverse suffix array: rank[i] = position of suffix i in SA
for i in range(n):
rank[sa[i]] = i
h = 0 # Current LCP value
for i in range(n):
if rank[i] > 0:
j = sa[rank[i] - 1] # Previous suffix in sorted order
while i + h < n and j + h < n and s[i + h] == s[j + h]:
h += 1
lcp[rank[i]] = h
if h > 0:
h -= 1 # LCP can decrease by at most 1 between consecutive positions
return lcp
s = "banana"
sa = build_suffix_array(s)
lcp = build_lcp_array(s, sa)
print(f"SA: {sa}") # [5, 3, 1, 0, 4, 2]
print(f"LCP: {lcp}") # [0, 1, 3, 0, 0, 2]
Pattern Searching in O(m log n)
With a suffix array, binary search finds all occurrences of pattern P in string S:
Example
def search_pattern(s: str, sa: list, pattern: str) -> list:
"""
Find all occurrences of pattern in s using binary search on SA.
Time: O(m log n) where m = len(pattern), n = len(s)
"""
n = len(s)
m = len(pattern)
# Binary search for leftmost occurrence
lo, hi = 0, n
while lo < hi:
mid = (lo + hi) // 2
suffix = s[sa[mid]:sa[mid] + m]
if suffix < pattern:
lo = mid + 1
else:
hi = mid
left = lo
# Binary search for rightmost occurrence
hi = n
while lo < hi:
mid = (lo + hi) // 2
suffix = s[sa[mid]:sa[mid] + m]
if suffix <= pattern:
lo = mid + 1
else:
hi = mid
right = lo
# All matches are in SA[left:right]
if left >= right:
return [] # Pattern not found
return sorted(sa[left:right]) # Starting positions of all occurrences
s = "banana"
sa = build_suffix_array(s)
print(search_pattern(s, sa, "an")) # [1, 3] — "an" appears at index 1 and 3
print(search_pattern(s, sa, "na")) # [2, 4]
print(search_pattern(s, sa, "xyz")) # []
Compare with KMP: Both O(n + m) for single search, but suffix array enables O(m log n) for multiple pattern searches without preprocessing each pattern separately.
Longest Repeated Substring
The longest repeated substring is found by looking at the maximum value in the LCP array.
Example
def longest_repeated_substring(s: str) -> str:
"""
Find the longest substring that appears at least twice.
Uses suffix array + LCP array.
Time: O(n log n) for SA construction + O(n) for LCP
"""
sa = build_suffix_array(s)
lcp = build_lcp_array(s, sa)
max_lcp = 0
max_idx = 0
for i in range(1, len(s)):
if lcp[i] > max_lcp:
max_lcp = lcp[i]
max_idx = i
if max_lcp == 0:
return "" # No repeated substring
return s[sa[max_idx]:sa[max_idx] + max_lcp]
print(longest_repeated_substring("banana")) # "ana" or "an" — depends on tie
print(longest_repeated_substring("abcabc")) # "abc"
print(longest_repeated_substring("abcde")) # "" — no repeated substring
Count Distinct Substrings
Total substrings of length-n string = n*(n+1)/2. Duplicate substrings are counted by the LCP array.
python example
def count_distinct_substrings(s: str) -> int:
"""
Count distinct non-empty substrings.
Total substrings - duplicates (counted by LCP array).
"""
n = len(s)
sa = build_suffix_array(s)
lcp = build_lcp_array(s, sa)
total = n * (n + 1) // 2 # All substrings
duplicates = sum(lcp) # LCP values count duplicates
return total - duplicates
print(count_distinct_substrings("aaa")) # 3 — "a","aa","aaa"
print(count_distinct_substrings("abab")) # 7 — "a","b","ab","ba","aba","bab","abab"
print(count_distinct_substrings("abcde")) # 15 — all 5*6/2 are distinctSuffix Array vs Suffix Tree
| Feature | Suffix Array | Suffix Tree |
|---|---|---|
| Space | O(n) — compact | O(n) — but large constant factor |
| Construction | O(n log n) simple | O(n) but very complex |
| Pattern search | O(m log n) | O(m) |
| Implementation | Manageable | Very complex |
| LCP queries | O(1) with sparse table | O(1) native |
| Practice | Widely used in competitive programming | Rare in practice |
In competitive programming: Suffix arrays with LCP arrays are the standard. Suffix trees are almost never implemented from scratch.
Summary
A suffix array is the sorted array of all suffix starting indices of a string.
- Construction: O(n² log n) naive, O(n log n) with prefix doubling
- LCP array: O(n) via Kasai's algorithm
- Pattern search: O(m log n) via binary search
- Longest repeated substring: Maximum value in LCP array
- Distinct substrings: n(n+1)/2 - sum(LCP)
Applications: DNA sequence analysis, data compression (BWT/bzip2), plagiarism detection, bioinformatics, full-text search engines.
*Next: Skip List*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Suffix Array — Construction, LCP Array, and String Search Applications.
Interview Use
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Search Terms
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