Surveillance System (Minimum Vertex Cover Problem)
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This document describes a surveillance system application that finds the minimum number of CCTV cameras needed to monitor a building or area. The application takes input of a building's layout and connectivity between locations. It uses a modified Alom algorithm to find the minimum vertex cover of this graph, representing the optimal camera placements. The output is a table showing camera locations and numbers, along with the total installation cost. The application aims to accurately and cost-effectively determine surveillance needs compared to human estimates.
![REFERENCES
[1] C. E. Leiserson, C. Stein, R. L. Rivest, and T. H. Cormen ”Introduction to
Algorithms McGraw-Hill, New York, 2nd edition: Local 2-Approximation
Algorithm”,2001.
[2] D.S. Johnson and M.R.Grarey, “Computers and Intractability-A guide to the
Theory of NP-Completeness, freeman: Greedy Algorithm for minimum vertex
cover”, 1978.
[3] Durgesh Pant, Kamlesh Chandra Purohit, and Sushil Chandra Dimri, “Clever
Greedy Minimum Vertex Cover Algorithm”, 2010.
[4] B. M. Monjurul Alom, Mohammad Abdur Rouf and Someresh Das, “Performance
Evaluation of Vertex Cover and Set Cover Problem using Optimal Algorithm”, 2011.
[5] Christian Laforest, Eric Angel and Romain Campigotto, “Algorithms for the
Vertex Cover Problem on Large Graphs”, 2010.
[6] Kaile Su, Qingliang Chen and Shaowie Cai, “ELWS-A new local search for
minimum vertex cover”, 2007.
[7] Marija Milanovic, “Solving the generalized vertex cover algorithm by genetic
algorithm”, 2008.
[8] Ge Xia, Iyad A.Kanj and Jianer Chen, “Improved Parameterized Upper Bounds
for Vertex Cover”, 2001.](https://image.slidesharecdn.com/presentation-140528142124-phpapp01/85/Surveillance-System-Minimum-Vertex-Cover-Problem-14-320.jpg)
Surveillance System (Minimum Vertex Cover Problem)
- 1. SURVEILLANCE SYSTEM SUBMITTED BY: Pranjal Kumar Srivastava (10103554) Saksham Saxena (10103621) SUBMITTED TO: Dr. Manish Kumar Thakur (Assistant Professor – JIIT)
- 2. INTRODUCTION In graph theory, a vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex of the set. The problem of finding a minimum vertex cover is a classical optimization problem in computer science and is a typical example of an NP- hard optimization problem that has an approximation algorithm. An algorithm that returns near-optimal solutions is called an approximation algorithm. In computer science, the vertex cover problem or node cover problem is one of Karp's 21 NP-complete problems. It is often used in complexity theory to prove NP-hardness of more complicated problems. Any user who wants to install CCTV cameras in his/her building can upload two files containing different positions of the building and their connections on the website. The algorithm returns the best possible solution in the form of table displaying the appropriate positions and the number of cameras required at these positions along with the total approximate cost of installation. User can also download all other possible solutions.
- 3. PROBLEM STATEMENT To develop an application such that a user will input the graph of a building or a locality to find minimum number of CCTV cameras or security guards required for the surveillance. Our application using the optimum algorithm will return all the best possible locations where the CCTVs can be installed or security guards can be placed. It will also give the number of cameras required and the total cost of installation.
- 4. BENEFITS/NOVELITY OF THE APPLICATION It helps remove the discrepancy in the result that occurs due to human error. It selects the minimum number of nodes from which all the edges can be accessed. This helps in placing cameras at minimum required places and helps in reducing the human effort and cost that is put in selecting these areas/places. This can also be used in placing minimum number of ATMs in the city so as to reduce the overall cost. Our proposed algorithm ”Modified Alom” gives all possible sets of minimum vertex cover, thus the user can opt for any one of them according to his/her constraints.
- 5. ARCHITECTURE OF APPLICATION User uploads its Location file of the building which gives aus all the possible locations to put the camera. User uploads its Connectivity file of the building which gives us the connectivity of all possible locations with each other. The algorithm form the graph from the input in the form of adjacency matrix. It then calculates the minimum vertex cover using the proposed Modified Alom algorithm keeping the constraints of costs and ranges in consideration and gives us the output in the form of a table.
- 9. TESTING REQUIRED Type of Test Will test be performed? Comments/ Explanations Software Component Requirement Testing Yes All the input requirements need to be fulfilled Graph.c, Connectivity.xls, Locations.xls, College_Graph.txt Unit Yes The application is mainly divided into data input, running of algorithm and display Input- College_Graph.txt Algorithm- Modified_Algorithm.c Output- College_Graph_Modified_Al om.xls Integration No All three components of application are independent Performance Yes Performance of the algorithms to be tested Analysis Table Stress Yes All the algorithms have been tested on graphs with approximately 1000 nodes. Security No At this point, security testing is not required. Load Yes Input is real time data
- 10. LIMITATION OF THE APPLICATION Algorithm is slow but accurate on large graphs. Application doesn’t consider the camera angle. It doesn’t give the direction in which the camera is to be installed. Any extra installation cost is not considered in the final cost displayed.
- 11. FINDINGS Approximation Algorithm: For minimum vertex cover problem, this algorithm randomly picks up an edge from the graph and includes both of its vertices in the vertex cover set. It then removes all the edges incident on those two vertices. It iteratively continues until no edges are left. Greedy Algorithm: For minimum vertex cover problem, the greedy algorithm randomly picks up an edge from the graph and includes its end vertex in the vertex cover set. It then removes all the edges incident on that vertex from the graph. It iteratively continues until no edges are left in the graph. Clever Greedy Algorithm: For minimum vertex cover problem, the clever greedy algorithm finds the vertex with the maximum degree in the graph and includes it in the vertex cover set. It then removes all the edges incident on that vertex from the graph. It iteratively continues until no edges are left in the graph. Sorted List Left Algorithm: For minimum vertex cover problem, the Sorted List Left algorithm selects the vertex belonging to the labelled graph that has at least one neighbor with a lower degree or a right neighbor with the same degree. Then the vertex is included in the vertex cover set. Alom Algorithm: For minimum vertex cover problem, the ‘Alom’ algorithm selects the vertex which has maximum number of edges incident to it. All the edges are discarded incident to that vertex. If more than one vertex have same maximum number of edges, this algorithm select that vertex which have at least one edge that is not covered by other vertices, which has maximum edge. This process is repeated until to cover all the vertices of the graph.
- 12. CONCLUSION Minimum Vertex Cover Algorithms are used in finding out the minimum number of nodes or vertices that can traverse each path in the graph. In this project, we have implemented five different algorithms for finding out the minimum vertex cover of 20 different graphs. The aim is to find the best possible algorithm that returns the minimum vertex cover. The algorithm proposed by us is a slight modification in the existing Alom Algorithm. The application is designed in such a way that it takes the input from the user. The input will be an excel file containing the locations and the connections of the locations. Algorithm is then run on this graph and gives the best possible solution to the minimum vertex cover problem. The most optimum solution is displayed in a table also showing the required number of cameras at each location. Graph is also displayed highlighting the positions where cameras are needed to be installed. All other possible solutions can also be downloaded in the form of excel sheet.
- 13. FUTURE WORK Angle of camera being installed can be considered. Direction of camera must also be taken into account. Other costs and charges should be added. Any other feature like giving information about the various camera installation mechanics can be given.
- 14. REFERENCES [1] C. E. Leiserson, C. Stein, R. L. Rivest, and T. H. Cormen ”Introduction to Algorithms McGraw-Hill, New York, 2nd edition: Local 2-Approximation Algorithm”,2001. [2] D.S. Johnson and M.R.Grarey, “Computers and Intractability-A guide to the Theory of NP-Completeness, freeman: Greedy Algorithm for minimum vertex cover”, 1978. [3] Durgesh Pant, Kamlesh Chandra Purohit, and Sushil Chandra Dimri, “Clever Greedy Minimum Vertex Cover Algorithm”, 2010. [4] B. M. Monjurul Alom, Mohammad Abdur Rouf and Someresh Das, “Performance Evaluation of Vertex Cover and Set Cover Problem using Optimal Algorithm”, 2011. [5] Christian Laforest, Eric Angel and Romain Campigotto, “Algorithms for the Vertex Cover Problem on Large Graphs”, 2010. [6] Kaile Su, Qingliang Chen and Shaowie Cai, “ELWS-A new local search for minimum vertex cover”, 2007. [7] Marija Milanovic, “Solving the generalized vertex cover algorithm by genetic algorithm”, 2008. [8] Ge Xia, Iyad A.Kanj and Jianer Chen, “Improved Parameterized Upper Bounds for Vertex Cover”, 2001.