Algorithms Illuminated is a DIY book series
by shadowrocket安卓下载,
inspired by online courses that are currently running on
the Coursera
and EdX
(Part 1/shadowrocket安卓下载) platforms.
There are four volumes:
Part 1: The Basics
Part 2: Graph Algorithms and Data Structures
Part 3: Greedy Algorithms and Dynamic Programming
Part 4: Algorithms for NP-Hard Problems
ios版shadowrocket下载:
Part 1;
Part 2;
Part 3;
Part 4.
Ordering info: Bookstores can order copies through Ingram. Individuals can order copies through Amazon (see links above).
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Instructors, book reviewers, and foreign publishers/translators can request an exam copy by contacting
the publisher at soundlikeyourselfpublishing@gmail.com.
shadowrocket官网 免费下载 Chinese (Posts & Telecom Press), Korean (Insight Publishing), Russian (Piter Publishing).
Stay updated:
Sign up for occasional email announcements about the book series, or follow algo_class on Twitter.
This page offers several
resources to help you replicate as much of the online course
experience as you like.
(Click on one of the following topics to expand.)
Videos (Part 1)
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Why Study Algorithms?
(Section 1.1)
Integer Multiplication
(Section 1.2)
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(Section 1.3)
MergeSort: Motivation and Example
(Section 1.4, part 1)
MergeSort: Pseudocode
(Section 1.4, part 2)
MergeSort: Analysis
(Section 1.5)
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(Section 1.6)
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(Section 2.1)
Big-O Notation
(Section 2.2)
Basic Examples
(Section 2.3)
Big-Omega and Big-Theta Notation
(Section 2.4)
Additional Examples
(Section 2.5)
The Divide-and-Conquer Paradigm
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Greedy Heuristics for Buying Back Licenses (Part 2)
(Section 24.2, Part 2)
Feasibility Checking (Part 1)
(Section 24.3, Part 1)
Feasibility Checking (Part 2)
(Section 24.3, Part 2)
Implementation as a Descending Clock Auction
(Section 24.4)
The Final Outcome
(Section 24.5)
A Field Guide to Algorithm Design
(Epilogue)
Videos (Bonus)
Karger's random contraction algorithm for graph cuts
Graphs and Minimum Cuts
Random Contraction Algorithm
Review of Conditional Probability
Analysis of the Random Contraction Algorithm
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Red-black trees
Red-Black Trees: The Basics
Insertion in a Red-Black Tree
More on hashing
Universal Hash Functions: Motivation
Universal Hash Functions: Definition and Example
Hash Table Performance with Chaining
Hash Table Performance with Open Addressing
A Greedy Algorithm for Optimal Caching
More on minimum spanning trees
Proof of the Cut Property
[See also Problem 15.7.]
State-of-the-Art and Open Questions
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State-of-the-art union-find implementations
Union-by-Rank
Analysis of Union-by-Rank
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Path Compression: The Hopcroft-Ullman Analysis (Part 1)
Path Compression: The Hopcroft-Ullman Analysis (Part 2)
The Ackermann Function
Path Compression: Tarjan's Analysis (Part 1)
Path Compression: Tarjan's Analysis (Part 2)
More on the Bellman-Ford algorithm
A Space Optimization
Internet Routing (Part 1)
Internet Routing (Part 2)
More on all-pairs shortest paths
A Reweighting Technique
Johnson's Algorithm (Part 1)
Johnson's Algorithm (Part 2)
More on approximately correct heuristic algorithms
A Greedy Heuristic Algorithm for the Knapsack Problem (Part 1) (see also Problem 20.3 in AI Part 4)
A Greedy Heuristic Algorithm for the Knapsack Problem (Part 2) (see also Problem 20.3 in AI Part 4)
A Greedy Heuristic Algorithm for the Knapsack Problem (Part 3) (see also Problem 20.3 in AI Part 4)
A Dynamic Programming Heuristic Algorithm for the Knapsack Problem (Part 1) (see also Problem 20.11 in AI Part 4)
A Dynamic Programming Heuristic Algorithm for the Knapsack Problem (Part 2) (see also Problem 20.11 in AI Part 4)
A Dynamic Programming Heuristic Algorithm for the Knapsack Problem (Part 3) (see also Problem 20.11 in AI Part 4)
More on local search
Local Search for the Maximum Cut Problem (Part 1) (see also Problem 20.14 in AI Part 4)
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Please report errors for Part 1, Part 2, Part 3, and Part 4.
Test Cases and Data Sets for Programming Projects
General advice: use the small test
cases to help debug your program before tackling the challenge data set.
Programming Problem 1.6: Karatsuba multiplication
Test cases:
For this problem, you can generate test cases just by plugging
numbers into a calculator. For example, 99,999 * 9,999 equals
999,890,001.
Challenge problem: What is the product of
3141592653589793238462643383279502884197169399375105820974944592
and
2718281828459045235360287471352662497757247093699959574966967627?
Programming Problem 3.5: Counting inversions
Sanity check: First, check that your algorithms counts 0
inversions for a sorted array, and n(n-1)/2 inversions for a reverse
sorted array (e.g., 28 inversions for [ 8 7 6 5 4 3 2 1 ]).
Test case:
This file contains 10
integers, representing a 10-element array. Your program should count
28 inversions in this array.
Challenge data set:
This file
contains all of the integers between 1 and 100,000
(inclusive) in some order, with no integer repeated.
The ith row of the file indicates the ith entry of
an array. How many inversions does this array have? (Obviously, to
get the most out of this assignment, you should implement the fast
divide-and-conquer algorithm from Section 3.2, rather than brute-force search.)
Programming Problem 5.6: QuickSort
Test case #1:
This file contains 10
integers, representing a 10-element array. Your program should count
25 comparisons if you always use the first element as the pivot,
31 comparisons if you always use the last element as the pivot,
and 21 comparisons if you always use the median-of-3 as the pivot (not counting the comparisons used to compute the pivot).
Test case #2:
This file contains 100
integers, representing a 100-element array. Your program should count
620 comparisons if you always use the first element as the pivot,
573 comparisons if you always use the last element as the pivot,
and 502 comparisons if you always use the median-of-3 as the pivot (not counting the comparisons used to compute the pivot).
Challenge data set:
This file
contains all of the integers between 1 and 10,000
(inclusive) in some order, with no integer repeated.
The ith row of the file indicates the ith entry of
an array. How many comparisons does QuickSort make on this input when the first element is always chosen as the pivot? If the last element is always chosen as the pivot? If the median-of-3 is always chosen as the pivot?
Programming Problem 6.5: Randomized Linear-Time Selection
Test case #1:
shadowrocket安卓下载 contains 10
integers, representing a 10-element array.
What is the median (i.e., the 5th-smallest element)?
(Solution: 5469.)
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This file contains 100
integers, representing a 100-element array.
What is the median (i.e., the 50th order statistic)? (Solution: 4715.)
Challenge data set:
Form an array in which the first element is the first 10 digits of pi,
the second element is the next 10 digits of pi, and so on.
(The digits of pi are
available here.)
Make the array as big as you can (perhaps 100,000 elements, or 1 million
elements, or ...).
What is the median of the array?
[Aside: do you think this array has any duplicate
elements?]
Bonus challenge: Implement the deterministic linear-time
selection algorithm from Section 6.3. For the challenge data set
above, compare the maximum array lengths solvable in a reasonable
amount of time (e.g., one hour) with the randomized and
deterministic linear-time selection algorithms.
Programming Problem 8.10: Computing Strongly Connected Components
Test case #1: A 9-vertex 11-edge graph.
Top 5 SCC sizes: 3,3,3,0,0
shadowrocket安卓下载 An 8-vertex 14-edge graph.
Top 5 SCC sizes: 3,3,2,0,0
Test case #3: An 8-vertex 9-edge graph.
Top 5 SCC sizes: 3,3,1,1,0
Test case #4: An 8-vertex 11-edge graph.
Top 5 SCC sizes: 7,1,0,0,0
Test case #5: A 12-vertex 20-edge graph.
Top 5 SCC sizes: 6,3,2,1,0
shadowrocket官网 免费下载 shadowrocket下载官网 describes the edges of a directed graph. Vertices are
labeled as positive integers from 1 to 875714. Each row indicates one
edge of the graph (the tail and head vertices, in that order).
For example, the eleventh row ("2 13019")
indicates that there is an edge directed from vertex 2 to vertex 13019.
What are the sizes of the biggest five strongly connected components?
Programming Problems 9.8 and 10.8: Implementing Dijkstra's Algorithm
Test case: shadowrocket安卓免费版 describes an undirected graph with 8 vertices (see below
for the file format). What are the shortest-path distances from
vertex 1 to every other vertex? (Answer, for vertices 1 through
8, in order: 0,1,2,3,4,4,3,2.)
Challenge data set:
This file contains an adjacency
list representation of an undirected graph with 200 vertices
labeled 1 to 200. Each row indicates the edges incident to the given
vertex along with their (nonnegative) lengths.
For example, the sixth row has a "6" as its first entry indicating
that this row corresponds to vertex 6. The next entry of
this row "141,8200" indicates that there is an undirected edge between vertex 6
and vertex 141 that has length 8200. The rest of the pairs in this row
indicate the other vertices adjacent to vertex 6 and the lengths of
the corresponding edges. Vertex 1 is the starting vertex.
What are the shortest-path distances from vertex 1 to the following
ten vertices?: 7,37,59,82,99,115,133,165,188,197.
Programming Problem 11.3: The Median Maintenance Problem
Test case: This file represents a stream of 10 numbers. What are the last 4 digits of the sum of the
kth medians? (See below for the definition of the kth median.) (Answer: 9335.)
Challenge data set: This
file contains a list of the integers from 1 to 10000 in unsorted
order; you should treat this as a stream of numbers, arriving one by
one. By the kth median, we mean the median of the first k numbers in the stream (the
((k+1)/2)th smallest number among the first k if k is odd, the (k/2)th smallest if k is even).
What are the last 4 digits of the sum of the kth medians (with k going from 1 to 10000)?
Which data structure makes your algorithm faster: two heaps, or a search tree?
Programming Problem 12.4: 2-SUM
Test case: This file describes an array with 9 integers. For how many target values t in the interval [3,10] are there distinct numbers x,y in the input array such that x+y=t?
(Note: ensuring distinctness requires a one-line addition to the algorithm in Section 12.2.2.) (Answer: 8)
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This file contains one million
integers, both positive and negative (possibly with
repetitions!), with the ith row specifying the ith entry of the input array.
For how many target values t in the interval [-10000,10000] are there distinct numbers x,y in the input array such that x+y=t?
Programming Problem 13.4: Greedy Scheduling
Test case: (contributed by Jeremy Brown) This file contains a list of 12 jobs with weights and lengths. It has the format: [number_of_jobs]
[job_1_weight] [job_1_length]
[job_2_weight] [job_2_length]
...
What is the weighted sum of completion times of the schedule output by the GreedyDiff and GreedyRatio algorithms? (Break ties in favor of jobs with larger weights.) (Answer: 68615 and 67247, respectively.)
Challenge data set:
Repeat the previous problem with the set of 10000 jobs listed in this file.
Programming Problem 14.6: Huffman Codes
In this problem the file format is:
[number_of_symbols]
[weight of symbol #1]
[weight of symbol #2]
... shadowrocket下载官网 (contributed by Rupendra Bandyopadhyay)
For the 10- and 15-symbol problem instances described in test case #1 and test case #2, what is the minimum and maximum length of a codeword in the corresponding optimal prefix-free code? (Answer: 2 and 5 for test case #1, 3 and 6 for test case #2.)
Challenge data set:
Repeat the previous problem with the 1000-symbol problem instance described in this file.
Programming Problem 15.9: Minimum Spanning Trees
In this problem the file format is:
[number_of_vertices] [number_of_edges]
[one_endpoint_of_edge_1] [other_endpoint_of_edge_1] [edge_1_cost]
[one_endpoint_of_edge_2] [other_endpoint_of_edge_2] [edge_2_cost]
...
Edge costs can be negative, and are not necessarily distinct. Test case: (contributed by Quentin Appleby)
What is the cost of an MST in the graph described in shadowrocket下载官网 (Answer: 14)
Challenge data set:
Repeat the previous problem for the graph described in this file. Which algorithm has a faster straightforward implementation, Prim's or Kruskal's algorithm? Which is faster, the heap-based implementation of Prim's algorithm or the union-find-based implementation of Kruskal's algorithm?
Programming Problem 16.6: Weighted Independent Set
In this problem, each file describes the weights of vertices in a path graph and has the format:
[number_of_vertices_in_path_graph]
[weight of first vertex]
[weight of second vertex]
... ios版shadowrocket下载 (contributed by Logan Travis)
What is the value of a maximum-weight independent set of the 10-vertex path graph described in this file, and which vertices belong to the MWIS? (Answer: 2617, and the vertices 2, 4, 7, and 10).
Challenge data set:
Repeat the previous problem for the 1000-vertex path graph described in this file.
Programming Problem 16.7: Knapsack
In this problem, each file describes an instance of the knapsack problem
and has the format:
[knapsack_size][number_of_items]
[value_1] [weight_1]
[value_2] [weight_2]
...
You can assume that all numbers are positive. You should assume that item weights and the knapsack capacity are integers. Test case:
What is the value of an optimal solution to the knapsack instance described in this file? (Answer: 2493893)
Challenge data set: Repeat the previous problem for the knapsack instance described in shadowrocket下载官网.
This instance is so big that the straightforward iterative implementation described in the book uses an infeasible amount of time and space. So you will have to be creative to compute an optimal solution. One idea is to go back to a recursive implementation, solving subproblems --- and, of course, caching the results to avoid redundant work --- only on an "as needed" basis. Also, be sure to think about appropriate data structures for storing and looking up solutions to subproblems.
Programming Problem 17.8: Sequence Alignment
This file describes an instance of the sequence alignment problem. The format of the file is:
1st line: length of X and length of Y
2nd line: gap cost and mismatch cost (the latter is the same for every pair of distinct symbols)
3rd line: X sequence
4th line: Y sequence
(Answer: the NW score is 224.)
Programming Problem 17.8: Optimal Binary Search Trees
This file describes an instance of the optimal binary search tree problem. The format of the file is:
1st line: number specifying the number n of keys
2nd line: n frequencies as comma-separated integer values
(Answer: the value of an optimal solution is 2780.)
Programming Problem 18.8: All-Pairs Shortest Paths
In this problem, each file describes a directed graph.
The first line of the file indicates the number of vertices and edges, respectively. Each subsequent line describes an edge (the first two numbers are its tail and head, respectively) and its length (the third number). NOTE: edge lengths might be negative, and the graphs may or may not be negative cycles. Test cases: (contributed by Matt Davis)
What is the shortest shortest path in the graphs decribed in this file and in this file? (I.e., the minimum, over choices of an origin s and a destination t, of a shortest s-t path in the graph.) (If the graph has a negative cycle, your algorithm should detect that fact.) (Answer: -2 and "contains a negative cycle," respectively.)
Challenge data set: Repeat the previous problem for the graphs described in the following files:
graph #1,
graph #2,
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Programming Problems 19.9, 20.15, 20.16, 21.14, and 21.15: The Traveling Salesman Problem
Input files have two possible formats. ios版shadowrocket下载
[number_of_vertices] [number_of_edges]
[one_endpoint_of_edge_1] [other_endpoint_of_edge_1] [edge_1_cost]
[one_endpoint_of_edge_2] [other_endpoint_of_edge_2] [edge_2_cost]
... Format #2: (for Euclidean instances)
The first line indicates the number of vertices. Each vertex is a point in the plane, and each subsequent line indicates the x- and y-coordinates of a single vertex. (The cost of the edge between two vertices is then the Eulidean distance between its endpoints.) Test case #1:
This file describes the instance in Quiz 19.2 (in format #1). (Optimal tour cost: 13) Test case #2:
This file describes the instance in Quiz 20.7 (in format #1). (Optimal tour cost: 23) ios版shadowrocket下载 (contributed by Eric Hulburd)
This file describes an 8-vertex Euclidean instance (in format #2). (Optimal tour cost (rounded): 12.36) Challenge data set: (for exact computation by e.g. Bellman-Held-Karp or a MIP solver) ios版shadowrocket下载 describes a 25-vertex Euclidean instance (in format #2). Bigger challenge data sets: (e.g., to experiment with greedy heuristics or local search) Countless examples here.
Programming Problem 24.5: Graph Coloring (via SAT solvers)
MiniSAT (or see recent SAT competitions for many more solvers).
Test cases: (to test out your SAT solvers) The file format is
[number_of_variables] [number_of_constraints]
[first_literal_of_constraint_1][second_literal_of_constraint_1]
[first_literal_of_constraint_2][second_literal_of_constraint_2]
...
(Both test cases have exactly two literals in each constraint.)
Each literal is described by a number denoting the variable and a "-" sign denoting logical "not". For example, the second line of the first test case file is "-16808 75250," which indicates the constraint "(not x_{16808}) or x_{75250}."
Test case #1 (answer: satisfiable)
Test case #2 (answer: unsatisfiable)
Challenge data sets (SAT): For challenging SAT instances, see recent SAT competitions.
Challenge data sets (graph coloring): Explore the benchmark instances here or derive a graph from the FCC Incentive Auction data here.
For additional test cases and to contribute your own,
visit this GitHub repo.
Mathematical Background
Lehman, Leighton, and Meyer,
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See also the shorter 2004 version.
Wikibook on Discrete Probability
Additional Links
A Second Course in Algorithms [a sequel course, on YouTube]
