#adjacency matrix representation of graph

[jeremy-ken-anderson]

An adjacency list in a graph is a series of numbers that tells you how many nodes are adjacent to each node. So if you imagine a 3x3 square of rooms that all connect with a door in the center of each wall, the corner rooms would have a value of 2 (the side rooms adjacent to them), the side rooms would have a value of 3 (the adjacent corner rooms and the center), and the center would have a value of 4 (being adjacent to all 4 side rooms).

An adjacency matrix for a graph is possibly more confusing, depending on how your brain works, but defaults to containing more info and has faster lookups in terms of computer processing. It would represent those 9 rooms as a 9x9 grid and mark each door you could go out of as a 1 instead of a 0. So

becomes

And you can see it's symmetrical (split down the line of Xs), because you can go through a door either direction. If these were streets in a city and the street going from intersection E to F was a one-way street, the E,F would be a 1 but the F,E would be a 0.

To get a 2-hop option - everything available in 2 jumps from each point, allowing for overlap - you do slightly different things depending on whether List or Matrix is your representation.

For a List, you have a nested for loop, grabbing the set of adjacent options in the outer loop, and asking for them to spit out a list of their adjacent options in the inner loop. Imagine a 4-square of rooms

J Q K A

the outer loop would say, What's adjacent to J? Q- What's adjacent to Q? J and A are adjacent to Q K- What's adjacent to K? J and A are adjacent to K What's adjacent to Q? J- What's adjacent to J? Q and K are adjacent to J A- What's adjacent to A? Q and K are adjacent to A and so on. So the 2-hop for J ends up with J,A,J,A, for Q it's Q,K,Q,K, for K it's Q,K,Q,K, and for A it's J,A,J,A.

For matrices you do Matrix Multiplication. For square matrices of the same size (which works perfectly for us because we're trying to square the matrix in the first place) you take the row and column that meet at each point in the matrix and multiply across. If you were squaring a matrix

your new A would be A*A + B*D + C*G. Your new B would be A*B + B*E + C*H.

So the square of

For A,A it's a,a(0)*a,a(0) + b,a(1)*a,b(1) ... + i,a(0)*a,i(0) = 2 For B,A it's a,a(0)*b,a(1) + b,a(1)*b,b(1) ... + i,b(0)*b,i(0) = 0

And this makes sense. Remember, this is representing how many paths there are to go from one space to another in exactly 2 jumps. A,A means "how many paths go from A back to A in 2 steps." You can see there are 2: A -> B -> A and A -> D -> A. There's no way to actually take 2 steps starting from B and get to A. Using this logic we can guess by looking at the "map" that B,H would give us a value of 1, because there's only one way to get from B to H in 2 hops.

If we do the same cross-section trick to multiply it out, we have 1*0 + 0*0 + 1*0 + 0*0 + 1*1 + 0*0 + 0*1 + 0*0 + 0*1 and sure enough, we have just one spot where the numbers match up.

[codingprolab]

50.004 – Introduction to Algorithms Homework Set 4

Question 1. Figure 1 shows a directed graph G. a b c d e f g Figure 1: A directed graph for use in Question 1 and Question 2. (i) Give the adjacency-list representation of G. When giving your answer, please order the vertices alphabetically. [2.5 points] (ii) Give the adjacency-matrix representation of G. When giving your answer, please order the vertices alphabetically. [2.5 points] Question 2.…

[programmingandengineering]

QUESTIONS (Write a program to find the adjacency list of a given directed graph)

Write a program to find the adjacency list of a given directed graph G which is represented as adjacency matrix. Input Format: The first line of the input contains a positive integer n, the number of vertices in the graph, in the range 1 to 1000. The next lines represents the Adjacency matrix representation of the given graph. Output Format: Then lines contain the adjacency list of each node…

[emilyratajkowski164395]

In network science, embedding refers to the process of transforming nodes, edges, or entire network structures into a lower-dimensional space while preserving the network's essential relationships and properties. Network embeddings are particularly valuable for machine learning applications, as they allow complex, non-Euclidean data (like a social network) to be represented in a structured, high-dimensional vector format that algorithms can process.

Building on the concept of embeddings in network science, these transformations unlock several advanced applications by enabling traditional machine learning and deep learning methods to operate effectively on graph data. A key advantage of embeddings is their ability to encode structural and relational information about nodes in a network into compact, dense vectors. This allows complex, sparse graphs to be represented in a way that preserves both local connections (like close friendships) and global structure (like communities within the network).

There are multiple approaches to generating these embeddings, each with its own strengths:

Random Walk-based Methods: Techniques like DeepWalk and node2vec use random walks to capture the network’s context for each node, similar to how word embeddings like Word2Vec capture word context. By simulating paths along the graph, these methods learn node representations that reflect both immediate neighbors and broader network structure.

Graph Neural Networks (GNNs): Graph neural networks, including variants like Graph Convolutional Networks (GCNs) and Graph Attention Networks (GATs), use neural architectures designed specifically for graph data. GNNs aggregate information from neighboring nodes, creating embeddings that adaptively capture the influence of each node’s surroundings. This is especially useful in tasks that require understanding both individual and community-level behavior, such as fraud detection or personalized recommendations.

Matrix Factorization Techniques: These methods, like LINE (Large-scale Information Network Embedding), decompose the graph adjacency matrix to find latent factors that explain connections within the network. Matrix factorization can be effective for representing highly interconnected networks, such as knowledge graphs, where the relationships between entities are intricate and abundant.

Higher-order Proximity Preserving Methods: Techniques like HOPE (High-Order Proximity preserved Embeddings) go beyond immediate neighbors, capturing higher-order structural relationships in the network. This approach helps model long-distance relationships, like discovering latent social or biological connections.

Temporal Network Embeddings: When networks evolve over time (e.g., dynamic social interactions or real-time communication networks), temporal embeddings capture changes by learning patterns across network snapshots, allowing predictive tasks on network evolution, like forecasting emerging connections or trends.

Network embeddings are powerful across disciplines. In financial networks, embeddings can model transaction patterns to detect anomalies indicative of fraud. In transportation networks, embeddings facilitate route optimization and traffic prediction. In academic and citation networks, embeddings reveal hidden relationships between research topics, leading to novel insights. Moreover, with visualization tools, embeddings make it possible to explore vast networks, highlighting community structures, influential nodes, and paths of information flow, ultimately transforming how we analyze complex interconnections across fields.

[myprogrammingsolver]

QUESTIONS (Write a program to find the adjacency list of a given directed graph)

Write a program to find the adjacency list of a given directed graph G which is represented as adjacency matrix. Input Format: The first line of the input contains a positive integer n, the number of vertices in the graph, in the range 1 to 1000. The next lines represents the Adjacency matrix representation of the given graph. Output Format: Then lines contain the adjacency list of each node…

[govindhtech]

Boost Intel CPU Speed for(GNN) Graph Neural Network Training

Graph Neural Network Training (GNN) Accelerated on Intel CPU with Hybrid Partitioning and Fused Sampling

High points

A novel graph sampling technique dubbed “fused sampling,” created by Intel Labs and AIA, may speed up the training of Graph Neural Networks (GNNs) on CPUs by up to two times. One of the most widely used libraries for GNN training, the Deep Graph Library (DGL), now includes the updated sample process.

With the use of a novel graph partitioning technique called “hybrid partitioning,” Intel Labs has been able to significantly accelerate the distributed training of Graph Neural Networks (GNNs) on huge networks. Popular graph benchmarks have seen epoch durations reduced by as much as 30%.

Using 16 2-socket computers, each with two 4th Gen Intel Xeon Scalable Processors (Sapphire Rapids), the combination of fused sampling and hybrid partitioning set a new CPU record for training GNNs on the well-known ogbn-papers100M benchmark, achieving a total FP32 training time of just 1.6 minutes.

In several graph-related tasks, including link prediction in recommendation graphs, molecular graph physical property prediction, and high-level semantic feature prediction of nodes in citation graphs and social networks, Graph Neural Networks (GNNs) have achieved state-of-the-art performance. In many fields, graphs may include billions of edges and millions of nodes.

Training over the whole graph at once may rapidly deplete memory. Sampling-based training is one of the most widely used techniques for training GNNs on big graphs: we randomly select a small portion of the graph (small enough to fit in available memory) for each training iteration and train the GNN on this graph sample. However, as a illustrates, the time required for the graph sampling during each iteration may easily eclipse the time for the forward and backward runs of the GNN.

The graph is often divided among many computers to speed up sampling-based training, as seen in the machines are in charge of producing their own graph samples and using them to train the GNN model. Each computer would need to speak with other machines in order to create a graph sample since the graph topology is divided across the devices. As we generate bigger graph samples, this communication cost will increase. When the GNN model includes additional layers, the graph sample size usually grows.

In the following, we outline two complimentary methods that tackle the significant communication cost associated with distributed sample-based training and the high sampling CPU sampling overhead now experienced by popular machine learning libraries.

1. Combining Sampling

Every training iteration must include graph sampling. Thus, it is essential to sample graphs as quickly as feasible. Popular GNN libraries, like DGL (a popular GNN training library), provide a typical sample pipeline that consists of several phases that each produce intermediate tensors that must be written to and subsequently read from memory.

 2. Adaptable partitioning

When a graph becomes too large to store in the memory of one training machine, it is often divided among many machines. The relevant graph data required for each machine to train the GNN model is requested and provided via inter-machine communication. We have noticed that the majority of the graph representation size is often occupied by the characteristics connected to the graph nodes. More than 90% of the RAM required to display the graph is often occupied by the node characteristics.

Inspired by this finding, we created a novel partitioning technique called hybrid partitioning, which, as splits the graph exclusively according to its node properties while reproducing the relatively tiny graph topology information (the graph’s adjacency matrix) across all training machines. Because the machines only need to communicate node characteristics, this results in a significant decrease in the number of communication rounds in distributed sampling based GNN training trials.

As a fused sampling in conjunction with hybrid partitioning resulted in a significant decrease in epoch durations for distributed sample-based GNN training. Even on its own, hybrid partitioning improves performance; when combined with fused sampling, it increases epoch times by more than two times. We obtain a record-breaking total FP32 training time of under 1.6 minutes on 16 2-socket computers by using hybrid partitioning and fused sampling.

Read more on Govindhtech.com

[positivelyapunovexponent-good]

Oooh this looks quite fun! I shall partake. I’m also a bit shy about tagging people but anyone who wants to is welcome to join in!

A couple notes on each of my choices because I have much to say about each of them!

Dungeon Meshi is one of the only pieces of media where I cannot pick a favorite character. At all. I love everyone in the main party + Falin and equal amount and agonised over which character to chose for this poll for frankly too long. Finally settled on Senshi because he’s the most iconic

It was also a little difficult to settle on The Archivist as my favorite character in TMA since the supporting characters are so good, but at the end of the day I cannot resist the incompetent asshole academic who is canonically described as a “grubby Jesus”

I have admittedly been somewhat liberal in what I have interpreted a “character” to be, in my inclusion of real life mathematician Paul Erdős in my poll. However, I stand by my decision, as many would describe him as “a character” in the sense he was quite eccentric. If you don’t believe me PLEASE read his Wikipedia page (or at least the ‘Personality’ section). Seriously it’s one of the few Wikipedia pages I’ve ever read that’s made me laugh out loud uncontrollably. Also - quirks aside- his mathematical work is very cool too! Look up the Erdos Distance problem if you want somewhere to start- mathematicians continue to be chipping away at it in the decades after Erdos’ death. My research advisor once attracted the attention of airport cops after he started jumping up and down with joy in the terminal upon receiving the news that his colleagues significantly improved the exponent of the Erdos distance problem bound. (Honestly he’s kinda a “character” too. )

Okay I now proceed to get even more liberal with the definition of “character” for the purposes of this poll. In particular, the Fourier characters of the cyclic group G = (Z_n, +) are the star blorbos of my thesis. This is because they are QUITE useful for computing the eigenvalues of the adjacency matrix of any Cayley graph on generating set S of G (specifically, the eigenvalues are the sums of Fourier characters evaluated at each of generators in S). Because they are such blorbos to me and literally called characters, I maintain they deserve a place in a character poll

Okay I finally return to a slightly more… ahem… traditional interpretation of “character” in a fandom context. The fandom in question here is my girlfriend’s D&D campaign, so not much of a broad audience. Which is a shame, because it’s WICKED good. If it was a piece of published media widely consumed on tumblr, the character of Ubiquity in particular would have a firmly cemented status as a tumblr sexyman in the most traditional possible sense. I hate the tragic grease stain of a man so much, but I hate him so affectionately that he’s also my favorite. Anyways, my girlfriend’s campaign has been going for years now, and it is by far the “fandom” I spend the most of my creative energy on thinking about- I could theorise for hours, it singlehandedly cured my art block, and I have multiple dumb AUs living rent free in my mind. All that is to say my girlfriend is an absolute world-building genius and one of my favorite writers

inspired by @1tbls et al— no one tagged me but it seemed fun and I want in

if you want to partake, make a poll with 5 of your favorite characters & tag 5 people

however I get Very Nervous about tagging folks in things like this so if you see this just consider yourself invited to join in

[coolravish]

Understanding vector, map and their uncommon implementation:Explanation with their Code

Hey guys,this is my first article on tumblr so suggestions are most welcomed, I thought of writing an article which contains precise explanation with precise code as i am not going to waste my time by speaking on why this is needed and all,So i begin my series of articles based on the responses i will be writing articles further on ds algo etc…Mainly i will try to cover implementation portion of it by using real problems as most of the sites lacks that information.In this article i am going to discuss on following topics:

Vector

Map

So lets explore and understand about them in simple words and in detail:

Vector:-Vector is basically a dynamic array.Now you might be wondering that what exactly we mean here by dynamic?

In a nutshell:-Dynamic means flexible,means size is not fixed.Like an array has constant/fixed size,so you may have question in your mind then what is the need of using vector.See it is useful in questions where we don’t have any idea about size and in array we cannot erase values at arbitrary position,whereas we can erase values in vector using vector erase function.

Now in programming it is believed that understanding the implementation part will help you in better way rather than wasting time on reading theory,

Implementation:

Vector Representation:-vector<int>v;vector<int>v(n),vector<int>v[n];

Now you saw that i wrote three representations,so what are the differences between them.

vector<int>v:-This represents that size of vector is not fixed here,it can change according to the situation. For eg-Initially size of vector is 0,and now if i did operation v.push_back(1),here push_back is used to insert value inside vector,so now size of vector will be 1,and similarly again i perform v.push_back(8),now size of vector is 8,and lets traverse the vector…so we will write for(i=0;i<v.size();i++){v[i]……}so now we can use it as an array…push_back inserts value at the end,like in above example v[0]=1,v[1]=8…Now i told you that we can erase values in vector..So if we want to erase value at desired position then apply v.erase(v.begin()+desired position).There are other ways of erasing also.

vector<int>v(n):-This is exactly same as int a[n].

vector<int>v[n]:-This means vector of vector,this one will better understand by example.

v[0].push_back(1);v[0].push_back(8);v[4].push_back(-1);

v[6].push_back(5);

So above we saw that v[0] is one vector and inside that one another vector is present of size 2 and its elements are 1 and 8 respectively.so if we want to traverse each element individually:-

for(i=0;i<v.size();i++)

{for(j=0;j<v[i].size();j++)

{

v[i][j]……//means whatever operation you want to perform can do…as v[i][j] is individual element like v[0][1]=8,its like 2d matrix.

}

}So above v.size()==3,as one is v[0],v[4],v[6],in nutshell we understand that inside one vector one other vector is present ,this is mainly used in graphs as when you will read about depth first search there you will get to know about adjacency list.

2)Map:-Very Powerful STL tool.

Lets understand map by example.Q)Say in a class we have marks of 10 students in maths exam-10,50,50,30,24,50,20,30,10,50…our question is that count no.of students corresponding to marks obtained?Means how many have scored 50,how many have scored 10 likewise……..

So if our question would have asked calculate frequency for 50 marks only then we could have solved it easily by running a loop and taking count of 50 marks students….but here we have to calculate frequency for each individual marks….so in cases like this our Map is helpful….

Implementation:

map<int,int>m1; This is representation of map.

for(i=0;i<n;i++){

m1[a[i]]++;

}

As you can see above m1[a[i]]++,so basically map is also an array which stores value in the form of key value pair…..here key is marks and value is count of students corresponding to that marks…like m1[50]=4,m1[24]=1 in above example likewise…..so you can see map is also an array but here index is not like array one,here we have one index as 24 ,one as 50 etc…and map stores index in sorted order so like when you will traverse map..then first of all m1[10] will come then m1[24] likewise…..

Now question comes how we will traverse the map?As for traversing the array,vector we have 0-index and in ordered form…so we simply run loop for(i=0;i<n;i++) and we can easily travel them ,but in the case of map as we saw that index was like 24,50,10,30 etc..and we don’t know them beforehand so how we will tackle them…..

For traversing map,set etc we use concept of iterator

What is iterator:In simple words we can say that it is a kind of pointer.Lets move on its implementation,Like here we will traverse map using iterator,its same as traversing array using simple for loop

Implementation:

map<int,int>::iterator it=m1.begin();

for(it=m1.begin();it!=m1.end();it++)

{

int k= it->first; //means it->first refers to map key which is 24,30,50 etc..

int r= it->second; //means it->second refers to map value that is m1[50] value,m1[24] value….say m1[50]=4…so it->first=50 and it->second =4..

cout<<No.of students k marks is r<<endl;

}

3. So we saw how to traverse map using iterator…and say if we want to find maximum number of students corresponding to one particular marks,then we will apply ….if(it->second>p){p=it->second;}….inside iterator loop we will use this condition….likewise anything we can do…And we will see more applications of map,set at the end after understanding pairs ,set etc……

Above one is known as ordered map as it stores unique keys…and all keys are in sorted manner….key refers to index in basic terms….Then we have unordered map which doesn’t store keys in sorted manner and then we have multimap which allows duplicating of keys,these two we will understand further in other articles..First we should be clear with basic implementation of map..Lets discuss one more final example to have better hold on implementation part of map…say we have string=”abaaabccaababa”

And our question of is maximum frequency of occurence of any substring of size 3.I hope you have idea about string and substring and if not then let me tell you in short…A substring of size 3 in above example can be-aba,aaa,baa,aab,bcc,cca,caa etc…means continuous characters of size 3 starting from any postion is substring …in simple words

Now lets move on implementation of this question….

map<string,int>m1; Here you might be wondering that why i wrote string as key,because we are going to take count of substring so key will be string and value is count so we wrote int as value…these things you need to keep in mind while implementing map.

for(i=0;i<n-2;i++)//n=s.size(),n-2 i guess you are clear as substring of size 3 so we need to take valid input or else error will be thrown…

{

m1[s.substr(i,3)]++;

}

So here in map we will have our map like m1[“aba”],m1[“caa”],m1[“baa”] likewise…..For maximum..lets traverse the loop

map<string,int>::iterator it=m1.begin();int p=0;

for(it=m1.begin();it!=m1.end();it++)

{

if(it->second>p)

{ p=it->second;

}

}

So answer of p will be 3 as m1[“aba”]=3…and one more thing if we are required to print substring then inside that if condition just add

string s1=it->first;

So we saw the basic implementation of map,then there is one more information map contains if we will write m1.size() it returns no. of unique keys our data contains,then we have this option also m1.erase(key),means if we write m1.erase(“aba”) then our m1[“aba”] …will be erased….and we can use one more function m1.find(“aba”)…lets see its implementation

map<string,int>::iterator it1 = m1.find(“aba”);

if(it1==m1.end())

{

//means key is not present }

else if(it1!=m1.end())

{

//means key is present

}

So using this m1.find() function we can check whether key is present or not,rest function you can study from this link…Like there is one function m1.upper_bound,lower_bound this all you can study from this link…

One thing i forgot to tell you that iterator can be written in other way also like

for(auto it:m1){}||for(auto it=m1.begin();it!=m1.end();it++)

Now you know we can travel map in reverse way also like as we know that we can travel array/vector in reverse manner using for(i=n-1;i≥0;i- -) similarly here we use for(auto it=m1.rbegin();it!=m1.rend();it- -)………

Then we can store map keys in descending order by using this function

map<int,int,greater<int>()>m1; Likewise there are many more operations which you will get to know while practicing.In my further articles i will try to cover some advanced implementation of maps by solving standardized questions,as i believe that by real problems we get better feel of any topic right?

3)Some Basic Thing but sometimes tricky

To understand this lets study about pairs first…

Pairs is also a kind of array where in value portion we can store two values…

Like say rishav has got 20 marks in maths and 30 marks in science,similarly rohit has got 15 marks in maths and 0 marks in science…so how we will store theses values

Implementation

pair<int,int>p1;

p1[0].first=20,p1[0].second=30;p1[1].first=15,p1[1].second=0;if 0-index is of rishav and 1-index is of rohit.

pairs we can use in many ways now lets look on deeper implementation of it….

vector<pair<pair<int,int>,int>>v;

So what this above representation implies….This means if we write

v.push_back({{20,30},40});({}is used for make_pair operation)

so v[0].first.first=20;

v[0].first.second=30;

v[1].second=40;

Likewise if we write

vector<pair<pair<int,int>,pair<int,int>>>v;

v.push_back({{20,0},{10,30}});

so in this way we can store value for above representation

which means v[0].first.first=20;

v[0].first.second=0;

v[1].second.first=10;

v[1].second.second=30;

so i guess you are getting some feeling by observing few examples above.

We can use pairs in map also like see.

map<pair<string,string>,int>m1;

m1.insert({{rohit,rishav},4});

m1[{rohit,rishav}]=4;

here key is {rohit,rishav} and value is 4.

map<int,vector<int>>m1;

what does this mean…

m1[1].push_back(4);

m1[1].push_back(-1);

m1[3].push_back(6);

m1[3].push_back(4);

so how we will traverse map in such case…

map<int,vector<int>>::iterator it=m1.begin();

for(it=m1.begin();it!=m1.end();it++)

{

for(j=0;j<it->second.size();j++)

{

if(it->second[j] ) //so this it->second[j] this term tells us the values present inside the vector like m1[1] contains -1,4 etc…..likewise…so in this way we can traverse the map…for this case

}

}

We can have many operations like this say

map<pair<int,int>,vector<int>>m1;

map<int,set<int>>m1;

map<int,multiset<int>>m1;

map<string,pair<string,int>>m1;

etc…..

Set/multiset we will discuss in further articles….

Suggestions are welcomed

[miseriathome]

Thinking in increasingly multiple and simultaneous dimensions is difficult and it’s a learned skill. People who do complex math seem to be aware of this, and will frequently strive to be able to conceptualize at will a 7th or 8th or 13th dimension.

The same holds true for thinking about matricies or identity. People who think of gender as a linear progression/spectrum with male on one end and female on the other have their minds blown when you add a second axis of highly concentrated gender to genderlessness onto the same graph.

So when you start being able to conceptualize masculinity-neutrality as an axis independent of femininity-neutrality, and you add in things like conformity-abjection and fluidity-stasis and nongender/genderlessness-definitive gender or concrete-ethereal... and you can hold points on these lines in your mind simultaneously and understand the relationships and interactions between them and how the sum of all those parts does create a unique coordinate,

that’s the matrix: being able to understand how dimensions fold over themselves and how their intersections matter and how one individual person’s experience is identified as a positionality to an infinite number of factors. And even beyond gender, there are so many more dimensions to be had with regards to race and religion and upbringing and education and politics and ability and spirituality and and and--

And to be fair, the axes I identified for gender aren’t even accurate, because to describe all spectrums as linearities would be a false representation. “Masculinity” in and of itself is more than just a two-ended sliding scale from masculine to neutral, and is itself comprised of infinite iterations, themselves identified through positionalities to other systems. The same applies to all other categories: race, wealth, kinship, trauma, empathy, pleasure, physicality, physique, neurodivergence, etc. And yes, it’s incredibly important to remember that a matrix of identity is about so much more than named systems of oppression, but rather incorporates any and all of the multiples factors, worldviews, beliefs, systems, adjacencies, and identifications which contribute to a person’s unimitable and individual identity... which yes, is constantly in flux as new life experiences, exposures, interactions, self-revelations, words, priorities, and reassessments develop.

[jeremy-ken-anderson]

A matrix representation of a graph of a square with doors going from 1-2 and back, 2-3 and back, 3-4 and back, and 4-1 and back, would look like this 0101 1010 0101 1010 Each node has a total degree of 4. In spite of using a door as an analogy the "degree" counts the fact that there's a path out to and in from those adjacent rooms. Looking at the matrix you can get the same result by adding up the number of 1s in the matching row and column. In row 3 there are two 1s, indicating the means to go out to rooms 2 and 4. In column 3 there are two 1s, indicating the ability to come in from rooms 2 and 4.

Now imagine the same square but it's all 1-way streets. You can only go from 1-2, from 2-3, from 3-4, and from 4-1. It looks like so: 0100 0010 0001 1000 You can see that the matrix still shows the correct information. Row 1 column 2 has a 1 in it, so you can go from 1 to 2. Row 4 column 1 has a 1 in it, so you can go from 4 to 1. And between row 1 and column 1 you have a total of two 1s, so the first node has a degree of two.

The adjacency list representation of the "doorways" is like this:

1 - (2,4) 2 - (1,3) 3 - (2,4) 4 - (1,3)

The list representation is nice when you've got big areas described, because you may have noticed that even with just a square there's a lot of needless 0s telling you "there's not a line between these two points going this direction." With the list you only bother to say something exists when it's something existing.

[codezclub]

Write a C Program for Creation of Adjacency Matrix

Creation of Adjacency Matrix Write a C Program for Creation of Adjacency Matrix. Here’s simple Program for adjacency matrix representation of graph in data structure in C Programming Language. Adjacency Matrix:  Adjacency Matrix is a 2D array of size V x V where V is the number of vertices in a graph. Let the 2D array be adj[][], a slot adj[i][j] = 1 indicates that there is an edge from vertex i…

[highgradewriters22]

What in your opinion is the single most important motivation for the development of hashing schemes while there already are other techniques that can be used to realize the same functionality provided by hashing  methods? 

Given a directed graph, described with the set of vertices and the set of edges,  · Draw the picture of the graph· Give an example of a path, a simple path, a cycle· Determine whether the graph is connected or disconnected· Give the matrix representation of the graph· Give the adjacency lists representation of the graphEssay Question: What in your opinion is the single most important motivation…

[myprogrammingsolver]

Simple Graph

For this computer assignment, you are to write a C++ program to implement several graph algorithms on simple graphs. These graphs are implemented with adjacency list representation. The `Graph` class is already defined in the header file `simplegraph.h` and included as part of this repository. The graph is created from a text file that contains the adjacency matrix representation of a graph. The…

[kitwallace]

Finding Eulerian circuits

As discussed previously, Eulerian circuits are useful in 3D printing because they enable continuous extrusion which increases the strength of the printed object . In addition the lack of a seam improves its visual appearance.  

However finding a Eulerian circuit in a design is not always easy, and finding the ‘best’ even harder.

For a rectangular grid  (3 x 2) 

a braiding program I developed a while ago finds a circuit.  It attempts to find a braid on an N x M grid.  If N and M are relatively prime, one rope (ie. a Eulerian circuit) completes the grid  but if not (like this 6 x 3), multiple ropes are needed.

This restriction on the size of this mesh is not helpful.  Since every N x M mesh of this style is Eulerian, they must all have a Eulerian circuit. This could  be constructed if we were able to join up the separate partial loops. Start somewhere and follow a chain of unvisited edges until you get back to the start.  If all edges have been visited, you are done. If not, go back to the last vertex with unvisited edges and start a new circuit at that point, with those points inserted into the current circuit .This is the Hierholzer algorithm.

Hierholzer algorithm

I searched for a pseudo code implementation but wasn’t satisfied with any that I found.  Any implementation depends heavily on the data structures used: for the graph itself in a form in which edges can be removed as they are visited and for the path as it is constructed, adding a vertex to the end and rotating the path to backtrack. 

One representation of a graph with V vertices and E edges is as a V x V adjacency matrix where adj[i,j] =1 indicates the edge i to j which will be symmetric across the diagonal if the graph (as here) is undirected.  Updating is straightforward but  navigating from vertex to vertex requires a scan through a row.  

Alternatively, the graph can be represented as an array of lists of the connected vertices  - easy to navigate, more compact for a sparse graph (as these are)  but harder to update.  

The JavaScript array structure can implement the vertex list and the evolving path. I found this is a useful summary of operators. push() and pop()  will add and remove the head of the list;  unshift() and shift() do the same at the tail of the list.  myArray[myArray.length - 1] and myArray[0] return the head and tail of the list respectively.

Minimizing turns in the circuit

The Hierholzer algorithm mechanically selects the next vertex by taking any vertex (the first or the last perhaps) from the list of possible vertices. This does find a circuit but not necessarily a good circuit.  For the purposes of 3D printing, good means a circuit with a minimum number of turns since straight runs will be stronger and faster.  To get a good, though not necessarily the best, circuit, I added a kind code to each edge and then choose a vertex in the same kind if possible.  For the rectangular mesh above, there are two kinds , coded +1 for the right diagonal, -1 for the left.  (note that we don’t need to distinguish between the direction of travel on a diagonal since we can’t reverse direction at a vertex as that edge has just been traversed)

JavaScript Implementation

In the Fottery suite, the algorithm is implemented as a method in a Graph class which includes the adjacency structure adj and supporting functions:

find_circuit(kind-0,start_vertex = 0 ) {    // assumes graph is connected and a euler circuit exists         var cpath =[];        var vertex, next_vertex, edge, edges;        cpath.push(start_vertex);        while (cpath.length < this.n_edges  ) {            vertex = cpath[cpath.length - 1];  // the last vertex in the constructed path            edges = this.adj[vertex];  // get its  adjacent nodes (untraversed edges)            if (edges.length == 0) {  // if no untraversed edges                cpath.pop();   // rotate back to the previous vertex                if (!(vertex == initial_vertex && cpath[0]==vertex)) { // dont  include the start at the end                  cpath.unshift(vertex);                 }            } else {                  edge = this.best_edge(edges,kind);                  next_vertex= edge[0];                kind = edge[1];                cpath.push(next_vertex);  // add  vertex to the path                this.remove_edge(vertex,next_vertex); // remove the traversed edge            }         }

       // get start back to the first in the path        while (cpath[0] != start_vertex)                  cpath.push(cpath.shift());        return cpath;

[damn discovered a use-case where this implementation fails!   

16 April 2024 - fixed edge case with the bold change ] 

Example

A  rectangular N x M  mesh like the one above is more useful if the vertices at the edges are connected to make a border.  This addition does not change the Eulerian nature of the graph, but introduces two more directions, horizonal and vertical.  This is the resultant 2 x 3 mesh.

and this a box constructed from rectangular meshes.  The sides are bevelled by insetting the top layer from the bottom by the thickness of the tile to make a 45 degree bevel.  Assembled with plastic glue (UHU).

Stars (stellated polygons)

This algorithm has been applied to other structures such as a triangular mesh and stars.   

A pentagram can be constructed with 5 vertices and 5 edges connecting pairs of vertices.This is a closed path so Eulerian.

A hexagram looks Eulerian, but if constructed by connecting the vertices of the star, the graph is not connected (two separate triangular paths).

To make a circuit so it can be printed continuously, we have to compute the 6 intersections and the 18 edges (of 3 kinds)  which result from the intersections, and then use the modified Hierholzer algorithm to find a good circuit.  

A fully connected star is only Eulerian if the number of vertices is odd - which makes for a kind of infill pattern.

Nearly Eulerian graphs

Structures which are Eulerian are rather limited. An hexagonal tiling is not Eulerian because vertices are order 3. However it and many others are nearly Eulerian in the sense that small modifications to the graph will make the graph Eulerian. The basic idea is to duplicate ( the minimum number of ) edges. If this is a problem with printing, the edge could be omitted or an additional edge inserted.

This problem  is called the Chinese Postman Problem or Route Inspection Problem and the subject of extensive research. 

This is work for another day -.April 2024 see Non-Eulerian graphs

References

Fleischner, Herbert (1991), "X.1 Algorithms for Eulerian Trails”   https://archive.org/details/euleriangraphsre0001flei/page/

Gregory Dreifus,Kyle Goodrick,Scott Giles,Milan Patel Path Optimization Along Lattices in Additive Manufacturing Using the Chinese Postman Problem June 2017  https://www.researchgate.net/publication/317701126_Path_Optimization_Along_Lattices_in_Additive_Manufacturing_Using_the_Chinese_Postman_Problem

Prashant Gupta, Bala Krishnamoorthy, Gregory Dreifus  Continuous Toolpath Planning in a Graphical Framework for Sparse Infill Additive Manufacturing  April 2021 https://arxiv.org/pdf/1908.07452.pdf

https://algorithms.discrete.ma.tum.de/graph-algorithms/directed-chinese-postman/index_en.html

https://www.stem.org.uk/resources/elibrary/resource/31128/chinese-postman-problems

https://ibmathsresources.com/2014/11/28/the-chinese-postman-problem/

http://www.suffolkmaths.co.uk/pages/Maths%20Projects/Projects/Topology%20and%20Graph%20Theory/Chinese%20Postman%20Problem.pdf

[felord]

50.004 Homework Set4 Solved

50.004 Homework Set4 Solved

Question 1. Figure 1 shows a directed graph G. Figure 1: A directed graph for use in Question 1 and Question 2. Give the adjacency-list representation of G. When giving your answer, please order the vertices alphabetically. [2.5 points] Give the adjacency-matrix representation of G. When giving your answer, please order the vertices alphabetically. [2.5 points] Question 2. Consider the directed…

[webtechschool]

Graph Representation | Adjacency List | Adjacency Matrix | Data Structures & Algorithms

https://bit.ly/GraphRepresentation