From 21d8d242f8e23d358676b83533d1a8cea43f5951 Mon Sep 17 00:00:00 2001 From: aheaton22 <117932053+aheaton22@users.noreply.github.com> Date: Fri, 1 Dec 2023 21:15:34 +0000 Subject: [PATCH 1/6] working on graphs update --- Algorithms/Graphs/readme.md | 103 ++++++++++++++++++++++++------------ 1 file changed, 68 insertions(+), 35 deletions(-) diff --git a/Algorithms/Graphs/readme.md b/Algorithms/Graphs/readme.md index e8c8ef0c..f4e68d16 100644 --- a/Algorithms/Graphs/readme.md +++ b/Algorithms/Graphs/readme.md @@ -1,57 +1,91 @@ # Graphs -A Graph is a non-linear data structure consisting of vertices and edges. The vertices are sometimes also referred to as nodes and the edges are lines or arcs that connect any two nodes in the graph. More formally a Graph is composed of a set of vertices( V ) and a set of edges( E ). The graph is denoted by G(E, V). +A graph is a data structure used in computer science and mathematics. Graphs are used to model pairwise relations between objects and are the most useful data structures for many real-world applications. For example, the airline route network is a graph in which the cities are the vertices, and the flight routes are the edges. It is a non-linear data structure consisting of a set of vertices and edges. +The vertices, also called nodes or points, is a finite but possibly changeable set of vertices and are written as a set of vertex names, e.g. V=(A, B, C, ..) -## Components of a Graph -- **[Vertices](#vertices)** - A vertex is a node of the graph. It can be denoted by any symbol such as V, U, X, Y, etc. A vertex may also be referred to as a node or a point. -- **[Edges](#edges)** - An edge is a connection between two nodes. It can be denoted by any symbol such as E, F, G, H, etc. An edge may also be referred to as a link or a line. -- **[Weight](#weight)** - A weight is a value associated with an edge. It can be denoted by any symbol such as W, X, Y, Z, etc. A weight may also be referred to as a cost. +The edges or connections between vertices are called edges. These edges are known as arcs, lines, and arrows. Edges are written as a set of unordered pairs, e.g. E= ((A,B), (A,C), ..). -## Types of Graphs -- [Undirected Graph](#undirected-graph) -- [Directed Graph](#directed-graph) -- [Weighted Graph](#weighted-graph) +These vertices and edges are what make up a graph. More formally, a Graph is defined as a set of vertices( V ), a set of edges( E ) and is denoted by G(E, V). -## Common Graph Representations -- [Adjacency Matrix](#adjacency-matrix) -- [Adjacency List](#adjacency-list) +### Example graph +Here is an example graph of a group of cities and plane flights in between them. + +Cities +New York (NY) +Los Angeles (LA) +Houston (H) +San Francisco (SF) +Atlanta (A) +Denver (D) +Flights +New York to Atlanta (NY, A) +Atlanta to Denver (A, D) +Atlanta to Houston (A, H) +Atlanta to Los Angeles (A, LA) +Denver to San Francisco (D, SF) +Denver to Huston (D, H) +San Francisco to Los Angeles (SF, LA) +Los Angeles to Houston (LA, H) -## Vertices -## Edges -## Weight -### Example -Pic Will be added +A graph representing the relation between cities and flights would look like this: -## Undirected Graph -An undirected graph is a graph where all the edges are bidirectional. That is, the undirected graph does not move in any direction. +V = (NY, LA, H, SF, A, D) -### Example -Pic Will be added +E = ((NY, A), (A, D), (A, H), (A, LA), (D, SF), (D, H), (SF, LA), (LA, H)) -## Directed Graph -A directed graph is a graph where all the edges are unidirectional. That is, the directed graph moves in a particular direction. +flights = G(E, V) -### Example -Pic Will be added +TODO add image of graph +(![alt text](graph.png "Title")) -## Weighted Graph -A weighted graph is a graph where all the edges have a weight associated with them. That is, the weighted graph has a value associated with each edge. -### Example -Pic Will be added -## Adjacency Matrix +## Graph Components +- Edges - An edge is a connection between two nodes. It can be denoted by any symbol such as E, F, G, H, etc. An edge may also be referred to as a link or a line. +- Weight - A weight is a value associated with an edge. It can be denoted by any symbol such as W, X, Y, Z, etc. A weight may also be referred to as a cost. +- Vertices - A vertex is a node of the graph. It can be denoted by any symbol such as V, U, X, Y, etc. Vertices also have a degree, or the number of edges connected to it. A vertex’s in-degree is defined as the number of edges that point to a vertex and the vertex’s out-degree is the number of edges that point to other vertices. There can be many different types of vertices such as: + - Isolated vertex - Has no edges pointing to the vertex, and it has no outgoing edges. Its in-degree and out-degree is zero. + TODO add example + - Source vertex - Has no edges point to the vertex, its in-degree is zero. + TODO add example + - Sink vertex - Has no outgoing edges, it’s out-degree is zero. + TODO add example + - [Articulation Points](Graphs/Articulation%20Points/readme.md) - A vertex that creates an undirected graph if it is removed from the graph. + TODO add example and fix explanation + + +## Types of Graphs +There are a few different types of graphs that are used in computer science. Some graph algorithms will only work on a specific type of graph. It is important to remember what kind of graph is +- Undirected Graph - A graph where edges of the graph are two-way paths or relations. +TODO add example +- Directed Graph - A graph where edges of the graph go only one way, usually marked with arrows. +TODO add example +- Weighted Graph - A graph where edges of the graph have costs or weights associated with them. +TODO add example +- [Tree Graphs](Graphs/Tree%20Based%20Algorithms/readme.md) - A graph with n vertices and n-1 edges where there exists only one path between vertices. +TODO add example + +## Common Graph Representations +In order to run algorithms on graphs we need to be able to store the vertex and edge data in a data structure. There are many different ways to represent a graph but the two most common ways are by using an adjacency matrix or list. + +### Adjacency Matrix An 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 to vertex j. Adjacency matrix for undirected graph is always symmetric. Adjacency Matrix is also used to represent weighted graphs. If adj[i][j] = w, then there is an edge from vertex i to vertex j with weight w. -### Example -Pic Will be added +#### Example +TODO add image -## Adjacency List +#### Adjacency Matrix code +TODO add example code + +### Adjacency List An adjacency list is a collection of unordered lists used to represent a finite graph. Each list describes the set of neighbors of a node in the graph. The number of lists is equal to the number of vertices in the graph. If there are n vertices, then there are n adjacency lists. An adjacency list is an array of linked lists. The size of the array is equal to the number of vertices. Let the array be array[]. An entry array[i] represents the linked list of vertices adjacent to the ith vertex. This representation can also be used to represent a weighted graph. The weights of edges can be represented as nodes with additional information. -### Example +#### Example +TODO - Pic Will be added + +#### Adjacency list code ```cpp // A C++ Program to demonstrate adjacency list // representation of graphs @@ -97,7 +131,6 @@ int main() return 0; } ``` -Pic Will be added ## Graph Topics 1. [Graph Traversal](Traversal%20Algorithms/readme.md) From ae6ad98d9c58826876197b61864a1dcbe15e9f45 Mon Sep 17 00:00:00 2001 From: aheaton22 <117932053+aheaton22@users.noreply.github.com> Date: Sat, 2 Dec 2023 00:02:16 +0000 Subject: [PATCH 2/6] working on examples --- Algorithms/Graphs/readme.md | 88 +++++++++++++++++++++---------------- Algorithms/readme.md | 2 +- 2 files changed, 50 insertions(+), 40 deletions(-) diff --git a/Algorithms/Graphs/readme.md b/Algorithms/Graphs/readme.md index f4e68d16..ec517f42 100644 --- a/Algorithms/Graphs/readme.md +++ b/Algorithms/Graphs/readme.md @@ -1,22 +1,20 @@ # Graphs -A graph is a data structure used in computer science and mathematics. Graphs are used to model pairwise relations between objects and are the most useful data structures for many real-world applications. For example, the airline route network is a graph in which the cities are the vertices, and the flight routes are the edges. It is a non-linear data structure consisting of a set of vertices and edges. +A graph is a data structure used in computer science and mathematics. Graphs are used to model pairwise relations between objects and are the most useful data structures for many real-world applications, such as the airline route network. Graphs are a non-linear data structure that allows us to more easily define complicated relations. A graph consists of a set of vertices and edges. -The vertices, also called nodes or points, is a finite but possibly changeable set of vertices and are written as a set of vertex names, e.g. V=(A, B, C, ..) +The vertices of a graph, also called nodes or points, are a finite but possibly changeable set of values and are written as a set of names, e.g. V=(A, B, C, ..). If we were to model the airline route network as a graph the cities would be represented as vertices. For example: -The edges or connections between vertices are called edges. These edges are known as arcs, lines, and arrows. Edges are written as a set of unordered pairs, e.g. E= ((A,B), (A,C), ..). +Cities +New York -> NY +Los Angeles -> LA +Houston -> H +San Francisco -> SF +Atlanta -> A +Denver -> D -These vertices and edges are what make up a graph. More formally, a Graph is defined as a set of vertices( V ), a set of edges( E ) and is denoted by G(E, V). - -### Example graph -Here is an example graph of a group of cities and plane flights in between them. +The set of graph vertices would be defined as: +V = (NY, LA, H, SF, A, D) -Cities -New York (NY) -Los Angeles (LA) -Houston (H) -San Francisco (SF) -Atlanta (A) -Denver (D) +The edges or connections between vertices are called edges. These edges are known as arcs, lines, and arrows. Edges are written as a set of unordered pairs, e.g. E= ((A,B), (A,C), ..). They represent the relations between values. In our airline example the graph edges would be the available flights between cites. Flights New York to Atlanta (NY, A) @@ -24,48 +22,60 @@ Atlanta to Denver (A, D) Atlanta to Houston (A, H) Atlanta to Los Angeles (A, LA) Denver to San Francisco (D, SF) -Denver to Huston (D, H) +Denver to Houston (D, H) San Francisco to Los Angeles (SF, LA) Los Angeles to Houston (LA, H) -A graph representing the relation between cities and flights would look like this: - -V = (NY, LA, H, SF, A, D) - +The set of graph edges would be defined as: E = ((NY, A), (A, D), (A, H), (A, LA), (D, SF), (D, H), (SF, LA), (LA, H)) -flights = G(E, V) - -TODO add image of graph -(![alt text](graph.png "Title")) - +These vertices and edges are what make up a graph. More formally, a graph is defined as a set of vertices( V ), a set of edges( E ) and is denoted by G(E, V). flights = G(E, V) +A graph representing the relation between cities and flights would look like this: +TODO add image of directed graph +(![alt text](graph.png "Title")) -## Graph Components -- Edges - An edge is a connection between two nodes. It can be denoted by any symbol such as E, F, G, H, etc. An edge may also be referred to as a link or a line. -- Weight - A weight is a value associated with an edge. It can be denoted by any symbol such as W, X, Y, Z, etc. A weight may also be referred to as a cost. -- Vertices - A vertex is a node of the graph. It can be denoted by any symbol such as V, U, X, Y, etc. Vertices also have a degree, or the number of edges connected to it. A vertex’s in-degree is defined as the number of edges that point to a vertex and the vertex’s out-degree is the number of edges that point to other vertices. There can be many different types of vertices such as: - - Isolated vertex - Has no edges pointing to the vertex, and it has no outgoing edges. Its in-degree and out-degree is zero. - TODO add example - - Source vertex - Has no edges point to the vertex, its in-degree is zero. - TODO add example - - Sink vertex - Has no outgoing edges, it’s out-degree is zero. - TODO add example - - [Articulation Points](Graphs/Articulation%20Points/readme.md) - A vertex that creates an undirected graph if it is removed from the graph. - TODO add example and fix explanation +With this graph we can now answer questions like: Can I get to San Francisco from New York? How many layovers are there between Atlanta and Denver? Can I get to Houston from Los Angeles? How many different flights are there from New York to Los Angeles? We can look at the graph and count the edges and vertices as we move from one city to another. But instead of calculating solutions by hand we can use algorithms like Dijkstra's, Bellman-Ford, or Kruskal's to give us the answers to those questions. ## Types of Graphs There are a few different types of graphs that are used in computer science. Some graph algorithms will only work on a specific type of graph. It is important to remember what kind of graph is + - Undirected Graph - A graph where edges of the graph are two-way paths or relations. TODO add example -- Directed Graph - A graph where edges of the graph go only one way, usually marked with arrows. +- Directed Graph - Also called a Digraph, is a graph where edges of the graph go only one way, usually marked with arrows. TODO add example -- Weighted Graph - A graph where edges of the graph have costs or weights associated with them. +- Weighted/Unweighted Graph - A graph where edges of the graph have costs or weights associated with them. TODO add example - [Tree Graphs](Graphs/Tree%20Based%20Algorithms/readme.md) - A graph with n vertices and n-1 edges where there exists only one path between vertices. TODO add example +- Connected and Disconnected Graph - A graph were not all of the nodes are connected. +TODO add example + - Complete Graph - A graph where all of the nodes are connected to all of the other nodes. +- Acyclic and Cyclic Graphs - A graph that has a cycle. A cycle is a path that starts and ends at the same vertex. +- Sparse Graph +TODO add example +- Dense Graph +TODO add example + + +## Graph Components +The typed of edges and vertices a graph has can vary. Here is a brief explanation of some of them. +- Edges - An edge is a connection between two nodes. It can be denoted by any symbol such as E, F, G, H, etc. An edge may also be referred to as a link or a line. + - Weighted Edge - When a weight or cost value associated with an edge it is an weighted edge. The weight may also be referred to as a cost. This cost can be distance, money, time, etc. These weights can be zero, negative, positive, or infinite. It depends on the relations the graph is representing. + - Undirected - A undirected edge is an edge that allows for to two way connections. In other words, if vertex A is connected to B then B is connected to A. The flight graph shown above is undirected edges. + TODO fix example + - Directed - A directed edge is an edge that only goes one way. In other words, vertex A is connected to B but B is not connected to A. If we were to alter the flight list in the above example and change the flights to be one-way then the edges would become directed edges. + TODO fix example + +- Vertices - A vertex is a node of the graph. It can be denoted by any symbol such as V, U, X, Y, etc. Vertices also have a degree, or the number of edges connected to it. A vertex’s in-degree is defined as the number of edges that point to a vertex. The vertex’s out-degree is the number of edges that point to other vertices. The different types of vertices are defined by these degrees. + - Isolated vertex - Has no edges pointing to the vertex, and it has no outgoing edges. Its in-degree and out-degree is zero. If we were to add the city Oklahoma to our list of vertices it would be an isolated vertex as there are no flights going in or our of the city. + - Source vertex - Has no edges point to the vertex, its in-degree is zero. The city, New York, is source vertex as there are no flights going to the city. + - Sink vertex - Has no outgoing edges, it’s out-degree is zero. The city Houston is a sink vertex as there are no flights going out of that city. + - [Articulation Points](Graphs/Articulation%20Points/readme.md) - Also called a bridge, is vertex in an undirected graph that would created a disconnected graph if removed. + TODO add example + ## Common Graph Representations In order to run algorithms on graphs we need to be able to store the vertex and edge data in a data structure. There are many different ways to represent a graph but the two most common ways are by using an adjacency matrix or list. @@ -132,7 +142,7 @@ int main() } ``` -## Graph Topics +## Graph Topics and Algorithms 1. [Graph Traversal](Traversal%20Algorithms/readme.md) - [Breadth First Search](Traversal%20Algorithms/BreadthFirstSearch/readme.md) - [A* Search](Traversal%20Algorithms/AstarSearch/readme.md) diff --git a/Algorithms/readme.md b/Algorithms/readme.md index 0064fee6..713f206d 100644 --- a/Algorithms/readme.md +++ b/Algorithms/readme.md @@ -53,7 +53,7 @@ Graphs are used to model pairwise relations between objects and are the most use - Isolated vertex - Has no edges pointing to the vertex, and it has no outgoing edges. Its in-degree and out-degree is zero. - Source vertex - Has no edges point to the vertex, its in-degree is zero. - Sink vertex - Has no outgoing edges, it’s out-degree is zero. - - [Articulation Points](Graphs/Articulation%20Points/readme.md) - A vertex that creates an undirected graph if it is removed from the graph.. + - [Articulation Points](Graphs/Articulation%20Points/readme.md) - A vertex in an undirected graph that would created a disconnected graph if removed. ### Types of Graphs - Undirected Graph - A graph where edges of the graph are two-way paths or relations. From f08aa16cddd83ecaf80614e315d50a590475b9d3 Mon Sep 17 00:00:00 2001 From: aheaton22 <117932053+aheaton22@users.noreply.github.com> Date: Sat, 2 Dec 2023 03:50:17 +0000 Subject: [PATCH 3/6] updated explanations --- .../Graphs/Tree Based Algorithms/readme.md | 2 +- .../graphImages/directedFlightsGraph.png | Bin 0 -> 54982 bytes .../graphImages/disconnectedFlightsGraph.png | Bin 0 -> 52899 bytes .../graphImages/undirectedFlightsGraph.png | Bin 0 -> 58037 bytes .../weightedDirectedFlightsGraph.png | Bin 0 -> 56062 bytes Algorithms/Graphs/readme.md | 124 +++++++++++++----- 6 files changed, 90 insertions(+), 36 deletions(-) create mode 100644 Algorithms/Graphs/graphImages/directedFlightsGraph.png create mode 100644 Algorithms/Graphs/graphImages/disconnectedFlightsGraph.png create mode 100644 Algorithms/Graphs/graphImages/undirectedFlightsGraph.png create mode 100644 Algorithms/Graphs/graphImages/weightedDirectedFlightsGraph.png diff --git a/Algorithms/Graphs/Tree Based Algorithms/readme.md b/Algorithms/Graphs/Tree Based Algorithms/readme.md index 5f04919e..9633aadf 100644 --- a/Algorithms/Graphs/Tree Based Algorithms/readme.md +++ b/Algorithms/Graphs/Tree Based Algorithms/readme.md @@ -1,4 +1,4 @@ # TREE -You might be wondering why this data structure is being explained again as it is already present in the data structures directory. That is because this is a special kind of a data structure that can be called as a special form of a graph. A graph containg n nodes and n-1 edges can be called as a tree provided all the nodes are connected to each other. This is used to solve a variety of different problems especially in competitive programming. +You might be wondering why this data structure is being explained again as it is already present in the data structures directory. That is because this is a special kind of a data structure that can be called as a special form of a graph. A graph containing n nodes and n-1 edges can be called a tree provided all the nodes are connected to each other. 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zw3b~egcPD!K}wo8CtR}-(L0#mFQ|g5)gfz;N8qoA%IMZch7}AdOGy9(k3-b*)UMQzrN4<1v&R~aDRd<6Y zO7d5Yxnw*21|i*xLYB8X4z*j(eEw)ab*zi7bBr~e887eMV{WI@6-Vgg>37kE|X0>IU(>i$5hNtV5m*`#T+l=I0){H$T|1(=T$F=i#nw=Bd} zPdXz0H|Y43vbU&i*fx=U2Gy^|ROB(IJB}aKleMpg9irC7`;o_SWI0Dm+X@QB9-;@2 zLmvJ+^6Z>ghLSvmj@_owr#@Wc=`k4{me;_Pi_{Nw5}b!go;vI+wHKmhqB0GeA`-Wjjt4sk`MR6=5Y2#%M_VvjpJsS-18hizn`w z=7Vc-ez~F1@P)s~v2J7iP~#hg52s z=oY2pwyVNFuEvrEf?twMcLZ$}g`$lepc1G8_(SiP?yI^C|B$B#{)YUAwSe2;&m;J@ z@fVLIuq^m<6RbP@b^Rj`G_UwDYyh6%&mrJ3@Rz*h-@nXoc79!QZL(jBboU$b|CN With this graph we can now answer questions like: Can I get to San Francisco from New York? How many layovers are there between Atlanta and Denver? Can I get to Houston from Los Angeles? How many different flights are there from New York to Los Angeles? We can look at the graph and count the edges and vertices as we move from one city to another. But instead of calculating solutions by hand we can use algorithms like Dijkstra's, Bellman-Ford, or Kruskal's to give us the answers to those questions. ## Types of Graphs -There are a few different types of graphs that are used in computer science. Some graph algorithms will only work on a specific type of graph. It is important to remember what kind of graph is - -- Undirected Graph - A graph where edges of the graph are two-way paths or relations. -TODO add example -- Directed Graph - Also called a Digraph, is a graph where edges of the graph go only one way, usually marked with arrows. -TODO add example -- Weighted/Unweighted Graph - A graph where edges of the graph have costs or weights associated with them. -TODO add example -- [Tree Graphs](Graphs/Tree%20Based%20Algorithms/readme.md) - A graph with n vertices and n-1 edges where there exists only one path between vertices. -TODO add example -- Connected and Disconnected Graph - A graph were not all of the nodes are connected. -TODO add example - - Complete Graph - A graph where all of the nodes are connected to all of the other nodes. -- Acyclic and Cyclic Graphs - A graph that has a cycle. A cycle is a path that starts and ends at the same vertex. -- Sparse Graph -TODO add example -- Dense Graph -TODO add example +There are a few different types of graphs that are used in computer science. Note that some graph algorithms will only work on a specific types of graph. + +- Directed Graph - Also called a Digraph, is a graph where edges of the graph go only one way with the directions marked with arrows. The above graph is an example of a directed graph. + +Directed Flights Graph + + + +- Undirected Graph - A graph where edges of the graph are two-way paths or relations. If we were to update change the flghts examples to be round-trip or two way we would get an undirected graph. + +Undirected Flights Graph + + + +- Unweighted Graph - A graph were the edges of the graph do not have any weight or cost associated with them. Undirected graphs are used to model connections. like social networks, where the cost of connecting two vertices does not matter. The above graph is an example of an undirected graph. + +- Weighted Graph - A graph where edges of the graph have costs or weights associated with them. As we traverse the graph we can add up the costs of the edges to get the total cost of taking the route. From theses costs we can determine the shortest path or the cheapest route. If we add weights to our flight graph we get the following weighted graph. + +Weighted Flights Graph - ToDo-fix image, one edge is missing an image. + + +- Connected Graphs + - Undirected Connected and Disconnected Graphs - An undirected graph is considered connected if there exists a path between every pair of vertices. Otherwise it is disconnected. In a undirected connected graph you can get to very node in the graph from any other node. The flight graph is an undirected connected graph but if we remove the edge between NY and A it would create an undirected graph. + + Disconnected Flights Graph + + + + - Connected Directed Graphs - An directed graph can be strongly or weakly connected. A strongly connected digraph graph has as path from every vertex to every other other vertex. A weakly connected digraph is one where every vertex is connected to graph in some way. See [Kosaraju's Algorithm](Kosaraju%20Algorithm/readme.md) for calculating strongly connected components in a graph. If we ignore the directionality of the edges and make our graph undirected that is connected then the digraph is considered weakly connected. Our digraph flight example is not an strongly connected graph but it is weakly connected. to create a strongly connected graph we would have to add more connecting edges. + + Todo-Example? + +- Acyclic and Cyclic Graphs - Graphs can have cycles or paths that allow us to get back to the same node. In other words A cycle is a path that starts and ends at the same vertex. A graph is cyclic if is has one or more of these loops and acyclic and if it does not. Not all graphs have a cycle and some algorithms like Prims assume there are no cycles and algorithms like Floyd-Warshall cannot handle negative cycles (a cycle where all the edge costs are negative). + + Todo-Example? + +- [Tree Graphs](Graphs/Tree%20Based%20Algorithms/readme.md) - Trees are acyclic and connected graphs where the vertices are connected in such a way that there is only one path between vertices. A tree graph will n vertices and n-1 edges with each node connected to the tree. A set of disconnected tree graphs is called a forest. A tree can have either directed or undirected edges. + + Todo-Example? ## Graph Components The typed of edges and vertices a graph has can vary. Here is a brief explanation of some of them. - Edges - An edge is a connection between two nodes. It can be denoted by any symbol such as E, F, G, H, etc. An edge may also be referred to as a link or a line. - - Weighted Edge - When a weight or cost value associated with an edge it is an weighted edge. The weight may also be referred to as a cost. This cost can be distance, money, time, etc. These weights can be zero, negative, positive, or infinite. It depends on the relations the graph is representing. - - Undirected - A undirected edge is an edge that allows for to two way connections. In other words, if vertex A is connected to B then B is connected to A. The flight graph shown above is undirected edges. - TODO fix example - - Directed - A directed edge is an edge that only goes one way. In other words, vertex A is connected to B but B is not connected to A. If we were to alter the flight list in the above example and change the flights to be one-way then the edges would become directed edges. - TODO fix example + - Weighted Edge - When a weight or cost value associated with an edge it is an weighted edge. The weight may also be referred to as a cost. This cost can be distance, money, time, etc. These weights can be zero, negative, positive, or infinite. It depends on the relations the graph is representing. See Weighted Flights Graph for an example. + - Undirected - A undirected edge is an edge that allows for to two way connections. In other words, if vertex A is connected to B then B is connected to A. See Undirected Flights Graph for an example. + - Directed - A directed edge is an edge that only goes one way. In other words, vertex A is connected to B but B is not connected to A. If we were to alter the flight list in the above example and change the flights to be one-way then the edges would become directed edges. see Directed Flights Graph for an example. -- Vertices - A vertex is a node of the graph. It can be denoted by any symbol such as V, U, X, Y, etc. Vertices also have a degree, or the number of edges connected to it. A vertex’s in-degree is defined as the number of edges that point to a vertex. The vertex’s out-degree is the number of edges that point to other vertices. The different types of vertices are defined by these degrees. - - Isolated vertex - Has no edges pointing to the vertex, and it has no outgoing edges. Its in-degree and out-degree is zero. If we were to add the city Oklahoma to our list of vertices it would be an isolated vertex as there are no flights going in or our of the city. - - Source vertex - Has no edges point to the vertex, its in-degree is zero. The city, New York, is source vertex as there are no flights going to the city. - - Sink vertex - Has no outgoing edges, it’s out-degree is zero. The city Houston is a sink vertex as there are no flights going out of that city. - - [Articulation Points](Graphs/Articulation%20Points/readme.md) - Also called a bridge, is vertex in an undirected graph that would created a disconnected graph if removed. - TODO add example +- Vertices - A vertex is a node of the graph. It can be denoted by any symbol such as V, U, X, Y, etc. Vertices also have a degree, or the number of edges connected to it. A vertex’s in-degree is defined as the number of edges that point to a vertex. The vertex’s out-degree is the number of edges that point to other vertices. In an undirected graph the in and out degree is the same. The different types of vertices are defined by these degrees. + - Isolated vertex - No edges pointing to the vertex, and it has no outgoing edges. Its in-degree and out-degree is zero. In the Disconnected Flights Graph NY is an isolated vertex. + - Source vertex - No edges point to the vertex, its in-degree is zero. In the Directed Flights Graph, NY is a source vertex as there are no flights or edges going to the city. + - Sink vertex - Has no outgoing edges, it’s out-degree is zero. In the Directed Flights Graph, vertex H is a sink vertex as there are no flights going out of that city. + - [Articulation Points](Graphs/Articulation%20Points/readme.md) - Also called a bridge, is vertex in an undirected graph that would created a disconnected graph if removed. if we were to remove A and all its edges from the Directed Flights Graph we would get an disconnected graph. A is an articulation point. ## Common Graph Representations @@ -84,16 +101,53 @@ In order to run algorithms on graphs we need to be able to store the vertex and An 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 to vertex j. Adjacency matrix for undirected graph is always symmetric. Adjacency Matrix is also used to represent weighted graphs. If adj[i][j] = w, then there is an edge from vertex i to vertex j with weight w. #### Example -TODO add image +Adjacency matrix for the undirected and directed flight graphs + +Undirected +| | NY | LA | H | SF | A | D | +|----|----|----|----|----|----|----| +| NY | | | | | 1 | | +| LA | | | 1 | 1 | 1 | | +| H | | 1 | | | 1 | 1 | +| SF | | 1 | | | | 1 | +| A | 1 | 1 | 1 | | | 1 | +| D | | | 1 | 1 | 1 | | + +Weighted Directed - ToDo-fix to match image when its updated. +| | NY | LA | H | SF | A | D | +|----|----|----|----|----|----|----| +| NY | | | | | 3 | | +| LA | | | 4 | | | | +| H | | | | | | | +| SF | | 2 | | | | | +| A | | ? | 3 | | | | +| D | 2 | | 1 | 3 | | | #### Adjacency Matrix code -TODO add example code +TODO add example code +```cpp + +``` ### Adjacency List An adjacency list is a collection of unordered lists used to represent a finite graph. Each list describes the set of neighbors of a node in the graph. The number of lists is equal to the number of vertices in the graph. If there are n vertices, then there are n adjacency lists. An adjacency list is an array of linked lists. The size of the array is equal to the number of vertices. Let the array be array[]. An entry array[i] represents the linked list of vertices adjacent to the ith vertex. This representation can also be used to represent a weighted graph. The weights of edges can be represented as nodes with additional information. #### Example -TODO - Pic Will be added +Undirected +NY -> A +LA -> H +H -> +SF -> LA +A -> D, H, LA +D -> SF, H + +Weighted Directed ToDo-fix to match image when its updated. +NY -> A(3) +LA -> H(4) +H -> +SF -> LA(2) +A -> D(2), H(3), LA(?) +D -> SF(4), H(1) #### Adjacency list code ```cpp From 855d6a3710dba4e75375710aee7c4665c955745c Mon Sep 17 00:00:00 2001 From: aheaton22 <117932053+aheaton22@users.noreply.github.com> Date: Sat, 2 Dec 2023 04:13:09 +0000 Subject: [PATCH 4/6] added example code --- Algorithms/Graphs/readme.md | 34 ++++++++++++++++++++++++++++++++-- 1 file changed, 32 insertions(+), 2 deletions(-) diff --git a/Algorithms/Graphs/readme.md b/Algorithms/Graphs/readme.md index 66b39469..29e01e16 100644 --- a/Algorithms/Graphs/readme.md +++ b/Algorithms/Graphs/readme.md @@ -124,9 +124,39 @@ Weighted Directed - ToDo-fix to match image when its updated. | D | 2 | | 1 | 3 | | | #### Adjacency Matrix code -TODO add example code -```cpp +TODO test example code +``` +#include +#include +using namespace std; + +const int N = 5; // number of vertices +int main() +{ + vector> adjMatrix(N, vector(N, 0)); + + // Add edges + adjMatrix[0][1] = 1; + adjMatrix[0][4] = 1; + adjMatrix[1][2] = 1; + adjMatrix[1][3] = 1; + adjMatrix[1][4] = 1; + adjMatrix[2][3] = 1; + adjMatrix[3][4] = 1; + + // Print the matrix + for (int i = 0; i < N; i++) + { + for (int j = 0; j < N; j++) + { + cout << adjMatrix[i][j] << " "; + } + cout << endl; + } + + return 0; +} ``` ### Adjacency List From 800533b4f651b81eba5e739e7572a8992ba5f6ab Mon Sep 17 00:00:00 2001 From: aheaton22 <117932053+aheaton22@users.noreply.github.com> Date: Fri, 15 Dec 2023 00:00:59 +0000 Subject: [PATCH 5/6] updated example graph images --- .../directedCyclesFlightsGraph.png | Bin 0 -> 51905 bytes .../graphImages/directedFlightsGraph.png | Bin 54982 -> 51794 bytes .../graphImages/directedTreeFlightsGraph.png | Bin 0 -> 83967 bytes 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z=oY2pwyVNFuEvrEf?twMcLZ$}g`$lepc1G8_(SiP?yI^C|B$B#{)YUAwSe2;&m;J@ z@fVLIuq^m<6RbP@b^Rj`G_UwDYyh6%&mrJ3@Rz*h-@nXoc79!QZL(jBboU$b|CN NY -Los Angeles -> LA -Houston -> H -San Francisco -> SF -Atlanta -> A -Denver -> D +| Cities | +|-----------------------| +| New York ( N ) | +| Los Angeles ( L ) | +| Houston ( H ) | +| San Francisco ( S ) | +| Atlanta ( A ) | +| Denver ( D ) | The set of graph vertices would be defined as: -V = (NY, LA, H, SF, A, D) +V = (N, L, H, S, A, D) The edges or connections between vertices are called edges. These edges are known as arcs, lines, and arrows. Edges are written as a set of unordered pairs, e.g. E= ((A,B), (A,C), ..). They represent the relations between values. In our airline example the graph edges would be the available flights between cites. -Flights -New York to Atlanta (NY, A) -Atlanta to Denver (A, D) -Atlanta to Houston (A, H) -Atlanta to Los Angeles (A, LA) -Denver to San Francisco (D, SF) -Denver to Houston (D, H) -San Francisco to Los Angeles (SF, LA) -Los Angeles to Houston (LA, H) +| Flights | +|---------------------------------------| +| New York to Atlanta (N, A) | +| Atlanta to Denver (A, D) | +| Atlanta to Houston (A, H) | +| Atlanta to Los Angeles (A, L) | +| Denver to San Francisco (D, S) | +| Denver to Houston (D, H) | +| San Francisco to Los Angeles (S, L) | +| Los Angeles to Houston (L, H) | The set of graph edges would be defined as: -E = ((NY, A), (A, D), (A, H), (A, LA), (D, SF), (D, H), (SF, LA), (LA, H)) +E = ((N, A), (A, D), (A, H), (A, L), (D, S), (D, H), (S, L), (L, H)) -These vertices and edges are what make up a graph. More formally, a graph is defined as a set of vertices( V ), a set of edges( E ) and is denoted by G(E, V). flights = G(E, V) +These vertices and edges are what make up a graph. More formally, a graph is defined as a set of vertices( V ), a set of edges( E ) and is denoted by G(E, V). Flights = G(E, V) A graph representing the relation between cities and flights would look like this: @@ -43,119 +45,145 @@ There are a few different types of graphs that are used in computer science. Not - Directed Graph - Also called a Digraph, is a graph where edges of the graph go only one way with the directions marked with arrows. The above graph is an example of a directed graph. -Directed Flights Graph + Directed Flights Graph - + -- Undirected Graph - A graph where edges of the graph are two-way paths or relations. If we were to update change the flghts examples to be round-trip or two way we would get an undirected graph. +- Undirected Graph - A graph where edges of the graph are two-way paths or relations. If we were to update change the flights examples to be round-trip or two way we would get an undirected graph. -Undirected Flights Graph + Undirected Flights Graph - + - Unweighted Graph - A graph were the edges of the graph do not have any weight or cost associated with them. Undirected graphs are used to model connections. like social networks, where the cost of connecting two vertices does not matter. The above graph is an example of an undirected graph. -- Weighted Graph - A graph where edges of the graph have costs or weights associated with them. As we traverse the graph we can add up the costs of the edges to get the total cost of taking the route. From theses costs we can determine the shortest path or the cheapest route. If we add weights to our flight graph we get the following weighted graph. +- Weighted Graph - A graph where edges of the graph have costs or weights associated with them. As we traverse the graph we can add up the costs of the edges to get the total cost of taking the route. From theses costs we can determine the shortest path or the cheapest route. If we add weights to our flight graph we get the following weighted graph. Edge costs: N->A:3, A->D:2, A->H:3, A->L:2, D->S:4, D->H:1, S->L:2, L->H:4 -Weighted Flights Graph - ToDo-fix image, one edge is missing an image. + Weighted Flights Graph - + - Connected Graphs - - Undirected Connected and Disconnected Graphs - An undirected graph is considered connected if there exists a path between every pair of vertices. Otherwise it is disconnected. In a undirected connected graph you can get to very node in the graph from any other node. The flight graph is an undirected connected graph but if we remove the edge between NY and A it would create an undirected graph. + - Undirected Connected and Disconnected Graphs - An undirected graph is considered connected if there exists a path between every pair of vertices. Otherwise it is disconnected. In a undirected connected graph you can get to very node in the graph from any other node. The flight graph is an undirected connected graph but if we remove the edge between N and A it would create an disconnected graph. Disconnected Flights Graph - + - - Connected Directed Graphs - An directed graph can be strongly or weakly connected. A strongly connected digraph graph has as path from every vertex to every other other vertex. A weakly connected digraph is one where every vertex is connected to graph in some way. See [Kosaraju's Algorithm](Kosaraju%20Algorithm/readme.md) for calculating strongly connected components in a graph. If we ignore the directionality of the edges and make our graph undirected that is connected then the digraph is considered weakly connected. Our digraph flight example is not an strongly connected graph but it is weakly connected. to create a strongly connected graph we would have to add more connecting edges. + - Connected Directed Graphs - An directed graph can be strongly or weakly connected. A strongly connected digraph graph has as path from every vertex to every other other vertex. A weakly connected digraph is one where every vertex is connected to graph in some way. See [Kosaraju's Algorithm](Kosaraju%20Algorithm/readme.md) for calculating strongly connected components in a graph. If we ignore the directionality of the edges and make our graph undirected that is connected then the digraph is considered weakly connected. Our digraph flight example is not an strongly connected graph but it is weakly connected. To create a strongly connected graph we would have to add more connecting edges. - Todo-Example? +- Acyclic and Cyclic Graphs - Graphs can have cycles or paths that allow us to get back to the same node. In other words A cycle is a path that starts and ends at the same vertex. A graph is cyclic if is has one or more of these loops and acyclic and if it does not. Not all graphs have a cycle and some algorithms like Prims assume there are no cycles and algorithms like Floyd-Warshall cannot handle negative cycles (a cycle where all the edge costs are negative). If we reverse one of the edges in our digraph, A->H and make it H->A we would have a graph with cycles. -- Acyclic and Cyclic Graphs - Graphs can have cycles or paths that allow us to get back to the same node. In other words A cycle is a path that starts and ends at the same vertex. A graph is cyclic if is has one or more of these loops and acyclic and if it does not. Not all graphs have a cycle and some algorithms like Prims assume there are no cycles and algorithms like Floyd-Warshall cannot handle negative cycles (a cycle where all the edge costs are negative). + Cycles Flights Graph - Todo-Example? + -- [Tree Graphs](Graphs/Tree%20Based%20Algorithms/readme.md) - Trees are acyclic and connected graphs where the vertices are connected in such a way that there is only one path between vertices. A tree graph will n vertices and n-1 edges with each node connected to the tree. A set of disconnected tree graphs is called a forest. A tree can have either directed or undirected edges. +- [Tree Graphs](Graphs/Tree%20Based%20Algorithms/readme.md) - Trees are acyclic and connected graphs where the vertices are connected in such a way that there is only one path between vertices. A tree graph will n vertices and n-1 edges with each node connected to the tree. A set of disconnected tree graphs is called a forest. A tree can have either directed or undirected edges. If we remove edges from our flights diagraph, we will get a tree graph. - Todo-Example? + Tree Flights Graph + + ## Graph Components The typed of edges and vertices a graph has can vary. Here is a brief explanation of some of them. - Edges - An edge is a connection between two nodes. It can be denoted by any symbol such as E, F, G, H, etc. An edge may also be referred to as a link or a line. - - Weighted Edge - When a weight or cost value associated with an edge it is an weighted edge. The weight may also be referred to as a cost. This cost can be distance, money, time, etc. These weights can be zero, negative, positive, or infinite. It depends on the relations the graph is representing. See Weighted Flights Graph for an example. - - Undirected - A undirected edge is an edge that allows for to two way connections. In other words, if vertex A is connected to B then B is connected to A. See Undirected Flights Graph for an example. - - Directed - A directed edge is an edge that only goes one way. In other words, vertex A is connected to B but B is not connected to A. If we were to alter the flight list in the above example and change the flights to be one-way then the edges would become directed edges. see Directed Flights Graph for an example. + - Weighted Edge - When a weight or cost value associated with an edge it is an weighted edge. The weight may also be referred to as a cost. This cost can be distance, money, time, etc. These weights can be zero, negative, positive, or infinite. It depends on the relations the graph is representing. See Weighted Flights Graph for an example. + - Undirected - A undirected edge is an edge that allows for to two way connections. In other words, if vertex A is connected to B then B is connected to A. See Undirected Flights Graph for an example. + - Directed - A directed edge is an edge that only goes one way. In other words, vertex A is connected to B but B is not connected to A. If we were to alter the flight list in the above example and change the flights to be one-way then the edges would become directed edges. see Directed Flights Graph for an example. - Vertices - A vertex is a node of the graph. It can be denoted by any symbol such as V, U, X, Y, etc. Vertices also have a degree, or the number of edges connected to it. A vertex’s in-degree is defined as the number of edges that point to a vertex. The vertex’s out-degree is the number of edges that point to other vertices. In an undirected graph the in and out degree is the same. The different types of vertices are defined by these degrees. - - Isolated vertex - No edges pointing to the vertex, and it has no outgoing edges. Its in-degree and out-degree is zero. In the Disconnected Flights Graph NY is an isolated vertex. - - Source vertex - No edges point to the vertex, its in-degree is zero. In the Directed Flights Graph, NY is a source vertex as there are no flights or edges going to the city. + - Isolated vertex - No edges pointing to the vertex, and it has no outgoing edges. Its in-degree and out-degree is zero. In the Disconnected Flights Graph N is an isolated vertex. + - Source vertex - No edges point to the vertex, its in-degree is zero. In the Directed Flights Graph, N is a source vertex as there are no flights or edges going to the city. - Sink vertex - Has no outgoing edges, it’s out-degree is zero. In the Directed Flights Graph, vertex H is a sink vertex as there are no flights going out of that city. - - [Articulation Points](Graphs/Articulation%20Points/readme.md) - Also called a bridge, is vertex in an undirected graph that would created a disconnected graph if removed. if we were to remove A and all its edges from the Directed Flights Graph we would get an disconnected graph. A is an articulation point. + - [Articulation Points](Graphs/Articulation%20Points/readme.md) - Also called a bridge, is vertex in an undirected graph that would created a disconnected graph if removed. if we were to remove A and all its edges from the Directed Flights Graph we would get an disconnected graph. A is an articulation point. ## Common Graph Representations -In order to run algorithms on graphs we need to be able to store the vertex and edge data in a data structure. There are many different ways to represent a graph but the two most common ways are by using an adjacency matrix or list. +In order to run algorithms on graphs we need to be able to store the vertex and edge data in a data structure. There are many different ways to represent a graph but the two most common ways are by using an adjacency matrix or list. ### Adjacency Matrix -An 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 to vertex j. Adjacency matrix for undirected graph is always symmetric. Adjacency Matrix is also used to represent weighted graphs. If adj[i][j] = w, then there is an edge from vertex i to vertex j with weight w. +An 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 to vertex j. Adjacency matrix for undirected graph is always symmetric across the diagonal. Adjacency Matrix is also used to represent weighted graphs. If adj[i][j] = w, then there is an edge from vertex i to vertex j with weight w. #### Example -Adjacency matrix for the undirected and directed flight graphs +Adjacency matrix for the undirected and directed flight graphs shown above Undirected -| | NY | LA | H | SF | A | D | -|----|----|----|----|----|----|----| -| NY | | | | | 1 | | -| LA | | | 1 | 1 | 1 | | -| H | | 1 | | | 1 | 1 | -| SF | | 1 | | | | 1 | -| A | 1 | 1 | 1 | | | 1 | -| D | | | 1 | 1 | 1 | | - -Weighted Directed - ToDo-fix to match image when its updated. -| | NY | LA | H | SF | A | D | -|----|----|----|----|----|----|----| -| NY | | | | | 3 | | -| LA | | | 4 | | | | -| H | | | | | | | -| SF | | 2 | | | | | -| A | | ? | 3 | | | | -| D | 2 | | 1 | 3 | | | +| | N | L | H | S | A | D | +|---|---|---|---|---|---|---| +| N | | | | | 1 | | +| L | | | 1 | 1 | 1 | | +| H | | 1 | | | 1 | 1 | +| S | | 1 | | | | 1 | +| A | 1 | 1 | 1 | | | 1 | +| D | | | 1 | 1 | 1 | | + +Weighted Directed +Edge costs: N->A:3, A->D:2, A->H:3, A->L:2, D->S:4, D->H:1, S->L:2, L->H:4 +| | N | L | H | S | A | D | +|---|---|---|---|---|---|---| +| N | | | | | 3 | | +| L | | | 4 | | | | +| H | | | | | | | +| S | | 2 | | | | | +| A | | 2 | 3 | | | 2 | +| D | | | 1 | 4 | | | #### Adjacency Matrix code -TODO test example code -``` +```cpp +// A C++ Program to demonstrate adjacency matrices +// representation of graphs #include #include using namespace std; -const int N = 5; // number of vertices - -int main() +// A utility function to print the adjacency matrix +// representation of graph +void printGraph(vector> adj, int V) { - vector> adjMatrix(N, vector(N, 0)); - - // Add edges - adjMatrix[0][1] = 1; - adjMatrix[0][4] = 1; - adjMatrix[1][2] = 1; - adjMatrix[1][3] = 1; - adjMatrix[1][4] = 1; - adjMatrix[2][3] = 1; - adjMatrix[3][4] = 1; - - // Print the matrix - for (int i = 0; i < N; i++) - { - for (int j = 0; j < N; j++) + cout << "Adjacency matrix of graph\n"; + + //print out column headers + cout << " | "; + for (int h = 0; h < V; h++) { - cout << adjMatrix[i][j] << " "; + cout << h << " "; } cout << endl; - } + for (int h = 0; h < V; h++) + { + cout << "---"; + } + cout << endl; + + //print out matrix values + for (int r = 0; r < V; ++r) + { + cout << r << " | "; + for (int c = 0; c < V; c++) + { + cout << adj[r][c] << " "; + } + cout << endl; + } +} - return 0; +//Driver code +int main() +{ + const int V = 5; // number of vertices + + vector> adjMatrix(V, vector(V, 0)); + + // Add edges + adjMatrix[0][1] = 1; + adjMatrix[0][4] = 1; + adjMatrix[1][2] = 1; + adjMatrix[1][3] = 1; + adjMatrix[1][4] = 1; + adjMatrix[2][3] = 1; + adjMatrix[3][4] = 1; + printGraph(adjMatrix, V); + return 0; } ``` @@ -164,20 +192,20 @@ An adjacency list is a collection of unordered lists used to represent a finite #### Example Undirected -NY -> A -LA -> H -H -> -SF -> LA -A -> D, H, LA -D -> SF, H - -Weighted Directed ToDo-fix to match image when its updated. -NY -> A(3) -LA -> H(4) -H -> -SF -> LA(2) -A -> D(2), H(3), LA(?) -D -> SF(4), H(1) +N -> A +L -> H +H -> +S -> L +A -> D, H, L +D -> S, H + +Weighted Directed +N -> A(3) +L -> H(4) +H -> +S -> L(2) +A -> D(2), H(3), L(2) +D -> S(4), H(1) #### Adjacency list code ```cpp @@ -205,14 +233,14 @@ void printGraph(vector adj[], int V) << v << "\n head "; for (auto x : adj[v]) cout << "-> " << x; - printf("\n"); + cout << "\n"; } } // Driver code int main() { - int V = 5; + int const V = 5; vector adj[V]; addEdge(adj, 0, 1); addEdge(adj, 0, 4); From 9fb2cf65955b870dd05e7bab716c89bc9ffae88d Mon Sep 17 00:00:00 2001 From: aheaton22 <117932053+aheaton22@users.noreply.github.com> Date: Tue, 19 Dec 2023 00:41:30 +0000 Subject: [PATCH 6/6] proofreading fixes --- Algorithms/Graphs/readme.md | 66 ++++++++++++++++++++----------------- 1 file changed, 35 insertions(+), 31 deletions(-) diff --git a/Algorithms/Graphs/readme.md b/Algorithms/Graphs/readme.md index 04575b54..96c6ae86 100644 --- a/Algorithms/Graphs/readme.md +++ b/Algorithms/Graphs/readme.md @@ -1,7 +1,7 @@ # Graphs -A graph is a data structure used in computer science and mathematics. Graphs are used to model pairwise relations between objects and are the most useful data structures for many real-world applications, such as the airline route network. Graphs are a non-linear data structure that allows us to more easily define complicated relations. A graph consists of a set of vertices and edges. +A graph is a data structure used in computer science and mathematics. Graphs are used to model pairwise relations between objects and are the most useful data structures for many real-world applications, such as the airline route network. Graphs are a non-linear data structure that allows us to define complicated relations more easily. A graph consists of a set of vertices and edges. -The vertices of a graph, also called nodes or points, are a finite but possibly changeable set of values and are written as a set of names, e.g. V=(A, B, C, ..). If we were to model the airline route network as a graph the cities would be represented as vertices. For example: +The vertices of a graph, also called nodes or points, are a finite but possibly changeable set of values and are written as a set of names, i.e. V=(A, B, C, ..). If we were to model the airline route network as a graph the cities would be represented as vertices. For example: | Cities | |-----------------------| @@ -15,7 +15,7 @@ The vertices of a graph, also called nodes or points, are a finite but possibly The set of graph vertices would be defined as: V = (N, L, H, S, A, D) -The edges or connections between vertices are called edges. These edges are known as arcs, lines, and arrows. Edges are written as a set of unordered pairs, e.g. E= ((A,B), (A,C), ..). They represent the relations between values. In our airline example the graph edges would be the available flights between cites. +The edges or connections between vertices are called edges. These edges are known as arcs, lines, and arrows. Edges are written as a set of unordered pairs, i.e. E= ((A,B), (A,C), ..). They represent the relations between values. In our airline example the graph edges would be the available flights between cities. | Flights | |---------------------------------------| @@ -31,7 +31,9 @@ The edges or connections between vertices are called edges. These edges are know The set of graph edges would be defined as: E = ((N, A), (A, D), (A, H), (A, L), (D, S), (D, H), (S, L), (L, H)) -These vertices and edges are what make up a graph. More formally, a graph is defined as a set of vertices( V ), a set of edges( E ) and is denoted by G(E, V). Flights = G(E, V) +These vertices and edges are what make up a graph. More formally, a graph is defined as a set of vertices( V ), a set of edges( E ) and is denoted by G(E, V). + +Flights = G(E, V) = G( {(N, A), (A, D), (A, H), (A, L), (D, S), (D, H), (S, L), (L, H)}, {N, L, H, S, A, D}) A graph representing the relation between cities and flights would look like this: @@ -41,73 +43,75 @@ A graph representing the relation between cities and flights would look like thi With this graph we can now answer questions like: Can I get to San Francisco from New York? How many layovers are there between Atlanta and Denver? Can I get to Houston from Los Angeles? How many different flights are there from New York to Los Angeles? We can look at the graph and count the edges and vertices as we move from one city to another. But instead of calculating solutions by hand we can use algorithms like Dijkstra's, Bellman-Ford, or Kruskal's to give us the answers to those questions. ## Types of Graphs -There are a few different types of graphs that are used in computer science. Note that some graph algorithms will only work on a specific types of graph. +There are a few different types of graphs that are used in computer science. Note that some graph algorithms will only work on specific types of graphs. -- Directed Graph - Also called a Digraph, is a graph where edges of the graph go only one way with the directions marked with arrows. The above graph is an example of a directed graph. +- Directed Graph - Also called a Digraph, is a graph where edges of the graph have only one direction, these directions are marked with arrows. The above graph is an example of a directed graph. Directed Flights Graph -- Undirected Graph - A graph where edges of the graph are two-way paths or relations. If we were to update change the flights examples to be round-trip or two way we would get an undirected graph. +- Undirected Graph - A graph where edges of the graph are two-way paths or relations. If we were to update the flights examples to be round-trip or two way we would get an undirected graph. Undirected Flights Graph -- Unweighted Graph - A graph were the edges of the graph do not have any weight or cost associated with them. Undirected graphs are used to model connections. like social networks, where the cost of connecting two vertices does not matter. The above graph is an example of an undirected graph. +- Unweighted Graph - A graph where the edges of the graph do not have any weight or cost associated with them. Unweighted graphs are used to model connections, such as social networks, where the cost of connecting two vertices does not matter. The above graph examples are all unweighted graphs. + +- Weighted Graph - A graph where edges of the graph have costs or weights associated with them. As we traverse the graph, we can add up the costs of the edges to get the total cost of taking the route. From these costs we can determine the shortest path or the cheapest route. If we add weights to our flight graph, we get the following weighted graph. -- Weighted Graph - A graph where edges of the graph have costs or weights associated with them. As we traverse the graph we can add up the costs of the edges to get the total cost of taking the route. From theses costs we can determine the shortest path or the cheapest route. If we add weights to our flight graph we get the following weighted graph. Edge costs: N->A:3, A->D:2, A->H:3, A->L:2, D->S:4, D->H:1, S->L:2, L->H:4 + Edge costs: N->A: 3, A->D: 2, A->H: 3, A->L: 2, D->S: 4, D->H: 1, S->L: 2, L->H: 4 Weighted Flights Graph - Connected Graphs - - Undirected Connected and Disconnected Graphs - An undirected graph is considered connected if there exists a path between every pair of vertices. Otherwise it is disconnected. In a undirected connected graph you can get to very node in the graph from any other node. The flight graph is an undirected connected graph but if we remove the edge between N and A it would create an disconnected graph. + - Undirected Connected and Disconnected Graphs - An undirected graph is considered connected if there exists a path between every pair of vertices. Otherwise, it is disconnected. In an undirected connected graph, you can get to every vertex in the graph from any other vertex. The Undirected Flights Graph is also an undirected connected graph but if we remove the edge between N and A it would create a disconnected graph. Disconnected Flights Graph - - Connected Directed Graphs - An directed graph can be strongly or weakly connected. A strongly connected digraph graph has as path from every vertex to every other other vertex. A weakly connected digraph is one where every vertex is connected to graph in some way. See [Kosaraju's Algorithm](Kosaraju%20Algorithm/readme.md) for calculating strongly connected components in a graph. If we ignore the directionality of the edges and make our graph undirected that is connected then the digraph is considered weakly connected. Our digraph flight example is not an strongly connected graph but it is weakly connected. To create a strongly connected graph we would have to add more connecting edges. + - Connected Directed Graphs - A directed graph can be strongly or weakly connected. A strongly connected digraph has a path from every vertex to every other vertex. A weakly connected digraph is one where every vertex is connected to the graph in some way. See [Kosaraju's Algorithm](Kosaraju%20Algorithm/readme.md) for calculating strongly connected components in a graph. Our digraph flight example is not a strongly connected graph, but it is weakly connected. To create a strongly connected graph we would have to add more connecting edges. -- Acyclic and Cyclic Graphs - Graphs can have cycles or paths that allow us to get back to the same node. In other words A cycle is a path that starts and ends at the same vertex. A graph is cyclic if is has one or more of these loops and acyclic and if it does not. Not all graphs have a cycle and some algorithms like Prims assume there are no cycles and algorithms like Floyd-Warshall cannot handle negative cycles (a cycle where all the edge costs are negative). If we reverse one of the edges in our digraph, A->H and make it H->A we would have a graph with cycles. +- Acyclic and Cyclic Graphs - Graphs can have cycles or paths that allow us to get back to the same vertex. In other words, a cycle is a path that starts and ends at the same vertex. A graph is cyclic if it has one or more of these loops and acyclic if it does not. Not all graphs have a cycle, some algorithms assume that a graph has no cycles. For example, Prim’s algorithm assumes there are no cycles and Floyd-Warshall’s algorithm assumes there are no negative cycles (a cycle where all the edge costs are negative). If we reverse one of the edges in our digraph, A->H and make it H->A we would have a graph with cycles. (cycles in red and orange) Cycles Flights Graph -- [Tree Graphs](Graphs/Tree%20Based%20Algorithms/readme.md) - Trees are acyclic and connected graphs where the vertices are connected in such a way that there is only one path between vertices. A tree graph will n vertices and n-1 edges with each node connected to the tree. A set of disconnected tree graphs is called a forest. A tree can have either directed or undirected edges. If we remove edges from our flights diagraph, we will get a tree graph. +- [Tree Graphs](Graphs/Tree%20Based%20Algorithms/readme.md) - Trees are acyclic, connected graphs where the vertices are connected in such a way that there is only one path between vertices. A tree graph will have n vertices and n-1 edges with each vertex connected to the tree. A tree can have either directed or undirected edges. If we remove edges from our flights digraph, we will get a tree graph. (Both graphs are the same, just drawn in different ways) Tree Flights Graph ## Graph Components -The typed of edges and vertices a graph has can vary. Here is a brief explanation of some of them. -- Edges - An edge is a connection between two nodes. It can be denoted by any symbol such as E, F, G, H, etc. An edge may also be referred to as a link or a line. - - Weighted Edge - When a weight or cost value associated with an edge it is an weighted edge. The weight may also be referred to as a cost. This cost can be distance, money, time, etc. These weights can be zero, negative, positive, or infinite. It depends on the relations the graph is representing. See Weighted Flights Graph for an example. - - Undirected - A undirected edge is an edge that allows for to two way connections. In other words, if vertex A is connected to B then B is connected to A. See Undirected Flights Graph for an example. - - Directed - A directed edge is an edge that only goes one way. In other words, vertex A is connected to B but B is not connected to A. If we were to alter the flight list in the above example and change the flights to be one-way then the edges would become directed edges. see Directed Flights Graph for an example. - -- Vertices - A vertex is a node of the graph. It can be denoted by any symbol such as V, U, X, Y, etc. Vertices also have a degree, or the number of edges connected to it. A vertex’s in-degree is defined as the number of edges that point to a vertex. The vertex’s out-degree is the number of edges that point to other vertices. In an undirected graph the in and out degree is the same. The different types of vertices are defined by these degrees. - - Isolated vertex - No edges pointing to the vertex, and it has no outgoing edges. Its in-degree and out-degree is zero. In the Disconnected Flights Graph N is an isolated vertex. - - Source vertex - No edges point to the vertex, its in-degree is zero. In the Directed Flights Graph, N is a source vertex as there are no flights or edges going to the city. +Here is a brief explanation of the type of edges and vertices a graph may have. +- Edges - An edge is a connection between two vertices. It can be denoted by any symbol such as E, F, G, H, etc. An edge may also be referred to as a link or a line. + - Weighted Edge - When a weight or cost value is associated with an edge it is a weighted edge. The weight may also be referred to as a cost. This cost can be distance, money, time, etc. These weights can be zero, negative, positive, or infinite, depending on the relations the graph represents. See Weighted Flights Graph for an example. + - Undirected - A undirected edge is an edge that allows for to two-way connections. In other words, if vertex A is connected to B then B is connected to A. See Undirected Flights Graph for an example. + - Directed - A directed edge is an edge that only goes one way. In other words, vertex A is connected to B but B is not connected to A. See Directed Flights Graph for an example. + +- Vertices - A vertex is a node of the graph. It can be denoted by any symbol such as V, U, X, Y, etc. Vertices also have a degree, or the number of edges connected to it. A vertex’s in-degree is defined as the number of edges that point to a vertex. The vertex’s out-degree is the number of edges that point to other vertices. In an undirected graph the in and out degrees are the same. The different types of vertices are defined by these degrees. + - Isolated vertex - No edges pointing to the vertex, and it has no outgoing edges. Both the in-degree and out-degree are zero. In the Disconnected Flights Graph, vertex N is an isolated vertex. + - Source vertex - No edges point to the vertex, its in-degree is zero. In the Directed Flights Graph, vertex N is a source vertex as there are no flights or edges going to the city. - Sink vertex - Has no outgoing edges, it’s out-degree is zero. In the Directed Flights Graph, vertex H is a sink vertex as there are no flights going out of that city. - - [Articulation Points](Graphs/Articulation%20Points/readme.md) - Also called a bridge, is vertex in an undirected graph that would created a disconnected graph if removed. if we were to remove A and all its edges from the Directed Flights Graph we would get an disconnected graph. A is an articulation point. + - [Articulation Points](Graphs/Articulation%20Points/readme.md) - Also called a bridge, an articulation point is a vertex in an undirected graph that would create a disconnected graph if removed. If we were to remove A and all its edges from the Undirected Flights Graph, we would get a disconnected graph. A is an articulation point. ## Common Graph Representations -In order to run algorithms on graphs we need to be able to store the vertex and edge data in a data structure. There are many different ways to represent a graph but the two most common ways are by using an adjacency matrix or list. +To run algorithms on graphs we need to store the vertex and edge info in a data structure. There are many ways to represent a graph, but the two most common ways are by using an adjacency matrix or list. ### Adjacency Matrix -An 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 to vertex j. Adjacency matrix for undirected graph is always symmetric across the diagonal. Adjacency Matrix is also used to represent weighted graphs. If adj[i][j] = w, then there is an edge from vertex i to vertex j with weight w. +An 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 to vertex j. An Adjacency matrix for an undirected graph is always symmetric across the diagonal. An Adjacency Matrix can also represent weighted graphs. If adj[i][j] = w, where w is the cost associated with that edge, then there is an edge from vertex i to vertex j with weight w. #### Example -Adjacency matrix for the undirected and directed flight graphs shown above +Adjacency matrix for the undirected and directed flight graphs shown above: -Undirected +Undirected Flights | | N | L | H | S | A | D | |---|---|---|---|---|---|---| | N | | | | | 1 | | @@ -117,7 +121,7 @@ Undirected | A | 1 | 1 | 1 | | | 1 | | D | | | 1 | 1 | 1 | | -Weighted Directed +Weighted Directed Flights Edge costs: N->A:3, A->D:2, A->H:3, A->L:2, D->S:4, D->H:1, S->L:2, L->H:4 | | N | L | H | S | A | D | |---|---|---|---|---|---|---| @@ -188,10 +192,10 @@ int main() ``` ### Adjacency List -An adjacency list is a collection of unordered lists used to represent a finite graph. Each list describes the set of neighbors of a node in the graph. The number of lists is equal to the number of vertices in the graph. If there are n vertices, then there are n adjacency lists. An adjacency list is an array of linked lists. The size of the array is equal to the number of vertices. Let the array be array[]. An entry array[i] represents the linked list of vertices adjacent to the ith vertex. This representation can also be used to represent a weighted graph. The weights of edges can be represented as nodes with additional information. +An adjacency list is a collection of linked lists used to represent a graph. Each list describes the set of neighbors of a vertex in the graph. The number of lists is equal to the number of vertices in the graph. If there are n vertices, then there are n adjacency lists. The size of the array is equal to the number of vertices. Let the array be array[]. An entry array[i] represents the linked list of vertices adjacent to the i-th vertex. This representation can also be used to represent a weighted graph. The weights of edges can be represented as vertices with additional information. #### Example -Undirected +Undirected Flights N -> A L -> H H -> @@ -199,7 +203,7 @@ S -> L A -> D, H, L D -> S, H -Weighted Directed +Weighted Directed Flights N -> A(3) L -> H(4) H ->