Introduction

Some exercises and problems in Introduction to Algorithms (CLRS) 3rd edition. Never ever trust a single word of the repo.

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Notebook Summary

I Foundations

• 1 The Role of Algorithms in Computing

  • 1.1 Algorithms
  • 1.2 Algorithms as a technology
  • Problems

• 2 Getting Started

  • 2.1 Insertion sort
  • 2.2 Analyzing algorithms
  • 2.3 Designing algorithms
  • Problems

• 3 Growth of Functions

  • 3.1 Asymptotic notation
  • 3.2 Standard notations and common functions
  • Problems

• 4 Divide-and-Conquer

  • 4.1 The maximum-subarray problem
  • 4.2 Strassen's algorithm for matrix multiplication
  • 4.3 The substitution method for solving recurrences
  • 4.4 The recursion-tree method for solving recurrences
  • 4.5 The master method for solving recurrences
  • 4.6 Proof of the master theorem
  • Problems

• 5 Probabilistic Analysis and Randomized Algorithms

  • 5.1 The hiring problem
  • 5.2 Indicator random variables
  • 5.3 Randomized algorithms
  • 5.4 Probabilistic analysis and further uses of indicator random variables
  • Problems

II Sorting and Order Statistics

• 6 Heapsort

  • 6.1 Heaps
  • 6.2 Maintaining the heap property
  • 6.3 Building a heap
  • 6.4 The heapsort algorithm
  • 6.5 Priority queues
  • Problems

• 7 Quicksort

  • 7.1 Description of quicksort
  • 7.2 Performance of quicksort
  • 7.3 A randomized version of quicksort
  • 7.4 Analysis of quicksort
  • Problems

• 8 Sorting in Linear Time

  • 8.1 Lower bounds for sorting
  • 8.2 Counting sort
  • 8.3 Radix sort
  • 8.4 Bucket sort
  • Problems

• 9 Medians and Order Statistics

  • 9.1 Minimum and maximum
  • 9.2 Selection in expected linear time
  • 9.3 Selection in worst-case linear time
  • Problems

III Data Structures

• 10 Elementary Data Structures

  • 10.1 Stacks and queues
  • 10.2 Linked lists
  • 10.3 Implementing pointers and objects
  • 10.4 Representing rooted trees
  • Problems

• 11 Hash Tables

  • 11.1 Direct-address tables
  • 11.2 Hash tables
  • 11.3 Hash functions
  • 11.4 Open addressing
  • 11.5 Perfect hashing
  • Problems

• 12 Binary Search Trees

  • 12.1 What is a binary search tree?
  • 12.2 Querying a binary search tree
  • 12.3 Insertion and deletion
  • 12.4 Randomly built binary search trees
  • Problems

• 13 Red-Black Trees

  • 13.1 Properties of red-black trees
  • 13.2 Rotations
  • 13.3 Insertion
  • 13.4 Deletion
  • Problems

• 14 Augmenting Data Structures

  • 14.1 Dynamic order statistics
  • 14.2 How to augment a data structure
  • 14.3 Interval trees
  • Problems

IV Advanced Design and Analysis Techniques

• 15 Dynamic Programming

  • 15.1 Rod cutting
  • 15.2 Matrix-chain multiplication
  • 15.3 Elements of dynamic programming
  • 15.4 Longest common subsequence
  • 15.5 Optimal binary search trees
  • Problems

• 16 Greedy Algorithm

  • 16.1 An activity-selection problem
  • 16.2 Elements of the greedy strategy
  • 16.3 Huffman codes
  • 16.4 Matroids and greedy methods
  • 16.5 A task-scheduling problem as a matroid
  • Problems

• 17 Amortized Analysis

  • 17.1 Aggregate analysis
  • 17.2 The accounting method
  • 17.3 The potential method
  • 17.4 Dynamic tables
  • Problems

V Advanced Data Structures

• 18 B-Trees

  • 18.1 Definition of B-trees
  • 18.2 Basic operations on B-trees
  • 18.3 Deleting a key from a B-tree
  • Problems

• 19 Fibonacci Heaps

  • 19.1 Structure of Fibonacci heaps
  • 19.2 Mergeable-heap operations
  • 19.3 Decreasing a key and deleting a node
  • 19.4 Bounding the maximum degree
  • Problems

• 20 van Emde Boas Trees

  • 20.1 Preliminary approaches
  • 20.2 A recursive structure
  • 20.3 The van Emde Boas tree
  • Problems

• 21 Data Structures for Disjoint Sets

  • 21.1 Disjoint-set operations
  • 21.2 Linked-list representation of disjoint sets
  • 21.3 Disjoint-set forests
  • 21.4 Analysis of union by rank with path compression
  • Problems

VI Graph Algorithms

• 22 Elementary Graph Algorithms

  • 22.1 Representations of graphs
  • 22.2 Breadth-first search
  • 22.3 Depth-first search
  • 22.4 Topological sort
  • 22.5 Strongly connected components
  • Problems

• 23 Minimum Spanning Trees

  • 23.1 Growing a minimum spanning tree
  • 23.2 The algorithms of Kruskal and Prim
  • Problems

• 24 Single-Source Shortest Paths

  • 24.1 The Bellman-Ford algorithm
  • 24.2 Single-source shortest paths in directed acyclic graphs
  • 24.3 Dijkstra's algorithm
  • 24.4 Difference constraints and shortest paths
  • 24.5 Proofs of shortest-paths properties
  • Problems

• 25 All-Pairs Shortest Paths

  • 25.1 Shortest paths and matrix multiplication
  • 25.2 The Floyd-Warshall algorithm
  • 25.3 Johnson's algorithm for sparse graphs
  • Problems

• 26 Maximum Flow

  • 26.1 Flow networks
  • 26.2 The Ford-Fulkerson method
  • 26.3 Maximum bipartite matching
  • 26.4 Push-relabel algorithms
  • 26.5 The relabel-to-front algorithm
  • Problems

VII Selected Topics

• 27 Multithreaded Algorithms

  • 27.1 The basics of dynamic multithreading
  • 27.2 Multithreaded matrix multiplication
  • 27.3 Multithreaded merge sort
  • Problems

• 28 Matrix Operations

  • 28.1 Solving systems of linear equations
  • 28.2 Inverting matrices
  • 28.3 Symmetric positive-definite matrices and least-squares approximation
  • Problems

• 29 Linear Programming

  • 29.1 Standard and slack forms
  • 29.2 Formulating problems as linear programs
  • 29.3 The simplex algorithm
  • 29.4 Duality
  • 29.5 The initial basic feasible solution
  • Problems

• 30 Polynomials and the FFT

  • 30.1 Representing polynomials
  • 30.2 The DFT and FFT
  • 30.3 Efficient FFT implementations
  • Problems

• 31 Number-Theoretic Algorithms

  • 31.1 Elementary number-theoretic notions
  • 31.2 Greatest common divisor
  • 31.3 Modular arithmetic
  • 31.4 Solving modular linear equations
  • 31.5 The Chinese remainder theorem
  • 31.6 Powers of an element
  • 31.7 The RSA public-key cryptosystem
  • 31.8 Primality testing
  • 31.9 Integer factorization
  • Problems

• 32 String Matching

  • 32.1 The naive string-matching algorithm
  • 32.2 The Rabin-Karp algorithm
  • 32.3 String matching with finite automata
  • 32.4 The Knuth-Morris-Pratt algorithm
  • Problems

• 33 Computational Geometry

  • 33.1 Line-segment properties
  • 33.2 Determining whether any pair of segments intersects
  • 33.3 Finding the convex hull
  • 33.4 Finding the closest pair of points
  • Problems

• 34 NP-Completeness

  • 34.1 Polynomial time
  • 34.2 Polynomial-time verification
  • 34.3 NP-completeness and reducibility
  • 34.4 NP-completeness proofs
  • 34.5 NP-complete problems
  • Problems

• 35 Approximation Algorithms

  • 35.1 The vertex-cover problem
  • 35.2 The traveling-salesman problems
  • 35.3 The set-covering problem
  • 35.4 Randomization and linear programming
  • 35.5 The subset-sum problem
  • Problem