Minimum Spanning Tree (MST)

Teori graph dimulai pada tahun 1736 ketika seorang matematikawan Swiss, Leonhard Euler mempublikasikan tulisan yang berisi solusi untuk menyelesaikan masalah jembatan Konigsberg di Prussia (sekarang Kaliningrad di Russia)(Susanna, 2010). Sejak itu, penelitian terhadap graph terus mengalami perkembangan sei- ring dengan semakin bervariasinya masalah yang dihadapi. Salah satu diantaranya adalah permasalahan degree constrained minimum spanning tree (DCMST).

III. RESULT AND ANALYSIS The system being built on this research is algorithm MST application to recognize musical note. This application will recognize musical note based on the form and produce a long sound from the recognized musical note. 2. The musical note used in this research is on 6 form, which are, quarter note, half note, eighth note, sixteenth note, and thirty-second note.

Data flow Diagram Level- 2 proses 2 merupakan perincian dari proses MST menggunakan algoritma genetika. Tahapan pertama yang dilakukan setelah data koordinat di-input-kan adala menghitung jarak antar titik koordinat, kemudian membangkitkan populasi awal secara acak, mengevaluasi kromosom (menghitung bobot, nilai fitness, nilai probabilitas, dan nilai kumulatif), melakukan seleksi kromosom untuk dapat dijadikan kromosom induk pada crossover, melakukan persilangan/ rekombinasi, mutasi, dan melakukan pergantian kromosom yang dipilih sebagai bentuk pelestarian kromosom. Memilih hasil minimum dan menggambar tree yang terbentu yang kemudian dilaporkan ke pengguna.

Kemiskinan bagi masyarakat seringkali diartikan sebagai suatu keadaan dimana seseorang tidak memiliki uang, atau pendapatannya yang rendah bahkan tidak dapat memenuhi kebutuhan dasar hidup sehari-hari. Konsep tentang kemiskinan sangat beragam, mulai dari sekedar ketidakmampuan memenuhi kebutuhan komsumsi dasar dan memperbaiki keadaan, kurangnya kesempatan berusaha, hingga pengertian yang lebih luas yang memasukkan aspek sosial dan moral. Tetapi pada umumnya, ketika orang berbicara tentang kemiskinan, yang dimaksud adalah kemiskinan material. Dengan pengertian ini, maka seseorang masuk dalam kategori miskin apabila tidak mampu memenuhi standar minimum kebutuhan pokok untuk dapat hidup secara layak (Ridwan Muhtadin. 2013). Berbagai cara telah dilakukan pemerintah Indonesia, untuk dapat menanggulangi kemiskinan yang melanda masyarakat seperti pemberian bantuan dana sosial bagi masyarakat yang kurang mampu, yaitu dengan cara pemberian bantuan melalui program keluarga harapan (PKH).

In graph theory, the Rural Postman Problem (RPP) aims to determine the shortest route with minimum cost, in which only certain edges or arcs are necessarily traversed. One type of RPP is Stacker Crane Problem (SCP) which deals with a mixed graph but only arcs are necessarily traversed. SCP can be solved by using two heuristic algorithms, i.e. Largearcs and Smallarcs algorithms, which include other algorithms in their steps, such as: Dijkstra algorithm to determine shortest route, Hungaria method to determine minimum bipartite matching, Prim algorithm to determine minimum spanning tree, and van Aardenne-Ehrenfest & de-Bruijn algorithm and Fleury algorithm to determine Euler circuits. The shortest Euler circuit resulted from two heuristic algorithms can be used to find a solution of SCP. The application of SCP is illustrated in establishing the minimum distance route of catering delivery.

For both synthetic datasets, we used a mixture of linear and non- linear DR techniques, namely PCA, LLE, LE, SNE, and Isomap. The neighborhood parameter for nonlinear techniques was set to 12 . First, we constructed the neighborhood graph of datasets with respect to euclidean distances between points. We performed a pre-processing in order to ensure that the distances differ. This was done due to the fact that if the edge weights in a graph are unique then the minimum spanning tree is also unique. The Prim algorithm was used to compute the tree. Second, we applied di- mension reduction to both synthetic and real datasets. For syn- thetic datasets we reduced the dimension from 3 to 2 and for real datasets from 560 to 3 . For synthetic datasets, the position of points before dimension reduction were used for 3D visualiza- tion which is not important because we are interested in distances between points. For real dataset, the 3D position of points for visualization is actually the output of dimension reduction. The computed trees for different datasets and different dimension re- duction techniques are presented in Figures 5, 6, and 7. In these figures the blue points are data points, the green lines depict the edges of minimum spanning tree constructed from neighborhood graphs of data points before dimension reduction, and the red lines depict the edges of minimum spanning tree constructed after applying dimension reduction. In these visualization, if the min- imum spanning tree does not change we should not see and red lines. Therefore, red lines show how much and where the mini- mum spanning tree is changed during dimension reduction. For example, in figure 5, we see readily that LE and Isomap exhibit better performance than PCA, LLE, and SNE. Figure 6 shows that PCA, Isomap, and LE exhibit better performance than LLE and SNE. Additionally, LE has a weaker performance relative to PCA and Isomap. These two figures shows us that the performance of a specific dimension reduction may vary from dataset to dataset. For instance, PCA exhibits a better performance in dealing with Swiss-roll than S-curve datasets. The same process was carried out on real dataset and the results are depicted in Figure 7. As it might be understood, 2D figures can not truely show us the performance of a specific dimension reduction technique. For example, in Figure 7 it is hard to compare and evaluate the perfor- mance of PCA, LE, and SNE in dealing with real dataset (the face dataset). Therefore, we additionally visualized the results in an immersive 3D virtual environment (CAVE). Some images of the CAVE while the user is looking at the results are depicted in Fig- ure 8. In immersive visualization, the use can easily go inside the data and see where the distances are not preserved. Furthermore, by zooming in and out, the user get a local or global overview of the performance of the employed dimension reduction technique.

Universitas Islam Indonesia (UII) intergrated campus computer network built since 1995. Development of UII integrated campus computer network is using a star topology and fiber optic (FO) cable. Considering that the star topology is the topology that requires a lot of wires, this study was conducted to determine and examine how the application of graph on the FO cable network UII integrated campus in order to minimize the cost, because FO cable network can be modeled by a graph where the buildings as points, while FO cable that connects to each building as a line. This type of research that is used here is a case study, in which data collection by observation, interviews, and documentation. This study used 4 algorithms, that is Kruskal algorithm, Prim, Boruvka and Solin algorithm to find the Minimum Spanning Tree. Based on the research that has been done, the conclution about the troubleshooting steps of optimization UII integrated campus FO cable network based graph theory has been got. From the four algorithms obtained the most optimal results FO cable length 4.700 meters long and is 1.590 meters cable lines. While the results of observations made, it is known that the existing computer network in UII integrated campus has a cable length of 6.120 meters and 2.050 meters long track. The results of the analysis showed that the resulrs of the study 23.2% more optimal than the existing computer networks in UII integrated campus.

yang besar perbedaannya sehingga akan diperoleh hasil subtrees yang merepresentasikan region akhir yang terbentuk (Assuncao et al., 2006). Pengolahan data untuk pengelompokan dengan metode ‘K’luster Analysis by Tree Edge Removal (SKATER) dapat dilakukan di software R menggunakan package spdep, maptools, lattice, dan gstat (Bekti R. D., 2015) atau dapat dilakukan analisis dengan implementasi library TERRALIB pada software TERRAVIEW (Gilberto, L'ubia, Ferreira, De, Cartaxo, Souza , 2008).

Ant Colony System (ACS) adalah sebuah metodologi yang dihasilkan melalui pengamatan terhadap semut. Pada algoritma ACS, semut berfungsi sebagai agen yang ditugaskan untuk mencari solusi terhadap suatu masalah optimisasi. ACS telah diterapkan dalam berbagai bidang, salah satunya adalah untuk mencari solusi optimal pada Traveling Salesman Problem (TSP).Algoritma Prim merupakan salah satu algoritma yang bekerja secara greedy. Pada algoritma Prim akan membentuk minimum spanning tree langkah per langkah. Pada setiap langkah dipilih sisi graf G yang mempunyai bobot minimum dan terhubung dengan minimum spanning tree yang telah terbentuk.Tugas akhir ini membandingkan antara Algoritma Ant Colony System dan Algoritma Prim dalam pencarian rute terpendek pada lokasi wisata kota tua Jakarta dimana seluruh tempat wisata yang ada dapat dilalui.Dengan menggunakan algoritma Ant Colony System, seorang wisatawan harus menempuh jarak 5,28 km untuk mengunjungi seluruh lokasi wisata yang ada. Sedangkan dengan menggunakan algoritma Prim,seorang wisatawan harus menempuh jarak 6,12 km.