International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438 Volume 4 Issue 9, September 2015 www.ijsr.net Licensed Under Creative Commons Attribution CC BY A Survey on Graph Partitioning Techniques Santosh Nage 1 , Girish Potdar 2 1,2 Department of Computer Engineering, Pune Institute of Computer Technology, Savitribai Phule Pune University, Pune 411043, India Abstract: In real life, there are many problems like shortest path, graph coloring, travelling Salesmen problem (TSP) etc, thus providing solution to each problem is nearly or highly impossible with the help of traditional methods in reasonable amount of time. But it may be possible with the help of heuristic approach. It provides solution but don’t guarantee optimal solution. Graph partitioning problems are NP-Complete problems, partitioning graph into p-partitions using multilevel method, spectral method etc. for various purposes .Here we are studying several techniques to partition graph. Keywords: graph partition, spectral, NP-complete, multi-level 1. Introduction Graph partitioning is an important problem in the field of computer science as it has many applications in real life. To find shortest path in a given graph, partitioning will lead to find optimal shortest path using suitable heuristic [9]. Partitioning graph having exact solution is possible only with very small graph having limited vertices and edges. If there are „2n‟ vertices then total number of possibilities is (2n)! / (n!) 2 i.e. for 100 vertices 5 x 10 28 possibilities[4]. To overcome this problem heuristic graph partitioning performs well and deploying it on multi-core systems will give good graph partitions. 2. Graph partitioning Graph partitioning is dividing a graph into two or more parts based on certain condition. It can be defined as dividing vertices V into union of partitions P such that , disjoint set of V 1 ,V 2 ,…V n such that V 1 Π V 2 =Φ for i ≠ j. Graphs are partitioned based on number of vertices and edges. Balanced graphs that are having equal number of vertices on each side, and partitioning it is NP-Complete[5][7], so we need good heuristic and approximation algorithms for partitioning graph to get optimal partition which will lead for further application base. 3. Heuristic It is an approach towards finding solutions in the reasonable amount of time. Traditional methods aren‟t performing well over large space to give results in a reasonable amount of time. According to G. Potdar et.al [13] heuristic approach doesn‟t give guarantee to find optimal solution always. Many real world problems like NP-hard can be solved by using heuristic approach. Heuristic value is denoted as h(n) and formulated to find the total estimated cost. Heuristic function f(n)=g(n)+h(n) Where, f(n)- Total estimated cost g(n)- Cost up to node n from starting point h(n)- Estimated cost from n-node to goal. 4. Graph partitioning techniques Graphs are based on two basic geometric figures 4.1 With nodal co-ordinate 4.1.1 Partitioning Planar graph[15][24] A planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints Lipton-Tarjan proposed a partitioning algorithm in 1977. It is based on √n vertex separator theorem. Ungar(1951) having complexity O (√n log n), Hopcroft-Tarjan(1973) based on depth first search tried for partitioning planar graph. 4.1.2 Inertial Partitioning[5][15] This graph bisection algorithm is very simple: For a graph with 2D coordinates, it chooses a line such that half the nodes are on one side of the line, and half are on the other. For a graph with 3D coordinates, it chooses a plane with the same property. Figure.2: Inertial Graph Partitioning 4.1.3 Partitioning with random lines and circles[15][22] The Lipton-Tarjan planar graph separator theorem provides intuition for the success of inertial graph partitioning. It is generalized idea of nearest neighbor of planar graph to higher dimension. Paper ID: SUB158492 1656