cyberzhg / clrs

Some exercises and problems in Introduction to Algorithms 3rd edition.

Introduction

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

You can use TeX All the Things Chrome extension to read the Markdown files. Please let me know if you found wrong formatting since there are conflicts in the grammars of Markdown and TeX.

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

clrs's Issues

solutions for exercises in appendices?

Could you please also provide solutions for exercises in the appendices?

7-3 (d), How the last two lines work?

The equation became smaller rather than magnified.

The solution of exercise 11.2

the number of the second slot should be 20

Solution to 5-2 f

This is by far the most sane solution I've seen to this question.
All the other ones I've came across doesn't make any sense to me.

But how exactly do you go from the summation to (n+1)/(k+1)?
I believe they are equivalent because I've verified, at least emprically (by trying my python code with many different n and k) that the two approaches produce the same answers