Cactus Stack Approach with Dijkstra’s Algorithm

Cactus Stack barbel with Dijkstras AlgorithmAdvance Dijkstras Algorithm with Cactus Stack Implementation LogicIdea Proposed for the Cactus Stack Approach with Dijkstras AlgorithmPalak KalaniNayyar KhanAbstract This paper illustrates a possible approach to reduce the complexity and calculation burden that is norm all in ally encountered in the Shortest course of action Problems. We intend to give a new concept based on the look ahead values of the in put leaf nodes given in the shortest path line of work as proposed by the famous Djktras Algorithm. A new data structure called as Cactus Stack could be procedured for the same and we could try to minimize and the iteration back issues and traversals in a given graph by looking at the adjacency intercellular substance as well as the creation of a cactus stack which is linked in a listed manner. The calculation of the node values and the total value shall be the same as proposed by the ancient algorithmic ruleic program however by o ur concept we argon trying to reduce the complexity of calculation to a larger extent.Index TermsComp geniusnt, formatting, style, styling, insert. (key words)I. IntroductionWith the advancement in technology, there is increase in demand for development of industries to fulfill the requirements of the people. But in industries there ar a lot of chemicals and lubricants used for equipment some of these chemicals and lubricants argon volatile in nature and may cause accidents such as fire. However, fire extinguishers and other pr pointtive measures are also available at the security, still they are non proven executive and we have to flee the area as soon as possible.This research paper proposes a new algorithm which calculates the shortest path forget away of such problems. The proposed algorithm is based on the original Dijkstra algorithm. This algorithm gives us optimum root in less quantify and also reduces the calculation.1II. Dijkstra AlgorithmDijkstra was one of the most fo rceful promoter of computer programming as a scientific discipline. He has made contri neverthelession to the areas of operating systems, programming languages, including deadlock avoidance, contain the nonion of structured programming, and algorithms. We will now consider the general problem of finding the length of a shortest path between a and z in an undirected connected simple weighted graph. Dijkstras algorithm proceeds by finding the length of a shortest path from a to a commencement apex, the length of a shortest path from a to a second vertex, and so on, until the length of a shortest path from a to z is found. As aside benefit, this algorithm is easily extended to find the length of the shortest path from a to all other vertices of the graph, and not just to z.The algorithm relies on a series of iterations. A distinguished set of vertices is constructed by adding one vertex at each iteration. A studying procedure is carried out for iteration. In this labeling proced ure, a vertex w is labeled with the length of a shortest path from a tow that contains only vertices already in the distinguished set. The vertex added to the distinguished set is one with a minimal label among those vertices not already in the set.We now give the details of Dijkstras algorithm. It begins by labeling a with 0 and the other vertices with . We use the notation L0(a) = 0 and L0(v)=for these labels before any iterations have taken place (the subscript 0 stands for the 0th iteration). These labels are the lengths of shortest paths from a to the vertices, where the paths contain only the vertex a.(Because no path from a to a vertex different from a exists, is the length of a shortest pathBetween a and this vertex.)Dijkstras algorithm proceeds by forming a distinguished set of vertices. Let S k denote this set after k iterations of the labeling procedure. We begin with S0 = . The set Skis formed from Sk1 by adding a vertex u not in S k1 with the smallest label.Lk(a, v) = m inLk1(a, v),Lk1(a, u) + w(u, v),ALGORITHMProcedure Dijkstra(G weighted connected simple graph, withall weights positive)G has vertices a = v0, v1, . . . ,vn= z and lengths w(vi , vj )where w(vi , vj )=if vi , vj is not an edge in Gfori = 1 to nL(vi ) =L(a) = 0S =the labels are now initialized so that the label of a is 0 and allother labels are, and S is the empty setwhilez Su= a vertex not in S with L(u) minimalS =S uforall vertices v not in SifL(u) + w(u, v) thus L(v) = L(u) + w(u, v)this adds a vertex to S with minimal label and updates thelabels of vertices not in SreturnL(z) L(z) = length of a shortest path from a to z2IV. Limitations of Dijkstras AlgorithmAlthough Dijkstras algorithm is an effective algorithm but still there are a lot of circumspections. Some of these are discussed below.Presence of calculations bounteously.Solutions pleaded by this algorithm are not equitable.The reasons for changing paths and beading elements are not favoured.It is not explicit when the pr oblem is not closed loop or cyclic i.e. we are having a last element other than culture and last element is connected with only one element then we are not able to reach destinationIt distracts when both next nodes are same then which node we are going to choose for operation.We have to check distance or path after one step which is not favourable. For compositors case, if we want to go neemuch from indore and distance of Ujjain from indore is more than distance of devas from indore then according to dijkstra we should go to devas and check distance of desvas between neemuch and if we find more than Ujjain route then we again come back to indore.V. Solutions to mentioned limitationsFirst we use look ahead dijkstra A. PrecodeWnext = min Wvu ,WvwLook ahead Wnext ,Wua , WsubWfinal = minWnextWua, WnextWubReturn Wfinala?bMin_(i=1)2-(Ti)+Li-1-Problem solution Vi=VjMin(P1_(i=1)2wi, P2_(j=1)2wj) =next node.B. Proposed MethodCs1 pop(a)Cs2(b)Weight (a,b)Cs2(c)Weigth(a,c)Returnmin( Cs2(b),Cs 2(c))(V+Cs)+ look aheadC. Dijkstra with VFS traversal and cactus stackVFS(vertical first search)In the vfs we search in vertical regularise of cactus stackf(a,b) f(n,m)n=af(a,m)m=b,c,d,e..struct nodeint*prevint*nextint dataD. Complexity ComparisonIn dijkstras complexity is more whereas in proposed method i.e. shortest path with cactus stack is less. In dijkstras we lead to do calculation from each and every point but in proposed algorithm we need to do calculation from particular points which reduces calculations and complexities.E. Adjacency MatrixIn mathematics and computer science, an adjacency matrix is a means of representing which vertices (or nodes) of graph are conterminous to which other vertices.34Adjacency matrix of above graph isF. Weighted Adjacency MatrixThe matrix which represents graph with respect to its weight. Now we have to convert matrix into another matrix by using the following program.Adjacency Matrix isCODE include include define INF 9999int main( )intar r44 int cost44 = 7, 5, 0, 0,7, 0, 0, 2,0, 3, 0, 0,4, 0, 1, 0 int i, j, k, n = 4 for ( i = 0 i for ( j = 0 j if ( costij == 0 )arrij = INF elsearrij = costij printf ( Adjacency matrix of cost of edgesn ) for ( i = 0 i for ( j = 0 j printf ( %dt, arrij ) printf ( n ) for ( k = 0 k for ( i = 0 i for ( j = 0 j if ( arrij arrik + arrkj )arrij = arrik + arrkjNow we take an above example and use this steps to convert into another adjacency matrix.G. Traversal of Adjacency MatrixFor solving adjacency matrix, we take the 1st node and let out the haggling of that node if there is weight on any vertex that means that vertex is connected to 1st node with respective weight.Similarly we check for all nodes and master the matrix.If any vertex have weight infinity that means that vertex is not directly connected to the main vertex whose row is being traverse. If vertex have weight zero that means there is no self loop present.If users last node is not the last node of matrix then we simply traverse the matrix till the node entered by user then we will back traverse rest of node that is we will start traversing last node.VI. Cactus StackA cactus stack is a set of stacks organized in a taxonomic format as a tree in which each path from the root to any leaf constitutes a stack.A Cactus Stack act both like a Tree and a Stack. Like a stack, items can only be added to or removed from only one end that is vizor of the cactus stack like a Tree, nodes in the Cactus Stack may have parent child relationships. Cactus Stacks are traversed from the child nodes to the parent nodes rather than vice-versa, as in a Binary Search Tree. One of the strongest benefits of a cactus stack is that it allows parallel data structures to exist with the same root.A. Creation of Cactus StackLet us understand this matrix with the help of an example showed above. Series of steps should be as followingFirst, We should know the starting and ending point. Let us repeat that a is the starting point and f is the ending point. Then we put vertices of graph in cactus stack. Put a in cs1. Now a is connected to b and c. so b and c are put in cs2. b is connected to d and c is connected to e so we put d and e in cs3. Repeat same step till the last node is traversed. On traversing the adjacency matrix if two adjacent node has weight other than infinity and zero then we put that nodes in different cactus stacks.B. Linkage in Cactus StackAfter plotting the vertices in cactus stacks. If there is companionship between element of cs1, cs2 then we have to join them. Similarly we will join elements of each stack to its consecutive stack.VII. ConclusionWith the help of Cactus Stack and colligate describe for the shortest path the time complexity is reduced theoretically than the theory proposed by Dijkshtras Shortest path algorithm. Thus we conclude that time complexity is reduced.ReferencesList and number all bibliographical references in 9-point Times, single-spaced, at the end of your pape r. When referenced in the text, enclose the approval number in square brackets, for example 1. Where appropriate, include the name(s) of editors of referenced books. The template will number citations consecutively within brackets 1. 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