Bellman Ford Algorithm implementation in c++

BELLMAN FORD ALGORITHM IMPLEMENTATION IN C++

Shortest path algorithm: Elaborated

Shortest path algorithm

Program for Bellman Algorithm

By K Sreram

Email: sreramk@outlook.com

Shortest path algorithms are used in may real life applications, especially applications involving maps and artificial intelligence algorithms which are NP in nature. A graph is a collection of nodes connected by edges  and can be expressed as. Stands for vertices. These vertices and the connecting edges together can be imagined to form a three dimensional geometrical structure. In some cases, for each (here  are vertices which are connected) there is a weight value assigned to it. This can also be called as “distance” or “cost” of connecting the two nodes. The shortest path algorithm traces the minimum distance (or cost) between two nodes  which are either directly or indirectly connected. The weight values along each possible paths to the destination node from the source node are summed up, and the path with the minimum summation value is chosen as the shortest path.

We may represent a weighted graph as where the extra parameter represents the set of weight values across each edge. Weight of the edge. Usually, shortest path algorithms are of time complexity because every node needs to be visited at least once. In case of AI algorithms, the vertices are generated based on specific rules, and usually it’s a function of depth; because most nodes have more than one connections, we may also have non-polynomial relation between the depth and the number of nodes.

Algorithms like Dijkstra’s algorithm’s time complexity is (note, represents number of nodes and represents number of edges). Dijkstra’s algorithm holds only for the situations where. But from the definition of a weighted graph given above, weight values can also be negative. So a modified version of the algorithm which also solves problems with negative weight edge is required. But this modified algorithm would require a greater runtime-time complexity; Dijkstra’s algorithm is a greedy algorithm, which selects the best possible path to every node in the graph, originating from source. But Dijkstra’s algorithm fails to solve problems that have negative weight values. In scenarios where negative weight values are present, another algorithm called Bellman Ford algorithm is used. But its runtime complexity is. Which is far greater than Dijkstra’s algorithm. Generally,  unless there are disjoint nodes or graphs. Unless stated, the graphs discussed here are always assumed to not contain any disjoint sets or nodes.

Fundamentals of calculus chapter 4: More on differential equations

Fundamentals of calculus chapter 4: More on differential equations

Differential equations do not have immediate means to solve them. The reason is, because differential equations are defined from differentiation and so is its integral, it’s easier to find the differential equation rather than it’s solution. If our motive is purely to find the solution of differential equations, then our best approach will be to try and find how the expressions “behave” while trying to differentiate them. Then we may back trace this sequence of steps to find the original expression.

An optimal algorithm for the shortest path in the graph

An optimal algorithm to find the shortest path in the graph

Introduction:

Many real-life solutions rely on shortest path algorithms, especially games rely on them! If a graphic game character traces your moves along a complicated path and finds you, then it is obvious that the game implemented a graph algorithm to find the shortest path to approach you. Is it really that easy to find an optimal path to reach a point in a graph? No not exactly, as the upper bound of the Dijkstra’s algorithm is $O(|V|^2)$. If we assume your game map to be a big grid, then the number of cells in the grid is equal to the number $|V|$. But this is still the worst case, and for most times we don’t need to consider the worst case. Let me describe the Dijkstra’s algorithm. 

Fundamentals of calculus Chapter 3: differential equations of functions involving one independent variable.

Fundamentals of calculus

Chapter 3: differential equations of functions involving one independent variable.

“Everything big is built out of smaller units; and if the smaller units fail to cooperate the bigger ones ultimately fall”

“A good learner never accepts the educative information he comes across unless he knows for sure, that there is no practical and logical way to contradict it”

Calculus as a whole evolved to support something new, a differential equation. One marvellous fact about differential equations are they don’t have any constants to describe the graphical structure they represent. In the previous article ‘chapter 2’, we saw how differentiation eliminates a constant. This fact can be taken advantage of to frame differential equations that elaborate the actual rate of change of different quantities. For example if, $z = ax^2 + bx +c$ then $\frac{dz}{dx} = 2ax + b$ and $\frac{d^2z}{dx^2} = 2a$ and $\frac{d^3z}{dx^3} = 0$.

Fundamentals of calculus Chapter 2: Integration

Fundamentals of calculus

Chapter 2: Integration

Integration is the converse of differentiation. To understand this better, it must be known that during differentiation an important information is lost. Let $\frac{dz(x)}{dx} = z’(x)$. Then, we cannot get back $z(x)$ from $z’(x)$ because, while taking the difference $z(x+\Delta x) – z(x)$ only the information regarding the difference between a point in Z axis and a neighbourhood of that point is retained. But the actual orientation is lost. Throughout this article, unless stated $\Delta x$ is limited to zero. To see how this could cause the loss of a piece of information, let me elaborate this with a simple algebraic example. Consider $a-b = c$ to be true. Then we may write $a – (a-c) = c$, here $b=a-c$.

Fundamentals of calculus Chapter 1: Differentiation.

Fundamentals of calculus
Chapter 1: Differentiation.

Can we really call calculus an art? Moreover, can we call mathematics an art? Most mathematicians will agree to the answer: yes. We express ourselves, our creativity through mathematics. It’s not always about calculation. Read my article ‘why do we need mathematics?’ to better understand what I believe mathematics is really. I will not agree that the explanation I had given there was enough. But anyway, I believe mathematics is an art because it is not based on mere calculations but based on creating new rules, using them to express what is around us in a more precise and defined manner. Most of the time, mathematicians would define these rules not for the sake of answering questions that follow up with our reality. But they just do it, in hope that their discovery might come in handy to some scientist in the near future. Some don’t even care about its necessity and for the sake of not expressing themselves to be someone ignorant of the reality most people scramble through physics equations or some other kind of scientific material to find the right place for their equations or mathematical rules to fit in and prove to be beneficial.