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 priority queues — what they are, how they differ from regular queues, heap-based implementation, Python
Introduction
A priority queue is a special queue where each element has an associated priority. Elements are dequeued in order of their priority — not in the order they were inserted.
In a regular queue: first in, first out. In a priority queue: highest priority element is always dequeued first.
| Regular Queue | [Job1, Job2, Job3] (FIFO) |
| Dequeue | Job1 |
| Priority Queue | [(Job3, priority=10), (Job1, priority=3), (Job2, priority=7)] |
| Dequeue | Job3 (priority 10, highest) |
| Dequeue | Job2 (priority 7) |
| Dequeue | Job1 (priority 3, lowest) |
Operations and Complexity
| Operation | Description | Time |
|---|---|---|
insert(item, priority) | Add element with priority | O(log n) |
extract_min() or extract_max() | Remove and return highest-priority element | O(log n) |
peek_min() or peek_max() | View highest-priority without removing | O(1) |
is_empty() | True if empty | O(1) |
size() | Number of elements | O(1) |
decrease_key() | Update priority (used in Dijkstra) | O(log n) |
Python's heapq Module
Python provides a min-heap in heapq:
Example
import heapq
# Min-heap
heap = []
heapq.heappush(heap, 5)
heapq.heappush(heap, 1)
heapq.heappush(heap, 3)
heapq.heappush(heap, 7)
heapq.heappush(heap, 2)
print(heap) # [1, 2, 3, 7, 5] — heap-ordered (not fully sorted)
print(heap[0]) # 1 — peek minimum: O(1)
print(heapq.heappop(heap)) # 1 — extract minimum: O(log n)
print(heapq.heappop(heap)) # 2
# Build heap from list: O(n) (better than pushing n items: O(n log n))
nums = [5, 1, 3, 7, 2]
heapq.heapify(nums) # In-place heapify: O(n)
# Max-heap workaround: negate values
max_heap = []
heapq.heappush(max_heap, -5)
heapq.heappush(max_heap, -1)
heapq.heappush(max_heap, -8)
max_val = -heapq.heappop(max_heap) # -(-8) = 8 — maximum
# Priority queue with (priority, item) tuples:
pq = []
heapq.heappush(pq, (1, "low priority task"))
heapq.heappush(pq, (10, "critical task"))
heapq.heappush(pq, (5, "medium task"))
priority, task = heapq.heappop(pq)
print(f"Processing: {task} (priority={priority})") # critical task (priority=10)? NO!
# heapq is MIN-heap, so priority=1 comes first.
# For max-priority-queue behavior, use negative priorities:
pq2 = []
heapq.heappush(pq2, (-10, "critical task"))
heapq.heappush(pq2, (-5, "medium task"))
heapq.heappush(pq2, (-1, "low task"))
neg_p, task = heapq.heappop(pq2)
print(f"Processing: {task} (priority={-neg_p})") # critical task (priority=10)
Complete Priority Queue Class
Example
import heapq
from typing import Any
class MinPriorityQueue:
"""
Min-priority queue: lowest priority value = highest urgency.
Wraps Python's heapq with a clean interface.
"""
def __init__(self):
self._heap = [] # List of (priority, counter, item)
self._counter = 0 # Tiebreaker for equal priorities (FIFO among equals)
def insert(self, item: Any, priority: float) -> None:
"""Insert item with given priority. O(log n)."""
heapq.heappush(self._heap, (priority, self._counter, item))
self._counter += 1
def extract_min(self) -> Any:
"""Remove and return item with lowest priority value. O(log n)."""
if self.is_empty():
raise IndexError("Priority queue is empty")
priority, _, item = heapq.heappop(self._heap)
return item
def peek_min(self) -> Any:
"""View minimum-priority item without removing. O(1)."""
if self.is_empty():
raise IndexError("Priority queue is empty")
return self._heap[0][2]
def min_priority(self) -> float:
"""Get the minimum priority value. O(1)."""
if self.is_empty():
raise IndexError("Priority queue is empty")
return self._heap[0][0]
def is_empty(self) -> bool:
return len(self._heap) == 0
def size(self) -> int:
return len(self._heap)
def __len__(self) -> int:
return len(self._heap)
def __repr__(self) -> str:
return f"MinPQ({[(p, item) for p, _, item in self._heap]})"
Application 1 — K Largest Elements
Example
def k_largest_elements(nums, k):
"""
Find the k largest elements using a min-heap of size k.
Time: O(n log k), Space: O(k)
Much better than O(n log n) for sorting when k << n.
"""
heap = [] # Min-heap of size k
for num in nums:
heapq.heappush(heap, num)
if len(heap) > k:
heapq.heappop(heap) # Remove smallest to maintain k largest
return sorted(heap, reverse=True)
print(k_largest_elements([3,1,4,1,5,9,2,6,5,3], 4)) # [9, 6, 5, 5]
Alternative — k-th largest element:
Example
def kth_largest(nums, k):
"""Find k-th largest element. O(n log k)."""
heap = []
for num in nums:
heapq.heappush(heap, num)
if len(heap) > k:
heapq.heappop(heap)
return heap[0] # Top of min-heap of size k = k-th largest
print(kth_largest([3,2,1,5,6,4], 2)) # 5 (2nd largest)
Application 2 — Merge K Sorted Lists
Example
def merge_k_sorted(lists):
"""
Merge k sorted lists. O(n log k) where n = total elements.
"""
heap = [] # (value, list_index, element_index)
for i, lst in enumerate(lists):
if lst:
heapq.heappush(heap, (lst[0], i, 0))
result = []
while heap:
val, list_idx, elem_idx = heapq.heappop(heap)
result.append(val)
if elem_idx + 1 < len(lists[list_idx]):
next_val = lists[list_idx][elem_idx + 1]
heapq.heappush(heap, (next_val, list_idx, elem_idx + 1))
return result
print(merge_k_sorted([[1,4,7],[2,5,8],[3,6,9]])) # [1,2,3,4,5,6,7,8,9]
Application 3 — Task Scheduler (CPU Scheduling)
Example
import heapq
from collections import Counter
def least_interval(tasks, n):
"""
CPU Scheduler: Given tasks and cooldown n between same tasks,
find minimum time to complete all tasks.
Time: O(T log T) where T = unique task count.
"""
freq = Counter(tasks)
max_heap = [-f for f in freq.values()] # Max-heap (negate for max)
heapq.heapify(max_heap)
time = 0
while max_heap:
cycle = []
for _ in range(n + 1): # Process n+1 tasks per cycle
if max_heap:
cycle.append(-heapq.heappop(max_heap)) # Get most frequent
# Decrement frequencies
for f in cycle:
if f - 1 > 0:
heapq.heappush(max_heap, -(f - 1))
time += len(cycle) if not max_heap else n + 1
return time
print(least_interval(["A","A","A","B","B","B"], 2)) # 8
# A→B→_→A→B→_→A→B (where _ = idle)
Application 4 — Dijkstra's Shortest Path
Priority queue is the heart of Dijkstra's algorithm:
Example
import heapq
def dijkstra(graph, source):
"""
Dijkstra's shortest path algorithm.
graph: dict {node: [(neighbor, weight), ...]}
Time: O((V + E) log V)
"""
distances = {node: float('inf') for node in graph}
distances[source] = 0
# (distance, node)
pq = [(0, source)]
while pq:
dist, node = heapq.heappop(pq)
if dist > distances[node]:
continue # Outdated entry
for neighbor, weight in graph.get(node, []):
new_dist = dist + weight
if new_dist < distances[neighbor]:
distances[neighbor] = new_dist
heapq.heappush(pq, (new_dist, neighbor))
return distances
graph = {
'A': [('B', 1), ('C', 4)],
'B': [('C', 2), ('D', 5)],
'C': [('D', 1)],
'D': []
}
print(dijkstra(graph, 'A'))
# {'A': 0, 'B': 1, 'C': 3, 'D': 4}
Python's queue.PriorityQueue (Thread-Safe)
python example
from queue import PriorityQueue
pq = PriorityQueue()
pq.put((3, "medium")) # (priority, item)
pq.put((1, "high"))
pq.put((5, "low"))
while not pq.empty():
priority, item = pq.get()
print(f"{priority}: {item}")
# 1: high
# 3: medium
# 5: lowSummary
| Feature | Priority Queue |
|---|---|
| Order | By priority, not insertion order |
| Insert | O(log n) |
| Extract min/max | O(log n) |
| Peek | O(1) |
| Implementation | Binary heap (most common) |
| Python | heapq module (min-heap) |
| Thread-safe | queue.PriorityQueue |
Key applications:
- K largest/smallest elements: Min-heap of size k
- Merge k sorted lists: Always extract the current minimum across all lists
- Dijkstra's algorithm: Always process the unvisited node with smallest distance
- Task scheduling: Always process highest-priority task
- Huffman encoding: Always merge two lowest-frequency trees
*Next Lesson: Queue Applications — BFS, level-order traversal, 01-BFS, and more.*
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Priority Queue — Complete Guide from Concept to Implementation.
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, queues, priority, queue
Related Data Structures & Algorithms Topics
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