KLASIFIKASI GRAF POHON BINER LENGKAP LEVEL- n (n = 1,2,3) BERDASARKAN PEWARNAAN-f

Graf G adalah pasangan himpunan (V,E) dengan V merupakan himpunan berhingga tidak kosong dari objek-objek yang disebut titik dan E merupakan himpunan berhingga, mungkin kosong, pasangan tidak terurut dari titik-titik berbeda. Setiap pasangan titik disebut sisi. Pewarnaan sisi (edge-coloring) pada suatu graf G merupakan salah satu cara pewarnaan pada graf G sedemikian sehingga sisi-sisi yang bertetangga memiliki warna yang berbeda. \ud Penelitian ini akan mengkaji perumuman dari pewarnaan sisi, yaitu pewarnaan-f. Pewarnaan-f pada graf G(V,E) adalah pemberian warna pada sisi-sisi G sedemikian sehingga sisi-sisi yang bertetangga pada suatu titik v dapat memiliki warna yang sama paling banyak f(v) buah. Banyaknya warna minimum yang digunakan pada pewarnaan-f pada suatu graf G disebut f-kromatik indeks, dinotasikan dengan ?????^'???_f (G). \ud Misalkan ???_f (G)=maks{??? ???d(v)/f(v) ??? ???| v ?? V} maka batas bawah dan batas atas dari f-kromatik indeks pada suatu graf G memenuhi:\ud ???_f (G) ????????? ?????^'???_f (G)??? ???_f (G)+1\ud Jika ?????? ?????^'???_f (G)=???_f (G) maka graf G tergolong dalam c_f 1 dan jika ?????? ?????^'???_f (G)= ???_f (G)+1 maka graf G tergolong dalam c_f 2.\ud Pada penelitian ini diperoleh bahwa graf pohon biner lengkap level-n (n=1,2,3) tergolong dalam c_f 1

Optimal Online Edge Coloring of Planar Graphs with Advice

Using the framework of advice complexity, we study the amount of knowledge about the future that an online algorithm needs to color the edges of a graph optimally, i.e., using as few colors as possible. For graphs of maximum degree $\Delta$, it follows from Vizing's Theorem that $O(m\log \Delta)$ bits of advice suffice to achieve optimality, where $m$ is the number of edges. We show that for graphs of bounded degeneracy (a class of graphs including e.g. trees and planar graphs), only $O(m)$ bits of advice are needed to compute an optimal solution online, independently of how large $\Delta$ is. On the other hand, we show that $\Omega (m)$ bits of advice are necessary just to achieve a competitive ratio better than that of the best deterministic online algorithm without advice. Furthermore, we consider algorithms which use a fixed number of advice bits per edge (our algorithm for graphs of bounded degeneracy belongs to this class of algorithms). We show that for bipartite graphs, any such algorithm must use at least $\Omega(m\log \Delta)$ bits of advice to achieve optimality.Comment: CIAC 201

Finding Small Complete Subgraphs Efficiently

(I) We revisit the algorithmic problem of finding all triangles in a graph $G=(V,E)$ with $n$ vertices and $m$ edges. According to a result of Chiba and Nishizeki (1985), this task can be achieved by a combinatorial algorithm running in $O(m \alpha) = O(m^{3/2})$ time, where $\alpha= \alpha(G)$ is the graph arboricity. We provide a new very simple combinatorial algorithm for finding all triangles in a graph and show that is amenable to the same running time analysis. We derive these worst-case bounds from first principles and with very simple proofs that do not rely on classic results due to Nash-Williams from the 1960s. (II) We extend our arguments to the problem of finding all small complete subgraphs of a given fixed size. We show that the dependency on $m$ and $\alpha$ in the running time $O(\alpha^{\ell-2} \cdot m)$ of the algorithm of Chiba and Nishizeki for listing all copies of $K_\ell$, where $\ell \geq 3$, is asymptotically tight. (III) We give improved arboricity-sensitive running times for counting and/or detection of copies of $K_\ell$, for small $\ell \geq 4$. A key ingredient in our algorithms is, once again, the algorithm of Chiba and Nishizeki. Our new algorithms are faster than all previous algorithms in certain high-range arboricity intervals for every $\ell \geq 7$.Comment: 14 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:2105.0126

Tailored vertex ordering for faster triangle listing in large graphs

Listing triangles is a fundamental graph problem with many applications, and large graphs require fast algorithms. Vertex ordering allows the orientation of edges from lower to higher vertex indices, and state-of-the-art triangle listing algorithms use this to accelerate their execution and to bound their time complexity. Yet, only basic orderings have been tested. In this paper, we show that studying the precise cost of algorithms instead of their bounded complexity leads to faster solutions. We introduce cost functions that link ordering properties with the running time of a given algorithm. We prove that their minimization is NP-hard and propose heuristics to obtain new orderings with different trade-offs between cost reduction and ordering time. Using datasets with up to two billion edges, we show that our heuristics accelerate the listing of triangles by an average of 38% when the ordering is already given as an input, and 16% when the ordering time is included.Comment: 11 pages, 4 figures. Open-source C++ code available at: https://github.com/lecfab/vol

Listing k-cliques in Sparse Real-World Graphs

International audienceMotivated by recent studies in the data mining community which require to efficiently list all k-cliques, we revisit the iconic algorithm of Chiba and Nishizeki and develop the most efficient parallel algorithm for such a problem. Our theoretical analysis provides the best asymptotic upper bound on the running time of our algorithm for the case when the input graph is sparse. Our experimental evaluation on large real-world graphs shows that our parallel algorithm is faster than state-of-the-art algorithms, while boasting an excellent degree of parallelism. In particular, we are able to list all k-cliques (for any k) in graphs containing up to tens of millions of edges as well as all 10-cliques in graphs containing billions of edges, within a few minutes and a few hours respectively. Finally, we show how our algorithm can be employed as an effective subroutine for finding the k-clique core decomposition and an approximate k-clique densest subgraphs in very large real-world graphs

GraphMineSuite: Enabling High-Performance and Programmable Graph Mining Algorithms with Set Algebra

We propose GraphMineSuite (GMS): the first benchmarking suite for graph mining that facilitates evaluating and constructing high-performance graph mining algorithms. First, GMS comes with a benchmark specification based on extensive literature review, prescribing representative problems, algorithms, and datasets. Second, GMS offers a carefully designed software platform for seamless testing of different fine-grained elements of graph mining algorithms, such as graph representations or algorithm subroutines. The platform includes parallel implementations of more than 40 considered baselines, and it facilitates developing complex and fast mining algorithms. High modularity is possible by harnessing set algebra operations such as set intersection and difference, which enables breaking complex graph mining algorithms into simple building blocks that can be separately experimented with. GMS is supported with a broad concurrency analysis for portability in performance insights, and a novel performance metric to assess the throughput of graph mining algorithms, enabling more insightful evaluation. As use cases, we harness GMS to rapidly redesign and accelerate state-of-the-art baselines of core graph mining problems: degeneracy reordering (by up to >2x), maximal clique listing (by up to >9x), k-clique listing (by 1.1x), and subgraph isomorphism (by up to 2.5x), also obtaining better theoretical performance bounds

Edge-Coloring and f-Coloring for Various Classes of Graphs

In an ordinary edge-coloring of a graph each color... This paper gives efficient sequential and parallel algorithms to find ordinary edge-colorings and f-colorings for various classes of graphs such as bipartite graphs, planar graphs, and graphs having fixed degeneracy, tree-width, genus, arboricity, unicyclic index or thickness