Graph Theory - Ankur · PDF fileGraph Theory Author: CamScanner Subject: Graph Theory
Graph Theory
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Graph Theory Graph theory is the study of the properties of graph structures. It provides us with a language with which to talk about graphs.
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Graph Theory. Graph theory is the study of the properties of graph structures. It provides us with a language with which to talk about graphs. Degree. The degree of a vertex is the number of edges incident upon it. - PowerPoint PPT Presentation
Transcript of Graph Theory
Graph Theory
Graph theory is the study of the properties of graph structures. It provides us with a language with which to talk about graphs.
Degree
• The degree of a vertex is the number of edges incident upon it.
• The sum of the vertex degrees in any undirected graph is even (twice the number of edges).
• Every graph contains an even number of odd-degree vertices.
Connectivity
• A graph is connected if there is an undirected path between every pair of vertices.
• The existence of a spanning tree is sufficient to prove connectivity.
• The vertex (edge) connectivity is the smallest number of vertices (edges) which must be deleted to disconnect the graph.
Some terms
• articulation vertex biconnected
• bridge edge-biconnected
• Testing for articulation vertices or bridges is easy via brute force.
Cycles
• Eulerian cycle (path): a tour which visits every edge of the graph exactly once.
• Actually, it is a circuit, not a cycle, because it may visit vertices more than once.
• A mailman’s route is ideally an Eulerian cycle, so he can visit every street (edge) in the neighborhood once before returning home.
• An undirected graph contains an Eulerian cycle if it is connected and every vertex is of even degree.
• A Hamiltonian cycle is a tour which visits every vertex of the graph exactly once.
• The traveling salesman problem asks for the shortest such tour on a weighted graph.
Planer Graph
• Euler’s formula:
n-m+f=2
• n:# of vertices• m:# of edges• f: # of faces• Trees: m=n-1, f=1• Cubes: n=8, m=12, f=6
MST
• Kruskal’s algorithm:
starting from a minimal edge
• Prim’s algorithm:
starting from a given vertex– How about maximum spanning tree– and Minimum Product spanning tree
Kruskal’s Algorithm
• Algorithm Kruskal(G)• Input ： G=(V, E) 為無向加權圖 (undirected weighted graph) ，
其中 V={v0,…,vn-1} • Output ： G 的最小含括樹 (minimum spanning tree, MST)• T← //T 為 MST ，一開始設為空集合• while T 包含少於 n-1 個邊 do• 選出邊 (u, v) ，其中 (u, v)E ，且 (u, v) 的加權 (weight) 最
小• E←E-(u, v)• if ( (u, v) 加入 T 中形成循環 (cycle) ) then 將 (u, v) 丟棄• else T←T(u, v)• return T
Kruskal’s Algorithm -Construct MST
Prim’s algorithm
• Algorithm Prim(G)• Input ： G=(V, E) 為無向加權圖 (undirected weighted graph) ，
其中 V={v0,…,vn-1} • Output ： G 的最小含括樹 (minimum spanning tree, MST)• T← //T 為 MST ，一開始設為空集合• X←{vx} // 隨意選擇一個頂點 vx加入集合 X 中• while T 包含少於 n-1 個邊 do• 選出 (u, v)E ，其中 uX 且 vV-X ，且 (u, v) 的加權 (weight)
最小• T←T(u, v)• X←X{v}• return T
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One-to-all shorted path
• Dijkstra演算法 : Dijkstra 演算法屬於求取單一來源 (source) 至所有目標 (destination) 頂點的一至多 (one-to-all) 最短路徑演算法
All-pair shortest path
• Floyd-Warshall 的所有頂點對最短路徑 (all-pair shortest path) 演算法：
