Graph theory

Graph theory

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  Graph Theory 1
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  History of GraphTheory Leonard Euler 1736 Gustav Kirchhoff 1847 22
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  Graph Theory  Chemicalengineering  Electrical engineering  Civil Engineering  Architectural Engineering  Computer engineering  ........ Journal of Graph Theory network representing electricity markets 3
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  Graph Definition multiple members Loopsimple graph Adjacency and Incidence Degree of a node 44
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  Graph Definition  Walk Path  Trail  Cycle A path is a walk that does not include any edge and vertex twice. A walk is an alternating sequence of vertices and connecting edges. A cycle is a path that begins and ends on the same vertex. A trail is a walk that does not pass over the same edge twice. A trail might visit the same vertex twice, but only if it comes and goes from a different edge each time. 5 ‫متن‬ ‫ترجمه‬‫م‬ ‫شما‬ ‫به‬ ‫را‬ ‫امکان‬ ‫این‬ ‫ایرانیان‬ ‫تخصصی‬ ‫ترجمه‬ ‫در‬‫ی‬ ‫ترجمه‬ ‫راحتی‬ ‫به‬ ‫را‬ ‫خود‬ ‫های‬ ‫پروژه‬ ‫و‬ ‫پاورپوینت‬ ‫انواع‬ ‫که‬ ‫دهد‬ ‫کنید‬. ‫روی‬ ‫بر‬ ‫است‬ ‫کافی‬ ‫ترجمه‬ ‫سفارش‬ ‫برای‬ http://iraniantranslate.com/index.php‫کنید‬ ‫کلیک‬
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  Graph types  Connectedgraph  Complete graph  Bipartite graph  Complete bipartite graph  Hamiltonian graph  Directed graph  Planar graph  Weighted graph  Tree  Spanning Tree  Forrest e=n(n-1)/2 6
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  Tree  Shortest PathTree (SPT)  Dijkstra's algorithm  Minimum Spanning Tree (MST)  Kruskal's algorithm  Prim's algorithm  Reverse-delete algorithm  Borůvka's algorithm 7
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  Dijkstra's algorithm  Bothdirected and undirected graphs  All edges must have nonnegative weights  Graph must be connected How do we get from 2-151 to Logan? 8
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  Pseudocode of Dijkstra'salgorithm 9
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  Example of Dijkstra'salgorithm v s a b c d d.[v] 0 ∞ ∞ ∞ ∞ Prev.[v] null null null null null Color[v] w w w w w s b a c d 2 3 7 2 1 5 4 80 ∞∞ ∞∞ 5 10
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  Example of Dijkstra'salgorithm v s a b c d d.[v] 0 ∞ ∞ ∞ ∞ Prev.[v] null null null null null Color[v] b w w w w s b a c d 2 3 7 2 1 5 4 80 ∞ 𝟕 ∞∞ 𝟐 ∞ 5 11
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  Example of Dijkstra'salgorithm v s a b c d d.[v] 0 2 7 ∞ ∞ Prev.[v] null s s null null Color[v] b b w w w s b a c d 2 3 7 2 1 5 4 80 ∞𝟕 ∞ 5 𝟐 12
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  Example of Dijkstra'salgorithm v s a b c d d.[v] 0 2 7 ∞ ∞ Prev.[v] null s s null null Color[v] b b w w w s b a c d 2 3 7 2 1 5 4 80 ∞𝟕 ∞ 𝟓 𝟏𝟎 𝟕 5 13
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  Example of Dijkstra'salgorithm v s a b c d d.[v] 0 2 5 10 7 Prev.[v] null s a a a Color[v] b b w w w s b a c d 2 3 7 2 1 5 4 80 𝟏𝟎𝟓 𝟕 5 14
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  Example of Dijkstra'salgorithm v s a b c d d.[v] 0 2 5 10 7 Prev.[v] null s a a a Color[v] b b b w w s b a c d 2 3 7 2 1 5 4 80 𝟏𝟎𝟓 𝟕 𝟔 5 15
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  Example of Dijkstra'salgorithm v s a b c d d.[v] 0 2 5 6 7 Prev.[v] null s a b a Color[v] b b b w w s b a c d 2 3 7 2 1 5 4 80 𝟔𝟓 𝟕 5 16
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  Example of Dijkstra'salgorithm v s a b c d d.[v] 0 2 5 6 7 Prev.[v] null s a b a Color[v] b b b b w s b a c d 2 3 7 2 1 5 4 80 𝟔𝟓 𝟕 5 𝟏𝟏 17
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  Example of Dijkstra'salgorithm v s a b c d d.[v] 0 2 5 6 7 Prev.[v] null s a b a Color[v] b b b b b s b a c d 2 3 7 2 1 5 4 80 𝟔𝟓 𝟕 5 18
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  Example of Dijkstra'salgorithm v s a b c d d.[v] 0 2 5 6 7 Prev.[v] null s a b a Color[v] b b b b b s b a c d 2 3 1 5 0 𝟔𝟓 𝟕 19
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  Complexity of Dijkstra'salgorithm  binary heap vs binomial heap  Fibonacci heap O(V log V + E log V) O(V log V + E) 20
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  Kruskal's algorithm (MinimumSpanning Tree (MST)) Joseph Bernard Kruskal American mathematician, statistician, computer scientist and psychometrician 21
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  Pseudocode of Kruskal'salgorithm 22
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  Kruskal's algorithm A 5 G F E C B D 7 3 12 4 9 6 15 8 10 7 23
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  Kruskal's algorithm A 5 G F E C B D 7 3 12 4 9 6 15 8 10 7 24
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  Kruskal's algorithm A 5 G F E C B D 7 3 12 4 9 6 15 8 10 7 25
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  Kruskal's algorithm A 5 G F E C B D 7 3 12 4 9 6 15 8 10 7 26
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  Kruskal's algorithm A 5 G F E C B D 7 3 12 4 9 6 15 8 10 7 27
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  Kruskal's algorithm A 5 G F E C B D 7 3 12 4 9 6 15 8 10 7 28
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  Kruskal's algorithm A 5 G F E C B D 7 3 12 4 9 6 15 8 10 7 29
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  Kruskal's algorithm A 5 G F E C B D 7 3 12 4 9 6 15 8 10 7 30
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  Kruskal's algorithm A 5 G F E C B D 3 4 6 8 7∑𝑊= 33 31
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  Complexity of Kruskal'salgorithm  first sort the edges by weight using a comparison sort in O(E log E) time this allows the step "remove an edge with minimum weight from S" to operate in constant time. Next, we use a disjoint-set data structure (Union&Find) to keep track of which vertices are in which components. 32
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  Prim's algorithm  Prim'salgorithm is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph  This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized. 33
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  Prim's algorithm 1. Startat any node in the graph  Mark the Starting node as Reached.  Mark all the other nodes in the graph as unreached. Right now, the minimum Spanning Tree (MST) consists of the starting node. 2. Find an edge e with minimum cost in the graph that connects:  A reached node x to an unreached node y. 3. Add the edge e found in the previous step to the Minimum cost spanning Tree Mark the unreached node y as reached. 4. Repeat the step 2 and 3 until all nodes in the graph have become reached 34
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  Prim's algorithm 5 A B D F E G C7 3 12 4 9 6 15 8 10 7 StartingNode Reached node Unreached node 35
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  Prim's algorithm 5 A B D F E G C7 3 12 4 9 6 15 8 10 7 StartingNode Reached node Unreached node 36
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  Prim's algorithm 5 A B D F E G C7 3 12 4 9 6 15 8 10 7 StartingNode Reached node Unreached node 37
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  Prim's algorithm 5 A B D F E G C7 3 12 4 9 6 15 8 10 7 StartingNode Reached node Unreached node 38
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  Prim's algorithm 5 A B D F E G C7 3 12 4 9 6 15 8 10 7 StartingNode Reached node Unreached node 39
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  Prim's algorithm 5 A B D F E G C7 3 12 4 9 6 15 8 10 7 StartingNode Reached node Unreached node 40
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  Prim's algorithm 5 A B D F E G C7 3 12 4 9 6 15 8 10 7 StartingNode Reached node Unreached node 41
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  Prim's algorithm 5 A B D F E G C7 3 12 4 9 6 15 8 10 7 StartingNode Reached node Unreached node 42
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  Prim's algorithm 5 A B D F E G C7 3 12 4 9 6 15 8 10 7 StartingNode Reached node Unreached node 43
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  Prim's algorithm 5 A B D F E G C7 3 12 4 9 6 15 8 10 7 StartingNode Reached node Unreached node 44
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  Prim's algorithm 5 A B D F E G C7 3 12 4 9 6 15 8 10 7 StartingNode Reached node Unreached node 45
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  Prim's algorithm 5 A B D F E G C7 3 12 4 9 6 15 8 10 7 StartingNode Reached node Unreached node 46
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  Prim's algorithm 5 A B D F E G C7 3 12 4 9 6 15 8 10 7 StartingNode Reached node Unreached node 47
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  Prim's algorithm 5 A B D F E G C 3 4 6 8 7 StartingNode Reached node Unreached node ∑𝑊 = 33 48
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  Complexity of Prim'salgorithm  V={v1, v2, v3 ,…}  For v1 : time complexity is O(V)  So you have V steps (V vertexes to add) each taking cost V, which gives you O(V^2) 49
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  Graph Partitioning (GP) Usedto:  reduce complexity  parallelization Application problems like:  social network,  road network,  air traffic control,  image analysis  etc. 50
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  Common Applications ofGraph Partitioning  Finite Element Method / Scientific Computing  Customizable Route Planning 51
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  Graph Partitioning (GP) the graph partitioning problem asks for blocks of nodes V1,. . . ,Vk that partition the node set V An example graph that is partitioned into four blocks. The partition has 17 cut edges. The weights of the blocks are blue(10), red(8), green(7) and purple(8). 52
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  Graph Partitioning (GP) An abstract view of the partitioned graph is the so called quotient graph, where nodes represent blocks and edges are induced by connectivity between blocks, i.e. there is an edge in the quotient graph if there is an edge that runs between the blocks in the original, partitioned graph. An example is given in following Figure A graph that is partitioned into three blocks of size four on the left and its corresponding quotient graph on the right. There is an edge in the quotient graph if there is an edge between the corresponding blocks in the original graph. 53
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  Objective Functions ofGraph Partitioning (GP) we often seek to find a partition that minimizes (or maximizes) an objective. Probably the most prominent objective function is to minimize the total cut. A typical model of computation and communication. 54
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  References  A. Kaveh,Structural Mechanics: Graph and Matrix Methods, Research Studies Press (John Wiley), Exeter, U.K., 1992 (first edition), 1995 (second edition), 2004 (third edition)  Singh, Rishi Pal. "Application of Graph Theory in Computer Science and Engineering." International Journal of Computer Applications 104.1 (2014).  Shekhawat, Krishnendra. "Mathematical propositions associated with the connectivity of architectural designs." Ain Shams Engineering Journal (2015).  Grimm, Veronika, et al. "Transmission and generation investment in electricity markets: The effects of market splitting and network fee regimes." European Journal of Operational Research 254.2 (2016): 493-509. 55
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  Thanks for yourattention 56