East Los Angeles College, PJH 503
Excerpt: ... Introduction Critical path analysis Minimum spanning tree Mathematical Modelling Lecture 9 Networks II Phil Hasnip pjh503@york.ac.uk November 21, 2008 Phil Hasnip Mathematical Modelling Introduction Critical path analysis Minimum spanning tree Overview of Course Model construction dimensional analysis Experimental input tting Finding a best answer optimisation Tools for constructing and manipulating models networks, differential equations, integration Tools for constructing and simulating models randomness Real world difculties chaos and fractals The material in this lecture is not in the course textbook. Phil Hasnip Mathematical Modelling Introduction Critical path analysis Minimum spanning tree What is a network? A network is any system of interconnected locations Locations usually less important than the connections Using networks we can answer questions like What is the shortest/cheapest route between two points? Are there any bottle ...
CSU Long Beach, CECS 328
Excerpt: ... Lecture 18: Minimum Spanning Tree s Consider a problem in which roads are to be built that connect all four cities a, b, c, and d to one another. In other words, after the roads are built, it will be possible to drive from one city to another. The costs of building a road between any two cities are given in the following table. cities a b c d a b d d 30 20 50 50 10 75 Using this table, nd a set of roads of minimum cost that will connect the cities. The above problem represents an example of nding a minimum spanning tree of a network graph G = (V, E, c). A minimum spanning tree (mst) is a tree T G that is a subgraph of G for which eT c(e) is minimum. Minimum spanning tree s have been used in many dierent contexts, including weather data interpretation (the graph of which consists of selected points from various weather contour graphs, along with edges that connect points that are close in proximity) skin cancer research (the vertices of the complete graph consist of nuclei in the epithe ...
Cornell, ENGRI 1101
Excerpt: ... ENGRI 115 Engineering Applications of OR Fall 2007 The Minimum Spanning Tree Problem Prelab 1 Name: Objectives: Introduce students to the graph theoretic concept of spanning trees. Show three different combinatorial algorithms for solving the minimum spanning tree problem. Demonstrate a practical use of minimum spanning tree s. Key Ideas: graph subgraph, spanning subgraph, connected subgraph tree greedy algorithm minimality sensitivity analysis Prelab Exercise: Please write your answers on the back of this sheet. Consider the following input for the minimum spanning tree problem. 2 10 11 5 5 11 11 10 22 5 3 23 1 4 40 1. Find the minimum spanning tree in this graph, and give a very simple argument why it is optimal. 2. It turns out that our input was more complicated: only nodes 1 through 4 need to be connected. We may include node 5 if this yields a cheaper solution, but we don't need to. Node 5 is called a Steiner node. We wish to compute the minimum-cost tree that connects ...
University of Illinois, Urbana Champaign, CS 473
Excerpt: ... Algorithms Lecture 15: Minimum Spanning Tree s We must all hang together, gentlemen, or else we shall most assuredly hang separately. Benjamin Franklin, at the signing of the Declaration of Independence (July 4, 1776) It is a very sad thing that nowadays there is so little useless information. Oscar Wilde A ship in port is safe, but that is not what ships are for. Rear Admiral Grace Murray Hopper 15 15.1 Minimum Spanning Tree s Introduction Suppose we are given a connected, undirected, weighted graph. This is a graph G = (V, E) together with a function w : E IR that assigns a weight w(e) to each edge e. For this lecture, well assume that the weights are real numbers. Our task is to nd the minimum spanning tree of G, i.e., the spanning tree T minimizing the function w(T ) = w(e). eT To keep things simple, Ill assume that all the edge weights are distinct: w(e) = w(e ) for any pair of edges e and e . Distinct weights guarantee that the minimum spanning tree of the graph is unique. ...
Maryland, CMSC 132
Excerpt: ... CMSC 132 Quiz 3 Worksheet The next quiz of the course will be on Monday, April 24 during your lab (discussion) session. The following list provides more information about the quiz: The quiz will be a written quiz (no computer). Closed book, closed notes quiz. Answers must be neat and legible. We recommend that you use pencil and eraser. The following exercises cover the material to be included in this quiz. Solutions to these exercises will not be provided, but you are welcome to discuss your solutions with TAs and instructors during office hours. Exercises Using the following graph answers the questions that follow. 5 11 c 3 b 1 4 7 d a 6 12 2 2 14 e f 10 1. Minimum Spanning Tree (Using Prim's Algorithm) When several node choices are available, use alphabetical order to choose a node to process. a. Generate the minimum spanning tree for the above graph using Prim's algorithm. Use a as the start vertex. Run Prim's algorithm using b as the start vertex. Indicate the cost and predecessor for ea ...
Maryland, CMSC 132
Excerpt: ... ds to add and remove items to a queue without data races or deadlock. Minimum Spanning Tree s Using the following graph answers the questions that follow. 5 11 c 3 b 1 4 7 d a 6 12 2 2 14 e f 10 9. Minimum Spanning Tree (Using Prim's Algorithm) When several node choices are available, use alphabetical order to choose a node to process. a. Generate the minimum spanning tree for the above graph using Prim's algorithm. Use a as the start vertex. Run Prim's algorithm using b as the start vertex. Indicate the cost and predecessor for each node in the graph, after three nodes have been added to the set of processed nodes (set S in the lecture slides). You don't need to draw a tree. b. 10. Minimum Spanning Tree (Using Kruskal's Algorithm) a. b. Generate the minimum spanning tree for the above graph using Kruskal's algorithm. Run Kruskal's algorithm in the above graph and draw the graph we will have after only four edges have been processed. ...
Cornell, ENGRI 1101
Excerpt: ... ENGRI 1101: Engineering Applications of OR, Fall 2008 Prelab 3: The Minimum Spanning Tree Problem Name: Netid: Objectives: Introduce students to the graph theoretic concept of spanning trees. 1 Show three dierent combinatorial algorithms for solving the minimum spanning tree problem. Demonstrate a practical use of minimum spanning tree s. Key Ideas: graph, subgraph, spanning subgraph, connected subgraph, tree, greedy algorithm, minimality, sensitivity analysis Reading Assignment: Read Handout 4 on the minimum spanning tree problem. Prelab exercise: Please write your answer on the back of this sheet. Consider the following input for the minimum spanning tree problem. 2 10 11 5 5 11 11 10 22 5 3 23 1 4 40 1. Find the minimum spanning tree in this graph, and give a very simple argument why it is optimal. 2. It turns out that our input was more complicated: only nodes 1 through 4 need to be connected. We may include node 5 if this yields a cheaper solution, but we dont need to. Node 5 is c ...
Cornell, ENGRI 1101
Excerpt: ... ENGRI 115 Engineering Applications of OR Fall 2007 The Minimum Spanning Tree Problem Homework 1 Due date: Thursday, September 6, 2007 ; by 10.10am at the beginning of the lecture. Place your homework in the box that I will provide in the class room. References: Lecture notes, Recitation 1, Winston chapter 8.6. The first two problems are practice problems. DO NOT hand them in. Your TAs will be glad to help you if you have any questions. 1. Winston, page 459, Problem 1. 2. Winston, page 459, Problem 2. 3. For this problem we will consider Example 8, page 457458 in the Winston book (same example as in the lecture). (a) Is the optimum solution to this problem unique? If yes, explain why, if not, give a different optimum solution. (b) Considering the optimum spanning tree given on page 458 (Figure 49d), how much must the cost on edge {4, 5} increase in order to force it out of the optimum spanning tree? Which edge will enter to replace {4, 5} when this happens? (c) Considering again the solution on F ...
CSU Long Beach, CECS 328
Excerpt: ... Lecture 18: Minimum Spanning Tree s Consider a problem in which roads are to be built that connect all four cities a, b, c, and d to one another. In other words, after the roads are built, it will be possible to drive from one city to another. The costs of building a road between any two cities are given in the following table. cities a b c d a b d d 30 20 50 50 10 75 Using this table, nd a set of roads of minimum cost that will connect the cities. The above problem represents an example of nding a minimum spanning tree of a network graph G = (V, E, c). A minimum spanning tree (mst) is a tree T G that is a subgraph of G for which eT c(e) is minimum. Minimum spanning tree s have been used in many dierent contexts, including weather data interpretation (the graph of which consists of selected points from various weather contour graphs, along with edges that connect points that are close in proximity) skin cancer research (the vertices of the complete graph consist of nuclei in the epithe ...
Berkeley, CS 174
Excerpt: ... CS174 Spring 98 Lecture 19 Summary Minimum Spanning Tree s Remember the minimum spanning tree problem you are given a graph G with weighted edges (real values on each edge) and the goal is to find a spanning tree T whose total weight is minimal. The minimum spanning tree is the least expensive way to connect up all the nodes. In the graph below, the minimum spanning tree is shown with heavy shaded edges. 5 12 13 17 10 8 11 6 8 4 14 Minimum spanning tree s are useful for planning computer networks and for certain other kinds of resource and layout problems. Deterministic Algorithms You have hopefully seen two deterministic algorithms for computing the minimum spanning tree : Prim's and Kruskal's algorithms. Both are greedy algorithms. They build the spanning tree by adding at each step the minimum weight edge that satisfies a certain property. Specifically: Prim's Algorithm Input: G = (V,E) which is a connected graph Output: T, the minimum spanning tree T empty graph For i = 1 to |V| do Let e be the m ...
RIT, CS 233
Excerpt: ... Shortest paths single source [ Section 13.6 ] Given a positive-weighted graph G and its node s, compute the shortest distance from s to every other node. Question: how to do this for unweighted graphs ? Shortest paths single source Dijkstra's algorithm [ Section 13.6 ] - Why it works ? - Running time ? Minimum spanning tree s [ Section 13.7 ] Given is a weighted graph G, find a spanning tree T of G with the smallest possible weight (where weight of T is the sum of its edge weights). Minimum spanning tree s Kruskal's algorithm [ Section 13.7 ] Note: - A greedy algorithm - Why it works ? - Running time ? Minimum spanning tree s Prim-Jarnik algorithm [ Section 13.7 ] Note: - A greedy algorithm - Why it works ? - Running time ? ...
University of Illinois, Urbana Champaign, CS 473
Excerpt: ... Algorithms Lecture 12: Minimum Spanning Tree s We must all hang together, gentlemen, or else we shall most assuredly hang separately. Benjamin Franklin, at the signing of the Declaration of Independence (July 4, 1776) It is a very sad thing that nowadays there is so little useless information. Oscar Wilde Computers are useless. They can only give you answers. Pablo Picasso A ship in port is safe, but that is not what ships are for. Rear Admiral Grace Murray Hopper 12 12.1 Minimum Spanning Tree s Introduction Suppose we are given a connected, undirected, weighted graph. This is a graph G = (V, E) together with a function w : E IR that assigns a weight w(e) to each edge e. For this lecture, well assume that the weights are real numbers. Our task is to nd the minimum spanning tree of G, i.e., the spanning tree T minimizing the function w(T ) = eT w(e). To keep things simple, Ill assume that all the edge weights are distinct: w(e) = w(e ) for any pair of edges e and e . Distinc ...
UCSD, CSE 101
Excerpt: ... CSE 101 - A LGORITHMS - S UMMER 2000 Lecture Notes 8 Wednesday, July 26, 2000. 62 Todays material can be also found in Chapter 4 of the textbook (Skiena). 5.1 Minimum Spanning Tree A tree is a connected graph without cycles. The following is an immediate property of trees: Proposition 5.1 If T = VT ; ET is a tree then jVT j = jET j + 1. A spanning tree of G = V ; E is a tree T = VT ; ET such that VT = V and ET E . If G is a weighted graph, so there is a function w : E ! R, then a minimum spanning tree of G is a spanning tree of minimum total weight. 5.1.1 Prims Algorithm See subsection 4.7.1. in the textbook. 5.1.2 Kruskals Algorithm See subsection 4.7.2 in the textbook. 5.2 Shortest Paths See section 4.8 in the textbook. 5.2.1 Dijkstras Algorithm See subsection 4.8.1 in the textbook. ...
Wellesley College, CS 331
Excerpt: ... Updating Minimum Spanning Tree in Parallel Updating Minimum Spanning Tree in Parallel Updating Minimum Spanning Tree in Parallel Updating Minimum Spanning Tree in Parallel a = min {a,b} b = min {a,b} Updating Minimum Spanning Tree in Parallel a = min{a,b,d} b = min{a,b,d} d = min{a,b,d} Updating Minimum Spanning Tree in Parallel ...
Cornell, INFO 295
Excerpt: ... The Minimum Spanning Tree Problem (plagiarized from Kleinberg and Tardos, Algorithm design, pp 142149) Recall that a minimum spanning tree (V, T ) of a graph G = (V, E) with weighted links is a spanning tree with minimum total weight. This solves, for example, the problem of constructing the lowest cost network connecting a set of sites, where the weight on the link represents the cost. Such a tree can be found many greedy algorithms, including these three: 1) Kruskals algorithm: build a spanning tree by successively inserting edges in order of increasing cost, as long as each new added edge does not create a cycle. 2) Prims algorithm: start at a root node and grow a spanning tree by attaching successive least cost edges directly to the partial tree being created. 3) Reverse Delete algorithm: start with the full graph (V, E) and delete edges in order of decreasing cost, as long as doing so does not disconnect the graph. [Note that Kruskals algorithm was slightly misstated in the previous not ...
Berkeley, CS 17
Excerpt: ... CS174 Lecture 17 Minimum Spanning Tree s Remember the minimum spanning tree problem from CS170 you are given a graph G with weighted edges (real values on each edge) and the goal is to find a spanning tree T whose total weight is minimal. The minimum spanning tree is the least expensive way to connect up all the nodes. In the graph below, the minimum spanning tree is shown with heavy shaded edges. 5 12 17 10 8 4 8 13 11 6 14 Minimum spanning tree s are useful for designing networks: computer networks, communication networks, and distribution networks for utilities like electrical power, gas, water etc. The edges represent links between sites, and the edge weight is the cost of connecting those sites. The minimum spanning tree is the cheapest structure that connects all the sites. You usually want some redundancy (extra paths) in such a network but the min spanning tree is a good starting point for designing an inexpensive network. Deterministic Algorithms You have hopefully seen two deterministic alg ...
Berkeley, CS 174
Excerpt: ... CS174 Lecture 17 Minimum Spanning Tree s Remember the minimum spanning tree problem from CS170 you are given a graph G with weighted edges (real values on each edge) and the goal is to find a spanning tree T whose total weight is minimal. The minimum spanning tree is the least expensive way to connect up all the nodes. In the graph below, the minimum spanning tree is shown with heavy shaded edges. 5 12 13 17 10 8 11 6 8 4 14 Minimum spanning tree s are useful for designing networks: computer networks, communication networks, and distribution networks for utilities like electrical power, gas, water etc. The edges represent links between sites, and the edge weight is the cost of connecting those sites. The minimum spanning tree is the cheapest structure that connects all the sites. You usually want some redundancy (extra paths) in such a network but the min spanning tree is a good starting point for designing an inexpensive network. Deterministic Algorithms You have hopefully seen two deterministic alg ...
Pittsburgh, BUSQOM 0050
Excerpt: ... o destination. Example: Ray Design o Must transport beds, chairs, and other furniture items daily from factory to warehouse. o Involves going through several cities (nodes). o Find path with shortest distance in miles. LP Formulation: 3 3. Minimum Spanning Tree (MST) o Minimum-spanning tree problem determines path through network that connects all points. o Most common objective is to minimize total distance of all arcs used in path. o Example: determine best way to connect all of houses to electrical power in a way that minimizes total distance or length of power lines. Example: o Determine least expensive way to provide water and power to each house. o Distance between each house (in hundreds of feet) is shown on network. o Distance between houses 1 and 2 is 300 feet (shown by the 3 on arc connecting houses 1 and 2). o Minimum-spanning tree model to be used to determine minimum total length (of water pipes or power cables) needed to connect all the houses. 4 Procedure for solving MST: ...