Given a data structure specification such as a purely functional map with known complexity bounds, one has to pick between several implementations. There is some folklore on how to pick the right one, for example Red-Black trees are considered to be generally faster, but AVL trees have better performance on work loads with many lookups.
Is there a systematic presentation (published paper) of this knowledge (as relates to sets/maps)? Ideally I would like to see statistical analysis performed on actual software. It might conclude, for example, that there are N typical kinds of map usage, and list the input probability distribution for each.
Are there systematic benchmarks that test map and set performance on different distributions of inputs?
Are there implementations that use adaptive algorithms to change representation depending on actual usage?
These are basically research topics, and the results are generally given in the form of conclusions, while the statistical data is hidden. One can have statistical analysis on their own data though.
For the benchmarks, better go through the implementation details.
The 3rd part of the question is a very subjective matter, and the actual intentions may never be known at the time of implementation. However, languages like perl do their best to implement highly optimized solutions to every operation.
Following might be of help:
Purely Functional Data Structures by Chris Okasaki
http://www.cs.cmu.edu/~rwh/theses/okasaki.pdf
I have been reading parts of Introduction to Algorithms by Cormen et al, and have implemented some of the algorithms.
In order to test my implementations I wrote some glue code to do file io, then made some sample input by hand and some more sample input by writing programs that generate sample input.
However I am doubtful as to the quality of my own sample inputs -- corner cases; I may have missed the more interesting possibilities; I may have miscalculated the proper output; etc.
Is there a set of test inputs and outputs for various algorithms collected somewhere on the Internet so that I might be able to test my code? I am looking for test data reasonably specific to particular algorithms, rather than contest problems that often involve a problem solving component as well.
I understand that I might have to adjust my code depending on the format the input is collected in (e.g. The various constraints of the inputs; for graph algorithms, the representation of the graph; etc.) although, I am hoping that the change I would have to make would be reasonably trivial.
Edit:
Some particular datasets I am currently looking for are:
Lists of numbers
Skewed so that Quick sort performs badly.
Skewed so that Fibonacci Heap performs particularly well or poorly for specific operations.
Graphs (for which High Performance Mark has offered a number of interesting references)
Sparse graphs (with specific bounds on number of edges),
Dense graphs,
Since, I am still working through the book, if you are in a similar situation as I am, or you just feel the list could be improved, please feel free to edit the list -- some time soon, I may come to need datasets similar to what you are looking for. I am not entirely sure how editing privileges work, but if I have any say over it, I will try to approve it.
I don't know of any one resource which will provide you with sample inputs for all the types of algorithm that Cormen et al cover but for graph datasets here are a couple of references:
Knuth's Stanford Graphbase
and
the Stanford Large Network Dataset Collection
which I stumbled across while looking for the link to the former. You might be interested in this one too:
the Matrix Market
Why not edit your question and let SO know what other types of input you are looking for.
I am going to stick my head on the line and say that I do not know of any such source, and I very much doubt that such a source exists.
As you seem to be aware, algorithms can be applied to almost any sort of data, and so it would be fruitless to attempt to provide sample data.
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After reading Stevey Yegge's Get That Job At Google article, I found this little quote interesting:
Whenever someone gives you a problem, think graphs. They are the most fundamental and flexible way of representing any kind of a relationship, so it's about a 50–50 shot that any interesting design problem has a graph involved in it. Make absolutely sure you can't think of a way to solve it using graphs before moving on to other solution types. This tip is important!
What are some examples of problems that are best represented and/or solved by graph data structures/algorithms?
One example I can think of: navigation units (ala Garmin, TomTom), that supply road directions from your current location to another, utilize graphs and advanced pathing algorithms.
What are some others?
Computer Networks: Graphs model intuitively model computer networks and the Internet. Often nodes will represent end-systems or routers, while edges represent connections between these systems.
Data Structures: Any data structure that makes use of pointers to link data together is making use of a graph of some kind. This includes tree structures and linked lists which are used all the time.
Pathing and Maps: Trying to find shortest or longest paths from some location to a destination makes use of graphs. This can include pathing like you see in an application like Google maps, or calculating paths for AI characters to take in a video game, and many other similar problems.
Constraint Satisfaction: A common problem in AI is to find some goal that satisfies a list of constraints. For example, for a University to set it's course schedules, it needs to make sure that certain courses don't conflict, that a professor isn't teaching two courses at the same time, that the lectures occur during certain timeslots, and so on. Constraint satisfaction problems like this are often modeled and solved using graphs.
Molecules: Graphs can be used to model atoms and molecules for studying their interaction and structure among other things.
I am very very interested in graph theory and ive used it solved so many different kinds of problem. You can solve a lot of Path related problem, matching problem, structure problems using graph.
Path problems have a lot of applications.
This was in a career cup's interview question.
Say you want to find the longest sum of a sub array. For example, [1, 2, 3, -1] has the longest sum of 6. Model it as a Directed Acyclic Graph (DAG), add a dummy source, dummy destination. Connect each node with an edge which has a weight corresponding to the number. Now use the Longest Path algorithm in the DAG to solve this problem.
Similarly, Arbitrage problems in financial world or even geometry problems of finding the longest overlapping structure is a similar path problem.
Some obvious ones would be the network problems (where your network could have computers people, organisation charts, etc).
You can glean a lot of structural information like
which point breaks the graph into two pieces
what is the best way to connect them
what is the best way to reach one place to another
is there a way to reach one place from another, etc.
I've solved a lot of project management related problems using graphs. A sequence of events can be pictured as a directed graph (if you don't have cycles then thats even better). So, now you can
sort the events according to their priority
you can find the event that is the most crucial (that is would free a lot of other projects)
you can find the duration needed to solve the total project (path problem), etc.
A lot of matching problems can be solved by graph. For example, if you need to match processors to the work load or match workers to their jobs. In my final exam, I had to match people to tables in restaurants. It follows the same principle (bipartite matching -> network flow algorithms). Its simple yet powerful.
A special graph, a tree, has numerous applications in the computer science world. For example, in the syntax of a programming language, or in a database indexing structure.
Most recently, I also used graphs in compiler optimization problems. I am using Morgan's Book, which is teaching me fascinating techniques.
The list really goes on and on and on. Graphs are a beautiful math abstraction for relation. You really can do wonders, if you can model it correctly. And since the graph theory has found so many applications, there are many active researches in the field. And because of numerous researches, we are seeing even more applications which is fuelling back researches.
If you want to get started on graph theory, get a good beginner discrete math book (Rosen comes to my mind), and you can buy books from authors like Fould or Even. CLRS also has good graph algorithms.
Your source code is tree structured, and a tree is a type of graph. Whenever you hear people talking about an AST (Abstract Syntax Tree), they're talking about a kind of graph.
Pointers form graph structures. Anything that walks pointers is doing some kind of graph manipulation.
The web is a huge directed graph. Google's key insight, that led them to dominate in search, is that the graph structure of the web is of comparable or greater importance than the textual content of the pages.
State machines are graphs. State machines are used in network protocols, regular expressions, games, and all kinds of other fields.
It's rather hard to think of anything you do that does not involve some sort of graph structure.
An example most people are familiar: build systems. Make is the typical example, but almost any good build system relies on a Directed Acyclic Graph. The basic idea is that the direction models the dependency between a source and a target, and you should "walk" the graph in a certain order to build things correctly -> this is an example of topological sort.
Another example is source control system: again based on a DAG. It is used for merging, for example, to find common parent.
Well, many program optimization algorithms that compilers use are based on graphs (e.g., figure out call graph, flow control, lots of static analysis).
Many optimization problems are based on graph. Since many problems are reducable to graph colouring and similar problems, then many other problems are also graph based.
I'm not sure I agree that graphs are the best way to represent every relation and I certainly try to avoid these "got a nail, let's find a hammer" approaches. Graphs often have poor memory representations and many algorithms are actually more efficient (in practice) when implemented with matrices, bitsets, and other things.
OCR. Picture a page of text scanned at an angle, with some noise in the image, where you must find the space between lines of text. One way is to make a graph of pixels, and find the shortest path from one side of the page to the other, where the difference in brightness is the distance between pixels.
This example is from the Algorithm Design Manual, which has lots of other real world examples of graph problems.
One popular example is garbage collection.
The collector starts with a set of references, then traverses all the objects they reference, then all the objects referenced there and so on. Everything it finds is added into a graph of reachable objects. All other objects are unreachable and collected.
To find out if two molecules can fit together. When developing drugs one is often interested in seeing if the drug molecules can fit into larger molecules in the body. The problem with determining whether this is possible is that molecules are not static. Different parts of the molecule can rotate around their chemical bindings so that a molecule can change into quite a lot of different shapes.
Each shape can be said to represent a point in a space consisting of shapes. Solving this problem involves finding a path through this space. You can do that by creating a roadmap through space, which is essentially a graph consisting of legal shapes and saying which shape a shape can turn into. By using a A* graph search algorithm through this roadmap you can find a solution.
Okay that was a lot of babble that perhaps wasn't very understandable or clear. But my point was that graphs pop up in all kinds of problems.
Graphs are not data structures. They are mathematical representation of relations. Yes, you can think and theoretize about problems using graphs, and there is a large body of theory about it. But when you need to implement an algorithm, you are choosing data structures to best represent the problem, not graphs. There are many data structures that represent general graphs, and even more for special kinds of graphs.
In your question, you mix these two things. The same theoretical solution may be in terms of graph, but practical solutions may use different data structures to represent the graph.
The following are based on graph theory:
Binary trees and other trees such as Red-black trees, splay trees, etc.
Linked lists
Anything that's modelled as a state machine (GUIs, network stacks, CPUs, etc)
Decision trees (used in AI and other applications)
Complex class inheritance
IMHO most of the domain models we use in normal applications are in some respect graphs. Already if you look at the UML diagrams you would notice that with a directed, labeled graph you could easily translate them directly into a persistence model. There are some examples of that over at Neo4j
Cheers
/peter
Social connections between people make an interesting graph example. I've tried to model these connections at the database level using a traditional RDMS but found it way too hard. I ended up choosing a graph database and it was a great choice because it makes it easy to follow connections (edges) between people (nodes).
Graphs are great for managing dependencies.
I recently started to use the Castle Windsor Container, after inspecting the Kernel I found a GraphNodes property. Castle Windsor uses a graph to represent the dependencies between objects so that injection will work correctly. Check out this article.
I have also used simple graph theory to develop a plugin framework, each graph node represent a plugin, once the dependencies have been defined I can traverse the graph to create a plugin load order.
I am planning on changing the algorithm to implement Dijkstra's algorithm so that each plugin is weighted with a specific version, thus a simple change will only load the latest version of the plugin.
I with I had discovered this sooner. I like that quote "Whenever someone gives you a problem, think graphs." I definitely think that's true.
Profiling and/or Benchmarking algorithms and implementations in code.
Anything that can be modelled as a foreign key in a relational database is essentially an edge and nodes in a graph.
Maybe that will help you think of examples, since most things are readily modelled in a RDBMS.
You could take a look at some of the examples in the Neo4j wiki,
http://wiki.neo4j.org/content/Domain_Modeling_Gallery
and the projects that Neo4j is used in (the known ones)
http://wiki.neo4j.org/content/Neo4j_In_The_Wild .
Otherwise, Recommender Algorithms are a good use for Graphs, see for instance PageRank, and other stuff at
https://github.com/tinkerpop/gremlin/wiki/pagerank
Analysing transaction serialisability in database theory.
You can utilise graphs anywhere you can define the problem domain objects into nodes and the solution as the flow of control and/or data amongst the nodes.
Considering the fact that trees are indeed connected-acyclic graphs, there are even more areas you can use the graph theory.
Basically nearlly all common data structures like trees, lists, queues, etc, can be thought as type of graph, some with different type of constraint.
To my experiences, I have used graph intensively in network flow problems, which is used in lots of areas like telecommunication network routing and optimisation, workload assignment, matching, supply chain optimisation and public transport planning.
Another interesting area is social network modelling as previous answer mentioned.
There are far more, like integrated circuit optimisation, etc.
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What are the most common problems that can be solved with both these data structures?
It would be good for me to have also recommendations on books that:
Implement the structures
Implement and explain the reasoning of the algorithms that use them
The first thing I think about when I read this question is: what types of things use graphs/trees? and then I think backwards to how I could use them.
For example, take two common uses of a tree:
The DOM
File systems
The DOM, and XML for that matter, resemble tree structures.
It makes sense, too. It makes sense because of how this data needs to be arranged. A file system, too. On a UNIX system there's a root node, and branching down below. When you mount a new device, you're attaching it onto the tree.
You should also be asking yourself: does the data fall into this type of structure? Create data structures that make sense to the problem and the rest will follow.
As far as being easier, I think thats relative. Are you good with recursive functions to traverse a tree/graph? What if you need to balance the tree?
Think about a program that solves a word search puzzle. You could map out all the letters of the word search into a graph and check surrounding nodes to see if that string is matching any of the words. But couldn't you just do the same with with a single array? All you really need to do is move an index to check letters to the left and right, and by the width to check above and below letters. Solving this problem with a graph isn't difficult, but it can create a lot of extra work and difficulty if you're not comfortable with using them - of course that shouldn't discourage you from doing it, especially if you are learning about them.
I hope that helps you think about these structures. As for a book recommendation, I'd have to go with Introduction to Algorithms.
Circuit diagrams.
Compilation (Directed Acyclic graphs)
Maps. Very compact as graphs.
Network flow problems.
Decision trees for expert systems (sic)
Fishbone diagrams for fault finding, process improvment, safety analysis. For bonus points, implement your error recovery code as objects that are the fishbone diagram.
Just about every problem can be re-written in terms of graph theory. I'm not kidding, look at any book on NP complete problems, there are some pretty wacky problems that get turned into graph theory because we have good tools for working with graphs...
The Algorithm Design Manual contains some interesting case studies with creative use of graphs. Despite its name, the book is very readable and even entertaining at times.
There's a course for such things at my university: CSE 326. I didn't think the book was too useful, but the projects are fun and teach you a fair bit about implementing some of the simpler structures.
As for examples, one of the most common problems (by number of people using it) that's solved with trees is that of cell phone text entry. You can use trees, not necessarily binary, to represent the space of possible words that can come out of any given list of numbers that a user punches in very quickly.
Algorithms for Java: Part 5 by Robert Sedgewick is all about graph algorithms and datastructures. This would be a good first book to work through if you want to implement some graph algorithms.
Scene graphs for drawing graphics in games and multimedia applications heavily use trees and graphs. Nodes represents objects to be rendered, transformations, controls, groups, ...
Scene graphs usually have multiple layers and attributes which mean that you can draw only some node of a graph (attributes) in a specified order (layers). Depending on the kind of scene graph you have it can have two parralel structures: declarations and instantiation. Th
#DavidJoiner / all:
FWIW: A new version of the Algorithm Design Manual is due out any day now.
The entire course that he Prof Skiena developed this book for is also available on the web:
http://www.cs.sunysb.edu/~algorith/video-lectures/2007-1.html
Trees are used a lot more in functional programming languages because of their recursive nature.
Also, graphs and trees are a good way to model a lot of AI problems.
Games often use graphs to facilitate finding paths across the game world. The graph representation of the world can have algorithms such as breadth-first search or A* in order to find a route across it.
They also often use trees to represent entities within the world. If you have thousands of entities and need to find one at a certain position then iterating linearly through a list can be inefficient, especially if you need to do it often. Therefore the area can be subdivided into a tree to allow it to be searched more quickly. Just as a linear space can be efficiently searched with a binary search (and thus divided into a binary tree), 2D space can be divided into a quadtree and 3D space into an octree.