Tree Partitioning Problem Research Articles

Serverless application composition leveraging function fusion: Theory and algorithms

Serverless computing is a novel cloud computing paradigm enabling flexible, cost-efficient and fine-granular development of cloud-native applications, although without any guarantees on scheduling or execution times. Thus, various high-level solutions have been proposed in recent years to find proper configurations of individual FaaS functions, while considering user-given QoS requirements. However, the ever-increasing complexity of invocation patterns among stateless functions, externalized management of intermediate states, and diverse public cloud resources pose new challenges to the composition of highly data-intensive serverless applications. In this paper, we fill this gap by proposing novel algorithms based on the emerging function fusion technique, along with the related cost/performance models of composite functions supporting implicit instance parallelization and internal state propagation. We prove the NP-completeness of the underlying latency-constrained tree partitioning problem, and design a bicriteria approximation scheme and a greedy heuristic to derive cost-efficient deployment configurations in polynomial time. With the help of extensive simulations using synthetic call graphs generated from public cloud traces, we demonstrate the applicability and superior runtime performance of our proposed methods compared to state-of-the-art solutions. In addition, we showcase that further cost-reduction of up to 3–6 % can be achieved compared to the optimal partitioning with the allowance of tolerable latency violations.

Heuristic Tree-Partition-Based Parallel Method for Biophysically Detailed Neuron Simulation.

Biophysically detailed neuron simulation is a powerful tool to explore the mechanisms behind biological experiments and bridge the gap between various scales in neuroscience research. However, the extremely high computational complexity of detailed neuron simulation restricts the modeling and exploration of detailed network models. The bottleneck is solving the system of linear equations. To accelerate detailed simulation, we propose a heuristic tree-partition-based parallel method (HTP) to parallelize the computation of the Hines algorithm, the kernel for solving linear equations, and leverage the strong parallel capability of the graphic processing unit (GPU) to achieve further speedup. We formulate the problem of how to get a fine parallel process as a tree-partition problem. Next, we present a heuristic partition algorithm to obtain an effective partition to efficiently parallelize the equation-solving process in detailed simulation. With further optimization on GPU, our HTP method achieves 2.2 to 8.5 folds speedup compared to the state-of-the-art GPU method and 36 to 660 folds speedup compared to the typical Hines algorithm.

Sum-of-Max partition under a Knapsack constraint

Sequence partition problems arise in many fields, such as sequential data analysis, information transmission, and parallel computing. In this paper, we study the following partition problem variant: given a sequence of n items 1,…,n, where each item i is associated with weight wi and another parameter si, partition the sequence into several consecutive subsequences, so that the total weight of each subsequence is no more than a threshold w0, and the sum of the largest si in each subsequence is minimized.This problem admits a straightforward solution based on dynamic programming, which costs O(n2) time and can be improved to O(nlogn) time easily. Our contribution is an O(n) time algorithm, which is nontrivial yet easy to implement. We also study the corresponding tree partition problem. We prove that the problem on the tree is NP-complete and we present an O(w0n2) time (O(w02n2) time, respectively) algorithm for the unit weight (integer weight, respectively) case.

Flopp: Extremely Fast Long-Read Polyploid Haplotype Phasing by Uniform Tree Partitioning.

Resolving haplotypes in polyploid genomes using phase information from sequencing reads is an important and challenging problem. We introduce two new mathematical formulations of polyploid haplotype phasing: (1) the min-sum max tree partition problem, which is a more flexible graphical metric compared with the standard minimum error correction (MEC) model in the polyploid setting, and (2) the uniform probabilistic error minimization model, which is a probabilistic analogue of the MEC model. We incorporate both formulations into a long-read based polyploid haplotype phasing method called flopp. We show that flopp compares favorably with state-of-the-art algorithms—up to 30 times faster with 2 times fewer switch errors on 6 × ploidy simulated data. Further, we show using real nanopore data that flopp can quickly reveal reasonable haplotype structures from the autotetraploid Solanum tuberosum (potato).

The Complexity of Tree Partitioning

Given a tree T on n vertices, and $$k, b, s_1, \ldots , s_b \in \mathbb {N}$$ , the Tree Partitioning problem asks if at most k edges can be removed from T so that the resulting components can be grouped into b groups such that the number of vertices in group i is $$s_i$$ , for $$i =1, \ldots , b$$ . The case where $$s_1=\cdots =s_b =n/b$$ , referred to as the Balanced Tree Partitioning problem, was shown to be $${\mathcal {NP}}$$ -complete for trees of maximum degree at most 5, and the complexity of the problem for trees of maximum degree 4 and 3 was posed as an open question. The parameterized complexity of Balanced Tree Partitioning was also posed as an open question in another work. In this paper, we answer both open questions negatively. We show that Balanced Tree Partitioning (and hence, Tree Partitioning) is $${\mathcal {NP}}$$ -complete for trees of maximum degree 3, thus closing the door on the complexity of Balanced Tree Partitioning, as the simple case when T is a path is in $${\mathcal {P}}$$ . In terms of the parameterized complexity of the problems, we show that both Balanced Tree Partitioning and Tree Partitioning are W[1]-complete parameterized by k. Using a compact representation of the solution space for an instance of the problem, we present a dynamic programming algorithm for the weighted version of Tree Partitioning (and hence for that of Balanced Tree Partitioning) that runs in subexponential-time $$2^{O(\sqrt{n})}$$ , adding a natural problem to the list of problems that can be solved in subexponential time. Finally, we extend this subexponential-time algorithm to the Weighted Graph Partitioning problem on graphs of treewidth $$o(n/\lg {n})$$ , and we also show an application of this subexponential-time algorithm for approximating the Weighted Graph Partitioning problem.

Balanced tree partition problems with virtual nodes

We study two variants of the Balanced Tree Partition (BTP for short) problem, whose goal is to find the minimum number of edges such that a partition of vertices into sets of equal size can be obtained after deleting those edges. We consider the BTP problem over trees with virtual nodes, which is motivated by the real applications in storing tree structured data distributively. Different from the traditional BTP problem, after deleting an edge from the tree, two virtual nodes must be added to provide the ability to recover the edge when computing, which will increase the size of some set in the partition and the set may become too large and need further splitting. Depending on whether or not to consider the partition set number k to be a parameter, to investigate the balanced tree partition methods over trees with virtual nodes, we formally defined two specific problems, \(k\)-VBTP and VBTP. The computational complexity and inapproximability of \(k\)-VBTP are analyzed, for the VBTP problem, a polynomial time algorithm is designed to find the optimal solution based on dynamic programming. Finally, the connections between \(k\)-VBTP and VBTP are built, and based the algorithm for VBTP, we design a heuristic algorithm for \(k\)-VBTP with performance guarantee.

Separation Algorithm for Tree Partitioning Inequalities

We consider the tree partition problem to partition the node set of a tree into subsets where the induced subgraph by each subset is connected and the total weight of nodes in a subset cannot exceed the capacity of the subset. We identify exponentially many valid inequalities for an integer programming formulation of the problem and develop a linear time separation algorithm for the valid inequalities.

Tree [formula omitted]-spanners in outerplanar graphs via supply demand partition

A treet-spanner of an unweighted graph G is a spanning tree T such that for every two vertices their distance in T is at most t times their distance in G. Given an unweighted graph G and a positive integer t as input, the Treet-spanner problem is to compute a tree t-spanner of G if one exists. This decision problem is known to be NP-complete even in the restricted class of unweighted planar graphs. We present a linear-time reduction from Treet-spanner in outerplanar graphs to the supply–demand tree partition problem. Based on this reduction, we obtain a linear-time algorithm to solve Treet-spanner in outerplanar graphs. Consequently, we show that the minimum value of t for which an input outerplanar graph on n vertices has a tree t-spanner can be found in O(nlogn) time.

A minimum spanning tree equipartition algorithm for microaggregation

In this paper, we propose a solution on microaggregation problem based on the hierarchical tree equi-partition (HTEP) algorithm. Microaggregation is a family of methods for statistical disclosure control of microdata, that is, for masking microdata, so that they can be released without disclose private information on the underlying individuals. Knowing that the microaggregation problem is non-deterministic polynomial-time-hard, the goal is to partition N given data into groups of at least K items, so that the sum of the within-partition squared error is minimized. The proposed method is general and it can be applied to any tree partition problem aiming at the minimization of a total score. The method is divisive, so that the tree with the highest ‘score’ is split into two trees, resulting in a hierarchical forest of trees with almost equal ‘score’ (equipartition). We propose a version of HTEP for microaggregation (HTEPM), that is applied on the minimum spanning tree (MST) of the graph defined by the data. The merit of the HTEPM algorithm is that it solves optimally some instances of the multivariate microaggregation problem on MST search space in . Experimental results and comparisons with existing methods from literature prove the high performance and robustness of HTEPM.

Rooted-Tree Decompositions with Matroid Constraints and the Infinitesimal Rigidity of Frameworks with Boundaries

As an extension of a classical tree-partition problem, we consider decompositions of graphs into edge-disjoint (rooted-)trees with an additional matroid constraint. Specifically, suppose that we are given a graph $G=(V,E)$, a multiset ${\bm R} = \{r_1,\dots, r_t\}$ of vertices in $V$, and a matroid ${\cal M}$ on ${\bm R}$. We prove a necessary and sufficient condition for $G$ to be decomposed into $t$ edge-disjoint subgraphs $G_1=(V_1,T_1), \dots, G_t=(V_t,T_t)$ such that (i) for each $i$, $G_i$ is a tree with $r_i\in V_i$, and (ii) for each $v\in V$, the multiset $\{r_i\in {\bm R} \mid v\in V_i\}$ is a base of ${\cal M}$. If ${\cal M}$ is a free matroid, this is a decomposition into $t$ edge-disjoint spanning trees; thus, our result is a proper extension of Nash-Williams' tree-partition theorem. Such a matroid constraint is motivated by combinatorial rigidity theory. As a direct application of our decomposition theorem, we present characterizations of the infinitesimal rigidity of frameworks with nongeneric “boundary,” which extend classical the Laman's theorem for generic 2-rigidity of bar-joint frameworks and Tay's theorem for generic $d$-rigidity of body-bar frameworks.

Heterochromatic tree partition problem in complete tripartite graphs

The heterochromatic tree partition number of an r -edge-colored graph G , denoted by t r ( G ) , is the minimum positive integer p such that whenever the edges of the graph G are colored with r colors, the vertices of G can be covered by at most p vertex disjoint heterochromatic trees. In this paper, we give an explicit formula for the heterochromatic tree partition number t r ( K n 1 , n 2 , n 3 ) of an r -edge-colored complete tripartite graph K n 1 , n 2 , n 3 . Given an r -edge-colored complete tripartite graph K n 1 , n 2 , n 3 , based on the proof of the formula, we can find at most t r ( K n 1 , n 2 , n 3 ) vertex disjoint heterochromatic trees which together cover all the vertices of K n 1 , n 2 , n 3 .

Minmax Tree Cover in the Euclidean Space

Let G = (V,E) be an edge-weighted graph, and let w(H) denote the sum of the weights of the edges in a subgraph H of G. Given a positive integer k, the balanced tree partitioning problem requires to cover all vertices in V by a set $\mathcal{T}$ of k trees of the graph so that the ratio α of $\max_{T\in \mathcal{T}}w(T)$ to w(T *)/k is minimized, where T * denotes a minimum spanning tree of G. The problem has been used as a core analysis in designing approximation algorithms for several types of graph partitioning problems over metric spaces, and the performance guarantees depend on the ratio α of the corresponding balanced tree partitioning problems. It is known that the best possible value of α is 2 for the general metric space. In this paper, we study the problem in the d-dimensional Euclidean space ℝ d , and break the bound 2 on α, showing that $\alpha <2\sqrt{3}-3/2 \fallingdotseq 1.964$ for d ≥ 3 and $\alpha <(13 + \sqrt{109})/12 \fallingdotseq 1.953$ for d = 2. These new results enable us to directly improve the performance guarantees of several existing approximation algorithms for graph partitioning problems if the metric space is an Euclidean space.

Scalable Delay-constrained Multicast Group Key Management

In the last few years, multicasting is increasingly used as an efficient communication mechanism for group-oriented applications in the Internet. Some multicast applications require confidentiality for transmitted data. So, a traffic encryption key is used to assure this confidentiality and has to be changed and distributed to all valid members whenever a membership change (join or leave) occurs in the group. The bandwidth used for re-keying operations could be high when the group size is large. To cope with this limitation, many solutions propose to organize group members into subgroups that use independent traffic encryption keys in order to mitigate the scope of key management and thereby to scale better to large groups. Unfortunately, these solutions require the decryption and re-encryption of multicast messages whenever they pass from one subgroup to another. Moreover, the decryption/re-encryption operations induce delays in packet delivery throughout the delivery path. In order to avoid delays in packet delivery and perturbations caused by re-keying, we propose in this paper an adaptive solution for key management which organizes group members into dynamic and homogeneous clusters according to the application level requirements. First, we show that partitioning the group into clusters of subgroups that use independent traffic encryption keys can be formulated as tree partitioning problem. Then, we propose a protocol to solve the problem with respect to the application requirements and membership behavior. We conducted several simulations of the proposed protocol and the obtained results show that our solution is efficient and achieves better performance trade-offs compared to other schemes.

Computing sharp bounds for hard clustering problems on trees

Clustering problems with relational constraints in which the underlying graph is a tree arise in a variety of applications: hierarchical data base paging, communication and distribution networks, districting, biological taxonomy, and others. They are formulated here as optimal tree partitioning problems. In a previous paper, it was shown that their computational complexity strongly depends on the nature of the objective function and, in particular, that minimizing the total within-cluster dissimilarity or the diameter is computationally hard. We propose heuristics that find good partitions within a reasonable time, even for instances of relatively large size. Such heuristics are based on the solution of continuous relaxations of certain integer (or almost integer) linear programs. Experimental results on over 2000 randomly generated instances with up to 500 entities show that the values (total within-cluster dissimilarity or diameter) of the solutions provided by these heuristics are quite close to the minimum one.

Vertex partitions of r-edge-colored graphs

Let G be an edge-colored graph. The monochromatic tree partition problem is to find the minimum number of vertex disjoint monochromatic trees to cover the all vertices of G. In the authors' previous work, it has been proved that the problem is NP-complete and there does not exist any constant factor approximation algorithm for it unless P = NP. In this paper the authors show that for any fixed integer r � 5, if the edges of a graph G are colored by r colors, called an r-edge-colored graph, the problem remains NP-complete. Similar result holds for the monochromatic path (cycle) partition problem. Therefore, to find some classes of interesting graphs for which the problem can be solved in polynomial time seems interesting. A linear time algorithm for the monochromatic path partition problem for edge-colored trees is given.

Combining Tree Partitioning, Precedence, and Incomparability Constraints

The tree constraint partitions a directed graph into node-disjoint trees. In many practical applications that involve such a partition, there exist side constraints specifying requirements on tree count, node degrees, or precedences and incomparabilities within node subsets. We present a generalisation of the tree constraint that incorporates such side constraints. The key point of our approach is to take partially into account the strong interactions between the tree partitioning problem and all the side constraints, in order to avoid thrashing during search. We describe filtering rules for this extended tree constraint and evaluate its effectiveness on three applications: the Hamiltonian path problem, the ordered disjoint paths problem, and the phylogenetic supertree problem.

Polynomial algorithms for partitioning a tree into single‐center subtrees to minimize flat service costs

AbstractThis paper deals with the following graph partitioning problem. Consider a connected graph with n nodes, p of which are centers, while the remaining ones are units. For each unit‐center pair there is a fixed service cost and the goal is to find a partition into connected components such that each component contains only one center and the total service cost is minimum. This problem is known to be NP‐hard on general graphs, and here we show that it remains such even if the service cost is monotone and the graph is bipartite. However, in this paper we derive some polynomial time algorithms for trees. For this class of graphs we provide several reformulations of the problem as integer linear programs proving the integrality of the corresponding polyhedra. As a consequence, the tree partitioning problem can be solved in polynomial time either by linear programming or by suitable convex nondifferentiable optimization algorithms. Moreover, we develop a dynamic programming algorithm, whose recursion is based on sequences of minimum weight closure problems, which solves the problem on trees in O(np) time. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008

On a tree-partition problem

If T=(V,E) is a tree then – T denotes the additive hereditary property consisting of all graphs that does not contain T as a subgraph. For an arbitrary vertex v of T we deal with a partition of T into two trees T1, T2, so that V(T1)∩V(T2)={v}, V(T1)∪(T2)=V(T), E(T1)∩E(T2)=∅, E(T1)∪E(T2)=E(T), T[V(T1)\{v}]⊆E(T1) and T[V(T2)\{v}]⊆E(T2). We call such a partition a Tv−partition of T. We study the following em: Given a graph G belonging to –T. Is it true that for any Tv-partition T1, T2 of T there exists a partition {V1,V2} of the vertices of G such that G[V1]∈−T1 and G[V2]∈−T2? This problem provides a natural generalization of Δ-partition problem studied by L. Lovász ([L. Lovász, On decomposition of graphs. Studia Sci. Math. Hungar. 1 (1966) 237–238]) and Path Partition Conjecture formulated by P. Mihók ([P. Mihók, Problem 4, in: M. Borowiecki, Z. Skupien (Eds.), Graphs, Hypergraphs and Matroids, Zielona Góra, 1985, p. 86]). We present some partial results and a contribution to the Path Kernel Conjecture that was formulated with connection to Path Partition Conjecture.

A subexponential algorithm for the coloured tree partition problem

Given a tree of n vertices and a list of feasible colours for each vertex, the coloured tree partition problem ( CTPP) consists in partitioning the tree into p vertex-disjoint subtrees of minimum total cost, and assigning to each subtree a different colour, which must be feasible for all of its vertices. The problem is strongly NP -hard on general graphs, as well as on grid and bipartite graphs. This paper deals with the previously open case of tree graphs, showing that it is strongly NP -complete to determine whether a feasible solution exists. It presents reduction, decomposition and bounding procedures to simplify the problem and an exact algorithm of O ( np log 2 ( a p - 2 ) ) complexity (with a > 3 2 ) for the special case in which a vertex of each subtree is given.

Antenna avoidance in layer assignment

The sustained progress of very-large-scale-integration (VLSI) technology has dramatically increased the likelihood of the antenna problem in the manufacturing process and calls for corresponding considerations in the routing stage. In this paper, the authors propose a technique that can handle the antenna problem during the layer-assignment (LA) stage, which is an important step between global routing and detailed routing. The antenna-avoidance problem is modeled as a tree-partitioning problem with a linear-time-optimal-algorithm solution. This algorithm is customized to guide antenna avoidance in the LA stage. A linear-time optimal jumper-insertion algorithm is also derived. Experimental results on benchmark circuits show that the proposed techniques can lead to an average of 76% antenna-violation reduction and 99% via-violation reduction.