#adjacency matrix representation of graph in c

[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.…

[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…

[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…

[programmingsolver]

Computer Assignment 09 Simple Graph Solution

Computer Assignment 09 Simple Graph Solution

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 `assignment09.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…

[felord]

FIT1045 Algorithms and programming in Python, S2 Assignment 1 Solved

Objectives

The objectives of this assignment are: To demonstrate the ability to implement algorithms using basic data structures and operations on them. To gain experience in designing an algorithm for a given problem description and implementing that algorithm in Python.

Submission Procedure

Save your files into a zip file called yourStudentID yourFirstName yourLastName.zip Submit your zip file containing your solution to Moodle. Your assignment will not be accepted unless it is a readable zip file. Important Note: Please ensure that you have read and understood the university’s policies on plagiarism and collusion available at http://www.monash.edu.au/students/policies/academic-integrity.html. You will be required to agree to these policies when you submit your assignment. A common mistake students make is to use Google to find solutions to the questions. Once you have seen a solution it is often difficult to come up with your own version. The best way to avoid making this mistake is to avoid using Google. You have been given all the tools you need in workshops. If you find you are stuck, feel free to ask for assistance on Moodle, ensuring that you do not post your code. Marks: This assignment will be marked both by the correctness of your code and by an interview with your lab demonstrator, to assess your understanding. This means that although the quality of your code (commenting, decomposition, good variable names etc.) will not be marked directly, it will help to write clean code so that it is easier for you to understand and explain. This assignment has a total of 12 marks and contributes to 10% of your final mark. For each day an assignment is late, the maximum achievable mark is reduced by 10% of the total. For example, if the assignment is late by 3 days (including weekends), the highest achievable mark is 70% of 12, which is 8.4. Assignments submitted 7 days after the due date will normally not be accepted. Detailed marking guides can be found at the end of each task. Marks are subtracted when you are unable to explain your code via a code walk-through in the assessment interview. Readable code is the basis of a convincing code walk-through.

Task 1: Strings (3.5 Marks)

Create a Python module called strings.py. Within this module implement the following three tasks. For this module you may not use the Python sequence method s.count(x). You may not import any other libraries or modules.

Part A: Code Breaker (1 Mark)

Write a function decode(string, n) that takes in an encoded message as a string and returns the decoded message. Input: a string string and a positive integer n. Output: the decoded string. The string is decoded by discarding the first n characters from the input string, keeping the next n characters, discarding the next n characters, and so on, until the input string is exhausted. There is no guarantee at which point in the decoding (keep or discard) the string will be exhausted. There is no guarantee that the length of string will be a multiple of n. Examples Calling decode(‘#P#y#t#h#o#n#’, 1) returns ‘Python’. Calling decode(‘AxYLet1x3’s T74codaa7e!’, 3) returns ‘Let’s code!’.

Part B: Pattern Finder (1 Mark)

Write a function pattern count(string, pat) that counts how many times a pattern occurs within a text. Input: a string string and a non-empty string pat, both comprising both upper and lower case alphanumeric and non-alphanumeric characters. Output: an integer count of the number of times pat occurs within string. The count is case sensitive. (See Example c.) Examples Calling pattern count(‘bcabcabca’, ‘abc’) returns 2. Calling pattern count(‘aaaa’, ‘aa’) returns 3. Calling pattern count(‘A long time ago in a galaxy far, far away...’, ‘a’) returns 8. Calling pattern count(‘If you must blink, do it now.’, ‘code’) returns 0.

Part C: Palindromes (1.5 Marks)

A palindrome is a word, phrase, or sequence that reads the same backwards as forwards. Write a function palindrome(string) that determines whether or not a given string is a palindrome. In order to receive full marks, the function must determine whether the alphanumerical characters create a palindrome (see Examples). Input: a string string comprising both upper and lower case alphanumeric and non-alphanumeric characters. Output: a boolean, True if the string when converted to lower case alphanumeric characters is a palindrome; otherwise False. Examples Calling palindrome(‘RacecaR’) returns True because ‘RacecaR’ reads the same backwards as forwards. Calling palindrome(‘66racecar77’) returns False because ‘66racecar77’ does not read the same backwards as forwards. Calling palindrome(‘Racecar’) returns True because ‘racecar’ reads the same backwards as forwards. Calling palindrome(‘Madam, I’m Adam’) returns True because ‘madamimadam’ reads the same backwards as forwards. Calling palindrome(‘#4Satire: Veritas4’) returns True because ‘4satireveritas4’ reads the same backwards as forwards.

Marking Guide (total 3.5 marks)

Marks are given for the correct behavior of the different functions: 1 mark for decode. 1 mark for pattern count. 5 marks if palindrome can correctly identify that a string with no processing is a palindrome (see Examples a and b), 1.5 marks if palindrome can correctly identify that a string when converted to lower case alphanumeric characters is a palindrome (see Examples c, d, and e).

Task 2: Friends (4.5 Marks)

You have been employed by Monash to analyse friendship groups on campus. Individuals have been allocated a number between 0 and the number of people on campus (minus one), and their friendships have been recorded as edges between numbered vertices. We assume friendships are always bi-directional. Monash has implemented this graph in a Python-readable format as both an adjacency matrix and an adjacency list. Create a Python module called friends.py. Within this module implement the following three tasks. Ensure you follow the inputs for each function correctly - one function takes an adjacency list, two functions take an adjacency matrix. You may not import any other libraries or modules.

Part A: Popular (1.5 Marks)

Write a function popular(graph list, n) that returns a list of people who have at least n friends. Each person is identified by the number of their vertex. Input: a nested list graph list that represents a graph as an adjacency list, that models the friendships at Monash; and a non-negative integer n. Output: a list of integers, where each integer is a person with at least n friends. If no person has at least n friends, return an empty list. The list may be in any order. Examples clayton_list = , , , , ] The example graph clayton list is provided for illustrative purpose. Your implemented function must be able to handle arbitrary graphs in list form. Calling popular(clayton list,2) returns . Calling popular(clayton list,3) returns . Calling popular(clayton list,0) returns .

Part B: Friend of a Friend (1.5 Marks)

Write a function foaf(graph matrix, person1, person2) that determines whether two people have exactly one degree of separation: that is, whether two (distinct) people are not friends, but have at least one friend in common. Input: a nested list graph matrix that represents a graph as an adjacency matrix, that models the friendships at Monash; an integer person1, and an integer person2, where 0 ≤person1, person2< number of people on campus, and person16= person2. Output: a boolean, True if the two people are not friends and they have at least one friend in common; otherwise False. Examples clayton_matrix = , , , , ] The example graph clayton matrix is provided for illustrative purpose. Your implemented function must be able to handle arbitrary graphs in matrix form. Calling foaf(clayton matrix,0,4) returns True as 0 and 4 are both friends with 2. Calling foaf(clayton matrix,0,3) returns False as 0 and 3 are friends directly. Calling foaf(clayton matrix,1,4) returns False as 1 and 4 have no friends in common.

Part C: Society (1.5 Marks)

Write a function society(graph matrix, person) that determines whether a person has two friends who are also friends with each other. Input: a nested list graph matrix that represents a graph as an adjacency matrix, that models the friendships at Monash; and an integer person, where 0 ≤person< number of people on campus. Output: a boolean, True if person has at least two friends who are also friends with each other; otherwise False. Examples Calling society(clayton matrix,0) returns True as 1 and 3 are both friends with 0. Calling society(clayton matrix,2) returns False as 2 is friends with 0 and 4, but 0 and 4 are not friends with each other.

Marking Guide (total 4.5 marks)

Marks are given for the correct behavior of the different functions: 5 marks for popular. 5 marks for foaf. 5 marks for society.

Task 3: Cards (4 Marks)

A friend of yours is developing a prototype for a website on which people can play poker. They have contacted you to help them implement some of the project’s backend: specifically, they would like some help telling what kind of hand a player has played. You have agreed to implement four functions to assist them. In the backend, your friend has decided to represent cards as tuples comprising an integer and a string: (rank, suit). For example, the seven of clubs would be represented as (7, ‘C’). As this is a prototype, your friend has told you that no face cards will be used (that is, rank is always an integer, where 2 ≤ rank ≤ 10, unless otherwise specified). There are four suits, with the following string representations: clubs: ‘C’, diamonds: ‘D’, hearts: ‘H’, spades: ‘S’. As this is poker, which is only played with one deck of cards, there will never be duplicates, and each hand will contain exactly five cards. Create a Python module called cards.py. Within this module implement the following four tasks. For this module you may not use the Python inbuilt function sorted() or the Python method list.sort(). You may not import any other libraries or modules.

Part A: Suits (1 Mark)

Write a function suits(hand) that sorts a given hand into the four suits. As there is no standard ranking of suits in poker, we will order alphabetically: clubs, diamonds, hearts, spades. Input: a list hand containing five cards. Output: a nested list containing four lists of cards, with each internal list corresponding to a suit. If there are no cards of that suit in the hand, the internal list for that suit is empty. Cards may be in any order within their suit-list.

Part B: Flush (1 Mark)

Write a function flush(hand) that determines whether a given hand is a flush. In poker, a flush is a hand that contains five cards of the same suit. Input: a list hand containing five cards. Output: a boolean, True if all five cards in hand are the same suit, False if not.

Part C: Straight (2 Marks)

Write a function straight(hand) that determines whether a given hand is a straight. In poker, a straight is a hand that contains five cards of sequential rank. The cards may be of different suits. Your friend has asked you an additional favour: they would like to use this function in various other games they may implement in the future, so instead of checking for a five-card straight, they would like you to check for an n card straight, where n is the number of cards in the hand. Like in poker, the cards may be of different suits. Input: a list hand containing at least three cards, where each card may have a rank of some number greater than zero. Output: a boolean, True if the ranks of the cards in hand may be ordered into an unbroken sequence with no rank duplicated, False if not.

Examples

Calling suits() returns , , , ]. As there are multiple clubs, the clubs cards may be in any order within their internal list. Calling flush() returns True. Calling flush() returns False. Calling straight() returns True. Calling straight() returns False. Calling straight() returns False. Calling straight() returns True as the ranks of the cards form an unbroken sequence, , with no rank duplicated.

Marking Guide (total 4 marks)

Marks are given for the correct behavior of the different functions: 1 mark for suits. 1 mark for flush. 1 mark for straight if it is able to handle a normal poker five-card hand with card ranks between 2 and 10 inclusive (see Examples d, e, and f); 2 marks for straight if it is able to handle an arbitrarily large hand with arbitrary ranks (see Example g). The functions may return True or False at your discretion for instances where person1 is person2. This case will not be marked. Read the full article

[naivelocus]

Protein Sequence Comparison Based on Physicochemical Properties and the Position-Feature Energy Matrix

Zhao, Y., Li, X. & Qi, Z. Novel 2D graphic representation of protein sequence and its application. J. Fiber Bioengineering and Informatics 7, 23–33 (2014).

Huang, D. & Yu, H. Normalized Feature Vectors: A novel alignment-free sequence comparison method based on the numbers of adjacent amino acids. IEEE/ACM Trans. Comput. Biol. 10, 457–467 (2013).

Gotoh, O. An improved algorithm for matching biological sequences. J. Mol. Biol. 162, 705–708 (1982).

Chakraborty, A. & Bandyopadhyay, S. FOGSAA: Fast optimal global sequence alignment algorithm. Sci. Rep. 3, 1746 (2013).

Feng, D. & Doolittle, R. F. Progresssive sequence alignment as a prerequisite to correct phylogenetic trees. J. Mol. Evol. 25, 351–360 (1987).

Bradley, R. K. et al. Fast Statistical Alignment. PLoS Comput. Biol. 5, e1000392 (2009).

Reinert, G., Chew, D., Sun, F. & Waterman, M. S. Alignment-free sequence comparison(I): Statistics and power. J. Comput. Biol. 16, 1615–1634 (2009).

Schwende, I. & Pham, T. D. Pattern recognition and probabilistic measures in alignment-free sequence analysis. Brief Bioinform 15, 354–368 (2014).

Borozan, I., Watt, S. & Ferretti, V. Integrating alignment-based and alignment-free sequence similarity measures for biological sequence classification. Bioinf. 31, 1396–1404 (2015).

Didier, G., Corel, E., Laprevotte, I., Grossmann, A. & Landès-Devauchelle, C. Variable length local decoding and alignment-free sequence comparison. Theor. Comput. Sci. 462, 1–11 (2012).

Nakashima, H., Nishikawa, K. & Ooi, T. The folding type of a protein is relevant to the amino acid composition. J. Biochem. 99, 152–162 (1986).

Chou, K. C. Some remarks on protein attribute prediction and pseudo amino acid composition (50th Anniversary Year Review). J. Theor. Biol. 273, 236–247 (2011).

Mohabatkar, H., Beigi, M. M., Abdolahi, K. & Mohsenzadeh, S. Prediction of allergenic proteins by means of the concept of Chou’s pseudo amino Aacid composition and a machine learning approach. Medicinal Chemistry 9, 133–137 (2013).

Zhong, W. & Zhou, S. Molecular science for drug development and biomedicine. Int. J. Molec. Sci. 15, 20072–20078 (2014).

He, P., Wei, J., Yao, Y. & Tie, Z. A novel graphical representation of proteins and its application. Physica A 391, 93–99 (2012).

Randić M. et al. Graphical representation of proteins. Chem. Rev. 111, 790–862 (2011).

Jiang, S., Liu, W. & Fee, C. H. Graph theory of enzyme kinetics: I. Steady state reaction system, Scientia Sinica 22, 341–358 (1979).

Yao, Y. et al. Analysis of similarity/dissimilarity of protein sequences. Proteins 73, 864–871 (2008).

Kuang, C., Liu, X., Wang, J., Yao, Y. & Dai, Q. Position-specific statistical model of DNA sequences and its application for similarity analysis. MATCH Commun. Math. Comput. Chem. 73, 545–558 (2015).

Sun, D., Xu, C. & Zhang, Y. A novel method of 2D graphical representation for proteins and its application. MATCH Commun. Math. Comput. Chem. 75, 431–446 (2016).

Xia, X. & Li, W. What amino acid properties affect protein evolution? J. Mol. Evol. 47, 557–564 (1998).

Qi, Z., Jin, M., Li, S. & Feng, J. A protein mapping method based on physicochemical properties and dimension reduction. Comput. Biol. Med. 57, 1–7 (2015).

Gutman, I. The energy of a graph. Ber. Math. Statist. Sekt. Forschungsz. Graz 103, 1–22 (1978).

Wu, H., Zhang, Y., Chen, W. & Mu, Z. Comparative analysis of protein primary sequences with graph energy. Physica A 43, 249–262 (2015).

Gutman, I., Li, X. & Zhang, J. Graph energy, in: Analysis of Complex Networks. From Biology to Linguistics, (ed. Dehmer, M. & Emmert-Streib, F.) 145–174 (Wiley-VCH, Weinheim, 2009).

Li, X., Shi, Y. & Gutman, I. Graph Energy (ed. Li, X., Shi, Y. & Gutman) (Springer. New York, 2012).

Zamyatin, A. A. Protein volume in solution. Prog. Biophys. Mol. Biol. 24, 107–123 (1972).

Chotia, C. The nature of the accessible and buried surfaces in proteins. J. Mol. Biol. 105, 1–14 (1975).

Randić, M. 2-D graphical representation of proteins based on physicochemical properties of amino acids. Chem. Phys. Lett. 444, 176–180 (2007).

Paola, L. D., Mei, G., Venere, A. D. & Giuliani, A. Exploring the stability of dimers through protein structure topology. Curr. Protein Peptide Sci. 17, 30–36 (2016).

Yu, L., Zhang, Y., Jian, G. & Gutman, I. Classification for microarray data based on K-means clustering combined with modified single-to-noise-ratio based on graph energy, J. Comput. Theor. Nanosci. 14, 598–606 (2017).

Emmert-Streib, F., Dehmer, M. & Shi, Y. Fifty years of graph matching, network alignment and comparison. Inform. Sci. 346–347, 180–197 (2016).

Dehmer, M., Emmert-Streib, F., Chen, Z., Li, X. & Shi, Y. Mathematical Foundations and Applications of Graph Entropy, (ed. Dehmer, M. et al.) (Wiley, 2016).

Yu, C., Deng, M. & Yau, S. S. DNA sequence comparison by a novel probabilistic method. Inf. Sci. 181, 1484–1492 (2011).

Cover, T. M. & Thomas, J. A. Elements of Informatiobn Theory, (ed. Wiley, J. & Sons) 2nd edition (Wiley, 1991).

Kullback, S. & Leibler, R. A. On information and sufficiency. Ann. Math. Stat. 22, 79–86 (2015).

Yu, C., Cheng, S., He, R. & Yau, S. S. Protein map: A alignment-free sequence comparison method based on various properties of amino acids. Gene 486, 110–118 (2011).

Emmert-Streib, F. & Dehmer, M. Information processing in the transcriptional regulatory network of yeast: Functional robustness. BMC Systems Biology 3 (2009).

Lam, W. & Bacchus, F. Learning Bayesian belief networks: An approach based on the MDL principle. Computat. Intell. 10, 269–293 (1994).

Xiao, X. et al. Using complexity measure factor to predict protein subcellular location. Amino Acids 28, 57–61 (2005).

Liao, B., Liao, B., Sun, X. & Zeng, Q. A novel method similarity analysis and protein sub-cellular localization prediction. Bioinf. 26, 2678–2683 (2010).

Mu, Z., Wu, J. & Zhang, Y. A novel method for similarity/dissimilarity analysis of protein sequences. Physica A 392, 6361–6366 (2013).

Chang, G. & Wang, T. Phylogenetic analysis of protein sequences based on distribution of length about common substring. Protein J. 30, 167–172 (2011).

Ford, M. J. Molecular evolution of transferrin: Evidence for positive selection in salmonids. Mol. Biol. Evol. 18, 639–647 (2001).

Davies, P. L., Baardsnes, J., Kuiper, M. J. & Walker, V. K. Structure and function of antifreeze proteins. Phil. Trans. R. Soc. Lond. B 357, 927–935 (2002).

Duman, J., Verleye, D. & Li, N. Site-specific forms of antifreeze protein in the beetle dendroides canadensis. J. Comp. Physiol. B 172, 547–552 (2002).

Graether, S. P. et al. Beta-helix structure and ice-binding properties of a hyperactive antifreeze protein from an insect. Nature 406, 325–328 (2000).

Graether, S. P. & Sykes, B. D. Cold survival in freeze intolerant insects: the structure and function of beta-helical antifreeze proteins. J. Biochem. 271, 3285–3296 (2004).

Altschul, S. F. et al. Gapped LAST and PSI-BLAST: a new generation of protein database search programs. Nucleic Acids Res. 25, 3389–3402 (1997).

Yau, S., Yu, C. & He, R. A protein map and its application. DNA Cell. Biol. 27, 241–250 (2008).

Xu, C., Sun, D., Liu, S. & Zhang, Y. Protein sequence analysis by incorporating modified chaos game and physicochemical properties into Chou’s general pseudo amino acid composition. J. Theor. Biol. 406, 105–115 (2016).

— Nature Scientific Reports

[codezclub]

C Program to find Path Matrix by powers of Adjacency matrix

Path Matrix by powers of Adjacency Matrix Write a C Program to find Path Matrix by powers of Adjacency Matrix. Here’s simple Program to find Path Matrix by powers of Adjacency Matrix 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…

[myprogrammingsolver]

Computer Assignment 09 Simple Graph Solution

Computer Assignment 09 Simple Graph Solution

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 `assignment09.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…