MASALAH PARTISI GRAF BOTTLENECK (BOTTLENECK GRAPH PARTITION PROBLEM)

ABSTRACT The Bottleneck graph partition problem is a partition of the vertices of an undirected edge weighted graph into two equally vertices sized sets, which are graph components, such that the maximum edge weight in the cut separing the two sets becomes minimum. In this thesis, the problem was solved by an optimum algorithm with running time 0(m+n3logn), where n is the number of vertices and m is the number of edges in the graph. This algorithm based on binary search tree using greedy method on prim\u27s algorithm. On validation the random graph is used for the computer\u27s program. Keywords : graph partition, bottleneck, greedy method, prim\u27s algorith

On Near-Linear-Time Algorithms for Dense Subset Sum

In the Subset Sum problem we are given a set of $n$ positive integers $X$ and a target $t$ and are asked whether some subset of $X$ sums to $t$. Natural parameters for this problem that have been studied in the literature are $n$ and $t$ as well as the maximum input number $\rm{mx}_X$ and the sum of all input numbers $\Sigma_X$. In this paper we study the dense case of Subset Sum, where all these parameters are polynomial in $n$. In this regime, standard pseudo-polynomial algorithms solve Subset Sum in polynomial time $n^{O(1)}$. Our main question is: When can dense Subset Sum be solved in near-linear time $\tilde{O}(n)$? We provide an essentially complete dichotomy by designing improved algorithms and proving conditional lower bounds, thereby determining essentially all settings of the parameters $n,t,\rm{mx}_X,\Sigma_X$ for which dense Subset Sum is in time $\tilde{O}(n)$. For notational convenience we assume without loss of generality that $t \ge \rm{mx}_X$ (as larger numbers can be ignored) and $t \le \Sigma_X/2$ (using symmetry). Then our dichotomy reads as follows: - By reviving and improving an additive-combinatorics-based approach by Galil and Margalit [SICOMP'91], we show that Subset Sum is in near-linear time $\tilde{O}(n)$ if $t \gg \rm{mx}_X \Sigma_X/n^2$. - We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with $t \ll \rm{mx}_X \Sigma_X/n^2$, then the Strong Exponential Time Hypothesis and the Strong k-Sum Hypothesis fail. We also generalize our algorithm from sets to multi-sets, albeit with non-matching upper and lower bounds

A note on the bottleneck graph partition problem

The bottleneck graph partition problem consists of partitioning the vertices of an undirected edge-weighted graph into two equally sized sets such that the maximum edge weight in the cut separating the two sets becomes minimum. In this short note, we present an optimum algorithm for this problem with running time O(n2), where n is the number of vertices in the graph. Our result answers an open problem posed in a recent paper by Hochbaum and Pathria (1996)

A note on the bottleneck graph partition problem

The bottleneck graph partition problem consists of partitioning the vertices of an undirected edge-weighted graph into two equally sized sets such that the maximum edge weight in the cut separating the two sets becomes minimum. In this short note, we present an optimum algorithm for this problem with running time O(n2), where n is the number of vertices in the graph. Our result answers an open problem posed in a recent paper by Hochbaum and Pathria (1996)

MASALAH PARTISI GRAF BOTTLENECK (BOTTLENECK GRAPH PARTITION PROBLEM)

ABSTRACT The Bottleneck graph partition problem is a partition of the vertices of an undirected edge weighted graph into two equally vertices sized sets, which are graph components, such that the maximum edge weight in the cut separing the two sets becomes minimum. In this thesis, the problem was solved by an optimum algorithm with running time 0(m+n3logn), where n is the number of vertices and m is the number of edges in the graph. This algorithm based on binary search tree using greedy method on prim's algorithm. On validation the random graph is used for the computer's program. Keywords : graph partition, bottleneck, greedy method, prim's algorith