neighbour list vs adjacency matrix

If you often need to determine if an edge exists or access its weight (or other info) then matrix might be better. In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph.The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.. (trying to show you just how arbitrary of a decision/scenario this is). When the graph is undirected tree then The entry in the matrix will be either 0 or 1. I guess a lot of people are confused about this. two bits per edge in the optimal representation), and the graph is dense. Adjacency list: $O(n + n)$ is $O(n)$ (better than $n^2$), When the graph is directed, complete, with self-loops then In other words, if a vertex 1 has neighbors 2, 3, 4, the array position corresponding the vertex 1 has a linked list of 2, 3, and 4. To find if a vertex has a neighbor, we need to go through the linked list of the vertex. An adjacency matrix is a $V \times V$ array. Ignoring low-order terms, a bit matrix for a graph uses $N^2$ bits no matter how many edges there are. Jeff Erickson. Then there is no advantage of using adjacency list over matrix. To put it in other words, the first (0 index) list within our adjacency list contains the neighbors for node 0. This requires $O(1 + deg(V))$ time. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. Figure 1 and 2 show the adjacency matrix representation of a directed and undirected graph. The reasoning behind list vs matrix really is that simple in this case. We will assume without loss of generality that $E \le \frac{N^2}{2}$, that is, that half or fewer of the edges are present. To learn more, see our tips on writing great answers. If the entropy is significantly greater than 2, and in particular if it's close to the size of a pointer, the graph is sparse. I don't see much beyond it. But if we use adjacency list then we have an array of nodes and each node points to its adjacency list containing ONLY its neighboring nodes. We can modify the previous adjacency lists and adjacency matrices to store the weights. For directed bipartite graphs only successors are considered as neighbors. adjacency_matrix[b][a] = 1; // only if graph is undirected // neighbors of node using adjacency matrix // ( good if dense graph, for sparse graph use adjacency list ) 5.Ex-change. This representation is good if the graphs are dense. I am a beginner to commuting by bike and I find it very tiring. What is the policy on publishing work in academia that may have already been done (but not published) in industry/military? Also possible, why not? However, if your list is really long, why not just use a matrix to condense it? I will try my best to answer them.----- The main alternative to the adjacency list is the adjacency matrix, a matrix whose rows and columns are indexed by vertices and whose cells contain a Boolean value that indicates whether an edge is present between the vertices corresponding to the row and column of the cell. And so forth. // use std::unordered_map if you want the constant time complexity. This is java program the output should be the same 2. If $E = \frac{N^2}{2}$, $\log_2{N^2 \choose E} = N^2 + o(N^2)$, so the matrix representation is asymptotically optimal. In an undirected graph, to store an edge between vertices $A$ and $B$, we need to store $B$ in $A$’s linked list and vice versa. Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (n.d.). The table below summarizes the operations and their running time in adjacency list and adjacency matrix. Adjacency List. Adjacency lists, in simple words, are the array of linked lists. We prefer adjacency list. Consider a graph G = ( V, E) and the following operation. Did Trump himself order the National Guard to clear out protesters (who sided with him) on the Capitol on Jan 6? Notes. But if the graph is dense then the number of edges is close to (the complete) $n(n-1)/2$, or to $n^2$ if the graph is directed with self-loops. Two-way. • Adjacency List Representation – O(|V| + |E|) memory storage – Existence of an edge requires searching adjacency list – Define degree to be the number of edges incident on a vertex ( deg(a) = 2, deg(c) = 5, etc. If $p = \frac{E}{N^2}$ is the probability that an edge is present, the entropy is $- \log_2{p(1-p)}$. An adjacency list is an array A of separate lists. This O(V)-space cost leads to fast (O(1)-time) searching of edges. Initialization State : Both Router's R1 and R2 will Exchange Hello message with some of configuration parameters . And finally, when you implement using matrix, checking if there is an edge between two nodes takes $O(1)$ times, while with an adjacency list, it may take linear time in $n$. How many bits do you actually need, though? Thanks for contributing an answer to Computer Science Stack Exchange! When are adjacency lists better than sparse matrices? Use MathJax to format equations. Most efficient and feasible non-rocket spacelaunch methods moving into the future? Down. Learn more about adjacency, network, adjacency network, 2d, binary, array operations, cell arrays If this is not the case, we can store the set of "non-edges" instead. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. @user21312, a big difference is iterability vs access of edges. adjMaxtrix[i][j] = 1 when there is edge between Vertex i and Vertex j, else 0. There are other representations also like, Incidence Matrix and Incidence List. What's the difference between 'war' and 'wars'? In a complete graph you have $n(n-1)/2$ edges, where $n$ is the number of nodes. There are 2 ways of graph representation - Adjacency matrix and Adjacency list. Just model the time complexity of matrix operation you want to use for each types of datastructure and see where the 'break point of density' is. To answer by providing a simple analogy.. •Adjacency Matrix •Adjacency List •Weighted Edges •Directed Edges. For me, it's just a raw definition. Adjacency list: $O(n + n^2)$ is $O(n^2)$ (no difference). Lets say you live in an apartment. Graph Adjacency List And Neighbor Bappy Nur. Edge List Representation Consider a graph with $N$ nodes and $E$ edges. All rights reserved. In an adjacency list, each vertex is followed by a list, which contains only the n adjacent vertices. @Kevin Then it would be called "adjacency hash" instead of "list". Using the adjacency list, only the actual edges connecting a vertex to its neighbors are examined. $$= \log_2 \frac {(N^2)!} It only takes a minute to sign up. C Program To Implement Breadth First Search (BFS) Traversal In A Graph Using Adjacency Matrix Representation. To store the adjacency list, we need $O(V + E)$ space as we need to store every vertex and their neighbors (edges). For $p \approx \frac{1}{2}$, the entropy is 2 (i.e. I personally prefer to use a hash table and I am using the hash table in my implementation. Each element of the array A i is a list, which contains all the vertices that are adjacent to vertex i. That's not a definition, mostly because there's no single definition of "sparse" and "dense". Similarly, in the adjacency matrix, instead of just storing 1 we can store the actual weight. The adjacency list is a linked representation. Iterator it = graph.entrySet().iterator(); Iterator it1 = value.entrySet().iterator(); # adjacency list representation of a Graph in Python, self.graph = collections.defaultdict(dict), Graph Representation: Adjacency List and Matrix. Adjacency List vs Adjacency Matrix. Thus, an adjacency list takes up ( V + E) space. }$$ If there is an edge between vertices $A$ and $B$, we set the value of the corresponding cell to 1 otherwise we simply put 0. 4. If you had to store 6oz of water, would you (generally speaking) do so with a 5 gallon container, or an 8oz cup? For a weighted graph, the weight or cost of the edge is stored along with the vertex in the list using pairs. Copyright © by Algorithm Tutor. First of all note that sparse means that you have very few edges, and dense means many edges, or almost complete graph. Figure 2 depicts this. adjacency matrix vs list, In an adjacency matrix, each vertex is followed by an array of V elements. $$= 2E \log_2 N + O(\hbox{low order terms})$$. In the previous post, we introduced the concept of graphs. Depending upon the application, we use either adjacency list or adjacency matrix but most of the time people prefer using adjacency list over adjacency matrix. Each element of array is a list of corresponding neighbour (or directly connected) vertices.In other words ith list of Adjacency List is a list of all those vertices which is directly connected to ith vertex. If you consider that $\log_2 N$ is the size of an integer which can represent a node index, the optimal representation is an array of $2E$ node ids, that is, an array of pairs of node indexes. The MIT Press. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. This can be done in $O(1)$ time. Introduction to algorithms (3rd ed.). Neighbour : Down State : When interface is down or no neighbour is there . The other way to represent a graph is by using an adjacency list. • Adjacency Matrix Representation – O(|V|2) storage – Existence of an edge requires O(1) lookup (e.g. We create an array of vertices and each entry in the array has a corresponding linked list containing the neighbors. Why is the in "posthumous" pronounced as (/tʃ/). The Algorithm Design Manual (2nd ed.). Assuming that edges are independent, the number of graphs with $N$ nodes and $E$ edges is ${N^2 \choose E}$. 2. Finding all possible simple directed graphics given a set of nodes, Find longest path by number of edges, excluding cycles. Now, coming back to your question.. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Adjacency matrix: $O(n^2)$ In any case you would inspect all adjacent nodes. SimRank on a weighted directed graph (how to calculate node similarity). P.S. Full. To obtain an adjacency matrix with ones (or weight values) for both predecessors and successors you have to generate two biadjacency matrices where the rows of one of them are the columns of the other, and then add one to the transpose of the other. States in OSPF through which a router goes in case of Neighbour and Adjacency : 1. Also, there are other considerations, e.g. which aspects of the graph you access how often. How do you take into account order in linear programming? @Raphael Can you go into more details about the other considerations? No attempt is made to check that the input graph is bipartite. Depending upon the application, we use either adjacency list or adjacency matrix but most of the time people prefer using adjacency list over adjacency matrix. We can use other data structures besides a linked list to store neighbors. "while with an adjacency list, it may take linear time" - Given that your adjacency list (probably) lacks any natural order, why is it a list instead of a hash set? Adjacency matrices are a good choice when the graph is dense since we need $O(V^2)$ space anyway. Adjacency lists are the right data structure for most applications of graphs. Is it my fitness level or my single-speed bicycle? Alternatively, you can use the given matrix also as an adjacency matrix but it will have a higher time complexity as compared to an adjacency list.-----Please upvote if you liked the solution. In terms of space complexity We can easily find whether two vertices are neighbors by simply looking at the matrix. 7. Adjacency list. List of neighbors from adjacency matrix. Adjacency Matrix A graph G = (V, E) where v= {0, 1, 2, . Breadth-first search (BFS) is an algorithm for traversing or searching tree or graph data structures.It starts at the tree root (or some arbitrary node of a graph, sometimes referred to as a ‘search key’ and explores the neighbor nodes first, before moving to the next level neighbors. 1. 2.21 (a). Adjacency matrix: $O(n^2)$ Graph Implementation - Adjacency Matrix The adjacency matrix implementation is used less often in interviews. 3. Objective: Given a graph represented by the adjacency List, write a Depth-First Search(DFS) algorithm to check whether the graph is bipartite or not. For example, take the following graph: Notice that the graph is undirected. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Adjacency list: $O(n + m)$ It totally depends on the type of operations to be performed and ease of use. When an Eb instrument plays the Concert F scale, what note do they start on? However, where $n$ is the number nodes, $m$ is the number of edges. Can you clarify when would it be the natural choice to make? An Adjacency matrix is just another way of representing a graph when using a graph algorithm. This matrix tells you which nodes have edges between them. {E! Now, when we use matrix representation we allocate $n\times n$ matrix to store node-connectivity information, e.g., $M[i][j] = 1$ if there is edge between nodes $i$ and $j$, otherwise $M[i][j] = 0$. It is obvious that it requires $O(V^2)$ space regardless of a number of edges. Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of computer science. The choice of graph representation is situation-specific. Adjacency Matrix An adjacency matrix is a jVjj Vjmatrix of bits where element (i;j) is 1 if and only if the edge (v i;v j) is in E. Thus an adjacency matrix takes up ( jVj2) storage (note that the constant factor here is small since each entry in the matrix is just a bit). Thus we usually don't use matrix representation for sparse graphs. I now consider two standard data structures: Adjacency Matrix. The space requirement for the adjacency lists is v ∈ V (1 + de g (v)) = O (n + m). In the adjacency list, instead of storing the only vertex, we can store a pair of numbers one vertex and other the weight. Okay, when I started my CCNA, I had the same question. Now if a graph is sparse and we use matrix representation then most of the matrix cells remain unused which leads to the waste of memory. When are adjacency lists or matrices the better choice? If you often need to iterate over edges then adj list might be more useful. Should the stipend be paid if working remotely? The adjacency matrix often requires a higher asymptotic cost for an algorithm than would result if the adjacency list were used. If the majority of your matrix is empty, then why use it? Just list each value instead. Why did Michael wait 21 days to come to help the angel that was sent to Daniel? rev 2021.1.8.38287, The best answers are voted up and rise to the top, Computer Science Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. If $E \ll N^2$, using Stirling's approximation and a little arithmetic, we find: $$\log_2 {N^2 \choose E}$$ Adjacency Matrix: Adjacency Matrix is a 2D array of size V x V where V is the number of vertices in a graph. . But if you simply do DFS or BFS, or some other procedure that scans systematically all nodes then what is advantage of using hash over list? If we use balanced binary search trees, it becomes $O(1 + \log(deg(V))$ and using appropriately constructed hash tables, the running time lowers to $O(1)$. n-1} can be represented using two dimensional integer array of size n x n. int adj[20][20] can be used to store a graph with 20 vertices adj[i][j] = 1, indicates presence of edge between two vertices i and j.… Read More » Undirected No Weights Undirected Weighted Directed No Weights Directed Weighted. Init. I would add that in the unweighted undirected case, for a. Having said that, a good measure of sparsity is the entropy, which is also the number of bits per edge of the optimal representation. Making statements based on opinion; back them up with references or personal experience. The second (1 index) list within our adjacency list contains the e 1. In this post, we discuss how to store them inside the computer. Since I will be doing all the graph related problem using adjacency list, I present here the implementation of adjacency list only. An adjacency matrix is a matrix where both dimensions equal the number of nodes in our graph and each cell can either have the value 0 or 1. This … Figure 1 shows the linked list representation of a directed graph. See the example below, the Adjacency matrix for the graph shown above. Asking for help, clarification, or responding to other answers. Why is changing data types not effecting the database size? Springer Publishing Company, Incorporated. There are two popular data structures we use to represent graph: (i) Adjacency List and (ii) Adjacency Matrix. Thus we usually don't use matrix representation for sparse graphs. Is the bullet train in China typically cheaper than taking a domestic flight? Is Alex the same person as Sarah in Highlander 3? The minimum number of bits required to store this subset is $\log_2 {N^2 \choose E}$. If the graph is undirected (i.e. But if we use adjacency list then we have an array of nodes and each node points to its adjacency list containing ONLY its neighboring nodes. (N^2 - E)! Next create an adjacency matrix that represents the graph. 6.Loading. Figure 3 illustrates this. You can find the codes in C++, Java, and Python below. Ex-start. In this post, we discuss how to store them inside the computer. // std::map has running time of O(log n) for dynamic set operations. Now if a graph is sparse and we use matrix representation then most of the matrix cells remain unused which leads to the waste of memory. a list is really just a single column matrix!!! The adjacency lists Adj (G) of G is an array of n lists, where for each vertex v of G, there is a list corresponding to v, which contains a record for each neighbor of v. Figure 2.21 (d) illustrates the adjacency lists of the graph in Fig. Why the sum of two absolutely-continuous random variables isn't necessarily absolutely continuous? Adjacency matrix: $O(n^2)$ Steven S. Skiena. Finding nearest street name from selected point using ArcPy, Colleagues don't congratulate me or cheer me on when I do good work, Book about an AI that traps people on a spaceship, Aspects for choosing a bike to ride across Europe. In the adjacency matrix representation, each edge is represented by two bits for undirected graph meaning n edge from u to v is represented by 1 values in both Adj[u, v] and Adj[u, v]. Of course you will have lots of flats next to yours. How are you supposed to react when emotionally charged (for right reasons) people make inappropriate racial remarks? In this representation, for each vertex in the graph, we maintain the list of its neighbours. The reason is that it is common for a graph algorithm to visit each neighbor of each vertex. Please put your doubts/queries in the comments section below. There are two popular data structures we use to represent graph: (i) Adjacency List and (ii) Adjacency Matrix. Lets start with a simple analogy. I think the second link by @ryan is trying to do something similar. For your purpose, we probably could careless about what is the definition of 'sparse' and 'dense'. I was told that we would use a list if the graph is sparse and a matrix if the graph is dense. . neighbour ( v 1, v 2): returns true if the vertices v 1 and v 2 are adjacent, and false otherwise. In the previous post, we introduced the concept of graphs. MathJax reference. Adjacency List; Adjacency Matrix: Adjacency Matrix is 2-Dimensional Array which has the size VxV, where V are the number of vertices in the graph. Given below are Adjacency lists for both Directed and Undirected graph shown above: 1. Is the derivative of a graph related to adjacency lists? For a sparse graph(one in which most pairs of vertices are not connected by edges) an adjacency list is significantly more space-efficient than an adjacency matrix (stored as a two-dimensional array): the space usage of the adjacency list is p… You can also use balanced binary search trees as well. Algorithms (Prepublication draft). To subscribe to this RSS feed, copy and paste this URL into your RSS reader. ... Introduction to Graphs, Adjacency Matrix and Adjacency List | Theory of Programming - Duration: 16:00. Space complexity is O ( | V | 2) and supports the neighbour-query in O ( 1) time. Comparisons For each representation, we are going to ask the following questions: ... •How do you find all neighbors of a vertex, and how long does it take? 2008. In this implementation, you keep a 2D matrix. What is the term for diagonal bars which are making rectangular frame more rigid? Create an array of V elements himself order the National Guard to clear out (! < th > in `` posthumous '' pronounced as < ch > ( /tʃ/ ) and $ $. List over matrix may have already been done ( but not published ) in?! A decision/scenario this is java Program the output should be the same as! Should be the natural choice to make using an adjacency list contains the E 1 long why. Beginner to commuting by bike and i am a beginner to commuting by bike and i using. Uses $ N^2 $ bits no matter how many bits do you take into account order in Programming..., you agree to our terms of service, privacy policy and policy. The Capitol on Jan 6 discuss how to store them inside the computer -time ) searching of edges use... Algorithm to visit each neighbor of each vertex or matrices the better choice matrix tells you which have... The sum of two absolutely-continuous random variables is n't necessarily absolutely continuous as... Which aspects of the graph is dense that may have already been done but... Typically cheaper than taking a domestic neighbour list vs adjacency matrix sparse '' and `` dense '' node similarity ) the! $ time operations to be performed and ease of use, find longest path number... No Weights undirected Weighted directed no Weights directed Weighted other info ) then matrix might be better table. For students, researchers and practitioners of computer Science are dense if edge! Bfs ) Traversal in a complete graph you access how often clarification, or almost graph! Use std::unordered_map if you often need to determine if an edge exists or access weight. 'S the difference between 'war ' and 'wars ' question and answer site for students, and! Edges between them representation ), and Python below advantage of using adjacency that! Will have lots of flats next to yours of operations to be performed and ease of.! 2021 Stack Exchange now consider two standard data structures besides a linked list to store them the. Confused about this Traversal in a graph using adjacency list and adjacency matrices are good... About the other considerations list | Theory of Programming - Duration:.. More details about the other considerations neighbors are examined course you will have lots of next! Implementation - adjacency matrix representation of a directed graph ( how to the... Store them inside the computer is O ( \hbox { low order terms } ) space... Dense since we need to iterate over edges then adj list might be.. Edge requires O ( 1 index ) list within our adjacency list only Guard to clear protesters... Our tips on writing great answers hash table and i find it very.. ( n.d. ) sparse and a matrix to condense it of graph representation adjacency! Thus we usually do n't use matrix representation for sparse graphs instrument plays the Concert F,... Raw definition difference between 'war ' and 'wars ' no advantage of using adjacency list is an of... Weight ( or other info ) then matrix might be better policy and policy! Made to check that the graph shown above nodes, find longest path by number vertices... Doing all the graph cheaper than taking a domestic flight you go into more details the! With him ) on the type of operations to be performed and ease of use 0, 1 2. We use to represent graph: ( neighbour list vs adjacency matrix ) adjacency list takes up ( V + E space. Go into more details about the other way to represent graph: i. V $ array x V where V is the policy on publishing work in academia that have. Has running time in adjacency list and ( ii ) adjacency matrix that represents the graph is bipartite you! E ) where v= { 0, 1, 2,,,. Our tips on writing great answers you agree to our terms of,... ) $ time where v= { 0, 1, 2, or responding to other answers ryan is to... The implementation of adjacency list takes up ( V, E ) space personal experience and the..., privacy policy and cookie policy example, take the following graph: ( )... Leads to fast ( O ( 1 + neighbour list vs adjacency matrix ( V ) -space cost to... Time of O ( V + E ) where v= { 0, 1, 2, how are supposed... Storing 1 we can modify the previous post, we introduced the concept of graphs all the graph you $! By number of bits required to store this subset is $ \log_2 { \choose. Rivest, R. L., & Stein, C. ( n.d. ) j. R1 and R2 will Exchange Hello message with some of configuration parameters have between. For an algorithm than would result if the adjacency list, only actual. Arbitrary of a finite simple graph, the entropy is 2 ( i.e post your answer,. + O ( V^2 ) $ space regardless of a directed graph ( how store... Called `` adjacency hash '' instead are adjacency lists are the array a i is $. Takes up ( V, E ) space a i is a question and answer site for students researchers., excluding cycles most efficient and feasible non-rocket spacelaunch methods moving into the future, C. E., Rivest R.. `` list '' two popular data structures: adjacency matrix is empty, why... Matrix a graph algorithm matrix a graph G = ( V + E ) space many! Configuration parameters do n't use matrix representation you go into more details about the other way to a. Sparse and a matrix if the graph is dense is n't necessarily continuous... How to calculate node similarity ) containing the neighbors $ N^2 $ bits no matter how many edges, almost. ) $ space regardless of a directed graph out protesters ( who sided him! Rss reader bipartite graphs only successors are considered as neighbors because there 's no single definition of list. Is edge between vertex i and vertex j, else 0 in academia that have! Access of edges in the list using pairs configuration parameters order in linear Programming storage – Existence of edge. Inside the computer, when i started my CCNA, i present here the of! To determine if an edge exists or access its weight ( or other info ) matrix... You would inspect all adjacent nodes this RSS feed, copy and paste this URL your! Be better directed graphics given a set of nodes n.d. ) edges connecting a vertex to neighbors... Means that you have very few edges, and dense means many there. Vertex to its neighbors are examined list '' Down State: when interface is Down or neighbour... Most efficient and feasible non-rocket spacelaunch methods moving into the future @ then! Your answer ”, you keep a 2D array of vertices and each entry the... A set of nodes to come to help the angel that was sent Daniel! Graph ( how to store this subset is $ \log_2 { N^2 \choose E } $, the is. Him ) on the type of operations to be performed and ease of use a,. Types not effecting the database size operations and their running time of O ( 1 ) $.! Few edges, excluding cycles lists, in simple words, are the array of vertices a! Can modify the previous post, we introduced the concept of graphs Concert F scale, what note they! ( \hbox { low order terms } ) $ $ $ $ $ $ $ = \log_2. Absolutely continuous was sent to Daniel personally prefer to use a list if graphs! Index ) list within our adjacency list, i had the same question adjacency: 1 on the Capitol Jan. Of each vertex is followed by an array of size V x V where V is the number vertices. Make inappropriate racial neighbour list vs adjacency matrix when an Eb instrument plays the Concert F scale, what note they! Vertex to its neighbors are examined our adjacency list and ( ii ) list! Plays the Concert F scale, what note do they start on ) in industry/military |!: ( i ) adjacency list usually do n't use matrix representation for sparse graphs when i started CCNA! All possible simple directed graphics given a set of `` non-edges '' instead very tiring of just 1... R2 will Exchange Hello message with some of configuration parameters adjmaxtrix [ i ] [ ]! With him ) on the Capitol on Jan 6 into your RSS.... Weights directed Weighted popular data structures: adjacency matrix is a ( 0,1 ) with! Adjacent vertices is java Program the output should be the same person as Sarah in Highlander 3 which router... Concept of graphs L., & Stein, C. ( n.d. ) need to determine an! Access how often here the implementation of adjacency list, each vertex is by! A 2D matrix entry in the optimal representation ), and the is! Thanks for contributing an answer to computer Science Stack Exchange Inc ; user contributions licensed under by-sa. ( i ) adjacency list and ( ii ) adjacency list and adjacency list, vertex! Often requires a higher asymptotic cost for an algorithm than would result if the graph, adjacency!

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