SE2205: Algorithms and Data Structures for Object-Oriented DesignLab Assignment 3Assigned: Mar 20, 2019; Due: April 8, 2019 @ 10:00 a.m.If you are working in a group of two, then indicate the associated student IDsand numbers in the Assignment3.java file as a comment in the header.1 ObjectivesIn this assignment, you will implement an algorithm that computes the maximum flow possible in a flownetwork digraph. A flow network can represent a network of pipelines having different flow rates. Pleaserefer to the diagram in Figure 1 for an example of a flow network. In a flow network, there are two specialnodes called the source and destination nodes (marked as S and D in Figure 1). Material such as oilfluids can propagate through the network starting from the source S and flow into the destination node D.Intermediate nodes (i.e. 1 to 7 in the example) can represent pumping stations.Figure 1: A sample flow network1.1 ConstraintsSince a flow network is a special kind of network subject to physical limitations, flow and capacity constraintsdo apply. Every edge ei,j or pipeline connected by vertices vi and vj has a maximum capacity of ci,j . Hence,due to this physical capacity limitation, any substance flowing through pipeline ei,j must have a flow offi,j that is between 0 and ci,j (i.e. 0 ≤ fi,j ≤ ci,j ). Referring back to Figure 1, e0,2 has a flow of f0,2 = 11and a capacity of 12. Since f0,2 ≤ c0,2, this is a valid flow.Next, suppose three pipelines or edges meet at a single node and two pipelines leave that node (considere2,5, e3,5, e4,5, e5,6, e5,7). Any substance flowing into a node must be equal to the substance flowing out ofthe node and this is called flow conservation. In Figure 1, f2,5 + f3,5 + f4,5 = 18 and f5,6 + f5,7 = 18 andmore generally this can be expressed as Pi∈Vfj,i for all vj with the exception of S and D asthese are source and sink nodes (generate or consume substance).Finally, there is the notion of back flow. For instance, if there exists positive flow fi,j on edge ei,j ,then it is possible for substance to flow in the opposite direction ej,i where 0 ≤ fj,i ≤ fi,j . If there is nosubstance in the pipeline then nothing can propagate in the opposite direction. However, if there is somematerial flowing through the pipe in one direction, then it is entirely possible for this material to flow in theopposite direction.1.2 General Algorithm for Finding Maximum Flow in the NetworkIf you attempt to increase the flows on edges of the flow network provided in Figure 1, you will notice that thisis not possible. Suppose you are increasing flow on e0,2 by 1 to 12. In order to satisfy flow conservation, the1flow on e2,5 must also increase by 1. This is impossible as this will violate the maximum capacity restrictionon e2,5 (i.e. maximum flow possible through this edge is 11 not 12). Hence, if you attempt to increase theflow like this on every edge, you will realize that this is not feasible. For this reason, all flows listed in Figure1 are the maximum possible flows throughout the flow network. The amount of substance leaving from node0 which is the source node is 23 and similarly the amount of substance entering the destination node 8 isalso 23. For this reason, the maximum flow throughout this flow network is 23! In this assignment, you willimplement a well known algorithm to compute the maximum flow throughout a flow network. In general,this algorithm is composed of the following steps: Find a path p from S to D that consists of no edges with flow that is equal to the full capacity of theedge (i.e. fi,j 6= ci,j ) Find the maximum flow that can be added to the path so that none of the edges in the path violatetheir flow capacity constraintsAdd this flow to every edge in the path pRepeat the above until no more paths exist in the graph from S to D to which more flow can be addedThis assignment is divided into two parts. In the first part, you will implement a breadth-first searchalgorithm to discover a path from S to D to which more flow can be added to all edges in this path. Thesepaths will be referred to as augmenting paths. In the second part, you will implement code that calls onthis path discovering algorithm repeatedly to add flow to the flow network until no more flow can be added.You can assume that you are provided with the definitions to the Graph ADT and Edge class (you canassume that all the functions provided in the ”Graphs” lecture slides are available). An Edge is composedof the following members: public int flow; public int flowCap;Edge represents an edge and stores the current flow in that edge and the maximum capacity of that edge inthe flow and flowCap members respectively.You will implement the function public int breadthFirstPathSearch(Graph FN, int s,int d) function. This function uses breadth-first search to find an augmenting path connecting vertices sand d to which flow can be added without infringing on capacity constraints of the network. Since this is abreadth-first search algorithm, you will have to use a queue in your implementation. You can assume thatall the interface functions associated with the LinkedListQueue are provided.In order to understand the general breadth-first path-finding algorithm, consider Figures 2, 3, 4. Figure2 illustrates the initial flow network which has no flow yet on any edges. The path-finding algorithm shouldfind a path connecting s and d which are source and destination nodes that consists of no edges with flowbeing equal to the maximum edge capacity. The breadth-first search algorithm attempts to find such a paththat has the shortest distance (i.e. smallest number of edges) from s to d. A breadth-first search algorithmwill start from s and examine all children/adjacent nodes of s (by enqueueing these nodes into a queue)one by one. For the network in Figure 2, node 1 will be examined first and checked to see if it has beenvisited or not and whether the current flow in the edge (v0, v1) has not reached maximum capacity. If theseconditions are met, node 1 is set to be visited and in the index corresponding to node 1 in the parent arrayis set to node 0 as node 0 is the parent of node 1 forming the edge (v0, v1) and then node 1 is enqueued intothe queue. All the children of node 0 are examined in a similar manner and enqueued into the queue if thechecking conditions are met.After examining all adjacent nodes of node 0, parent={-1, 0, 0, 0, -1, -1, -1, -1, -1}.The next node in the queue will be node 1 and its child node 4 is examined. As 4 is unvisited and has noflow, the parent of node 4 is set to node 1, marked as visited and enqueued into the queue. Proceeding inthis manner, parent of 5 is set to 2 and node 5 is enqueued into the queue. The next node examined is 3and its children are 5 and 7. Since 5 has already been visited, its parent will not change and it will notbe enqueued into the queue. However, since 7 is not visited, its parent is set to 3, marked as viSE2205留学生作业代做、Data Structures作业代写、代写Java课程设计作业、Java程序作业调试 代写sited andenqueued into the queue. This will proceed until all nodes in the graph are examined (i.e. the queue isempty). At this point, the parent array for the flow network in Figure 2 will be parent={-1, 0, 0, 0,2Figure 2: Initial network with no flowFigure 3: First Augmenting Path from S-D1, 2, 5, 3, 7}. Starting from the last index corresponding to node 8 which is the destination node, itis clear that the parent of 8 is 7, the parent of 7 is 3, the parent of 3 is 0. Hence, the augmenting path froms to d in which flow can be increased is 0-3-7-8 and this is the shortest available path from node 0 to 8 asillustrated in Figure 3.Figure 4: Second Augmenting Path from S-DSuppose that a flow of 5 has been added to the path 0-3-7-8. If the breadth-first search algorithm iscalled again to operate on this flow network, the parent of 7 will not be set to be 3 anymore as the flow on theedge (v3, v7) is 5 which is the maximum capacity of that edge (no more flow can be added without infringingon the capacity constraint of that edge). The parent of 7 in this case will be set to 5. The augmented pathresulting from this graph will then be 0-2-5-6-8 as illustrated in Figure 4. Hence, the path-finding algorithmwill successfully avoid the edge that is full. At the conclusion of the path search, the function will return 1if the destination node has been visited and 0 otherwise. If the return value is 0, then an augmenting path3does not exist that connects s to d. This breadth-first searching algorithm is called the Edmonds-Karpalgorithm. The implementation of this algorithm is summarized in the following: First set all elements in the visitedNodes array to 0 (to mark all nodes as unvisited) Initialize a queue and enqueue the starting node Begin a while-loop in which:– The queue is dequeued (say the dequeued node is vi)– For every node vj that is adjacent to the dequeued node vi, check if this adjacent node isunvisited and whether the current flow in the edge (vi, vj ) is such that more than 0 flow addedto the edge (i.e. ci,j ? fi,j > 0)– If these conditions are met, then set the parent of this adjacent node vj to be the dequeuednode vi and enqueue the adjacent node into the queue– Repeat the above until the queue is empty– Before the function terminates return 1 if d is visited and 0 otherwiseNext, you will implement the function public void maximizeFlowNetwork(Graph fN, int s,int t). This is the function that computes the maximum flow in the flow network fN. Following are thethree main components of this function: The breadthFirstPathSearch function is called to find an augmented path If such a path exists then the return value of this function is 1.(i.e. if node d is visited by the pathsearching algorithm). In this case, the maximum possible flow that can be added on all edges thatform the augmenting path is computed (this flow must heed the capacity constraints of the edge) The flow on all edges forming the augmenting path is incremented to the value computed in theprevious step. For every edge in which flow is added, ensure that you account for the back flow in theopposite direction All three steps above are repeated until no more flow can be added to the flow networkFor instance, suppose that the augmented path computed by the path searching algorithm is 0-2-5-7-8.From Figure 3, it is clear that the maximum flow that can be added/augmented to all edges in this pathis 10 due to edge (v5, v7) resulting in the flow network illustrated in Figure 4. This algorithm is called theFord-Fulkerson algorithm.Available Functions (Clarifications)If you have already completed the assignment, do not worry about this section. If you need more guidanceon the return types of the Graph ADT, then you can utilize the following function declarations (this can bedifferent): public int numVertices(); public Iterable> vertices(); public int numEdges(); public Iterable> edges(); public Edge getEdge(Vertex u,Vertex v); public Iterable> endVertices(Edge e); public Vertex opposite(Vertex v, Edge e); public int outDegree(Vertex v);4 public int inDegree(Vertex v); public Iterable> outgoingEdges(Vertex v); public Iterable> incomingEdges(Vertex v); public void insertVertex(Vertex v); public void insertEdge(Vertex u, Vertex v, E x); public void removeVertex(Vertex v); public void removeEdge(Edge e); public Vertex getVertex(int label);For the Edge ADT, the value that you are storing can be flow and flow capacity (combined into E).You can assume that the following Edge interface functions are available (if you have already made yourassumptions, then don’t worry about this). public int flowCapacity(); public int flow();Similarly, for the Vertex ADT, you can assume that the following interface functions are available: public int getLabel();For the functions: public void maximizeFlowNetwork(Graph fN, int s, int t) and publicint breadthFirstPathSearch(Graph FN, int s, int d), the last two arguments indicate thelabels/numbers assigned to the vertices (start and end nodes). You can utilize the function getVertex toobtain the reference to the vertex corresponding to the label. Or you can assume that the two argumentsare of Vertex type (change the input arguments type).GradingWe will evaluate the logic behind your function implementations (breadthFirstPathSearch andmaximizeFlowNetwork) (i.e. we will not run it). Marks will be deducted for syntax errors and logicalerrors.Code SubmissionYou will use git to submit this assignment. Create a new repository for LabAssignment3: Create a new folder SE2205B-LabAssignment3 and add the file Assignment3.java in whichyou will implement the two afore-mentioned functions. Open GitHub Desktop. Run File → Add local repository. Click Choose and browse for the folder SE2205B-LabAssignment3. You will get the following warning: This directory does not appear to be a Git repository.Would you like to create a repository here instead? Click on Create a Repository and then click on Create Repository. In the GitHub Desktop tool bar, click Publish repository, check the option Keep this codeprivate, choose your GitHub account, and then click Publish repository. Now a new repository called “SE2205B-LabAssignment3” should appear in your GitHub account. You will need to add the instructor and the TAs to this repository. For this,5– In the GitHub account click the repository SE2205B-LabAssignment3 and then click Setting– Click Collaborators & teams option.– At the bottom of the page and in the Collaborators section, enter the account id of the GitHubuser you would like to add and then click Add collaborator.– You need to add the following accounts to your repo:�ENSURE that your work satisfies the following checklist: You submit before the deadline6转自：http://www.7daixie.com/2019041013143550.html
