![Introduction
Topological sorting is a common operation performed on
directed acyclic graphs (dags). It arises in numerous
applications including task scheduling, serialization and
dependency determinations, and graph drawing. The goal of this
project is to manipulate graph representations, perform
topological sorting on a dag, and use the topological numbers in
a longest path computation as well as in a heuristic solving a
scheduling problem.
Degree-based Topological Sort
Given is a dag G=(V,E), V={0, 1, … ,n-1}, having n vertices
and m edges. The graph is represented in the form of adjacency
lists. For each vertex u, Aout[u] points to a list containing all
vertices v such that <u,v> is an edge from u to v and Ain[u]
points to a list containing all vertices v such that <v,u> is an
edge from v to u. The length of list Aout[u] is the out-degree of
node u and the length of list Ain[u] is the in-degree of node u.
In any dag there exists at least one vertex having in-degree 0
(called the sources) and at least one vertex with out-degree 0
(called the sinks). The dags used in the project can have an
arbitrary number of sources, but will always have one sink. The
dag shown in Figure 1 has two sources (vertices A and G) and
one sink (vertex F).
Figure 1: A 7-vertex dag with two sources and one sink.
Topological sorting assigns to each vertex an integer (also
called its topological number). See Chapter 4.2 in the text for
details. The topological numbers represent a linear ordering of
the vertices such that for every edge <u,v> vertex u appears
before v in the ordering. For the graph in Figure 1, the ordering
A G B D C E F is a topological ordering and the topological
numbers are A=0, B=2, C=4, D=3, E=5, F=6, G=1. Note that
topological orderings are not unique.
We note that the textbook presents topological sorting as an
application of DFS (Algorithm 4.5 in Chapter 4.2). Another](https://image.slidesharecdn.com/introductiontopologicalsortingisacommonoperationperformed-221122093203-5bab4e12/85/IntroductionTopological-sorting-is-a-common-operation-performed-docx-1-320.jpg)
![approach is a non-recursive implementation based on a degree-
driven graph exploration. In this project you need to implement
the degree-based topological sorting approach described below.
All source vertices are ready to be placed. For Figure 1, vertices
A and G can be placed right away. Once a vertex u has all the
vertices incident to its incoming edges placed, u can be placed.
This observation forms the basis of degree-based topological
sorting algorithm.
The algorithm uses a queue TSQ. Whether one uses a FIFO or a
LIFO (i.e., stack) queue does not matter for determining a
topological ordering (different queues will produce different
orderings). The topological numbers are maintained in an array,
say array T.
The high-level description of the degree-driven algorithm is as
follows:
1 Make all initializations; includes a variable count set to 0 and
placing all sources in queue TSQ.
2 u = dequeue() and we set T[u]=count
3 All of u’s out-going edges are examined. For each edge <u,v>,
reduce the number of edges into v by 1. Once all incoming
edges of vertex v have received topological numbers, this count
would be zero and v is placed into TSQ.
4 Repeat steps 2 to 3 until TSQ is empty.
The needed book-keeping details are yours to work out. In
particular, you need to decide how to manage the counts into
vertices used in step 3. Keep in mind that the running time of
step 3 needs to be proportional to the number of v's outgoing
edges.
You can assume that all sinks can be reached from at least one
source. The algorithm implementation needs to run in O(n+m)
time.
Longest Path Computation and Scheduling
Assume the dag describes a manufacturing process and each
vertex corresponds to one production step. Production step u
cannot start until all production steps associated with the
incoming edges incident to u have been completed. A](https://image.slidesharecdn.com/introductiontopologicalsortingisacommonoperationperformed-221122093203-5bab4e12/85/IntroductionTopological-sorting-is-a-common-operation-performed-docx-2-320.jpg)
![1:
In: 0 2
Out: 3
2:
In:
Out: 3 0 1
3:
In: 2 0 1
Out:
Represent each dag so that it maintains for each vertex the list
of out-edges as well as in-edges. You need to define a class
Digraph that you can adapt from the one given from Princeton
library. Add additional entries as needed.
For tasks 3 and 4, the generated schedule is represented as
described in Schedule.java. The data structure resembles the
structure of adjacency lists: it is an array of lists. The size of
the array equals the production span and the list at the j-th entry
of the array contains the steps executing simultaneously at time
j (listed in arbitrary order). Do not modify Schedule.java. You
can use methods in Schedule.java to construct a schedule.
You must implement the following methods of the DagUtilities
class:
· public static int[] topologicalSort(Digraph G)
· perform topological sort on dag G and assign topological
numbers.
· public static int longestPath(Digraph G)
· return the length of the longest path in the dag G.
· public static Schedule minProdSpanSchedule(Digraph G)
· for dag G, return a schedule minimizing the production span.
· public static Schedule spanKStations(Digraph G, int k)
· for dag G and integer k, return a schedule using at most k
stations.
· public static Digraph readGraphFromFile(File file)
· Read a dag from input file and return the dag
Testing
You have been provided with several example trees which you](https://image.slidesharecdn.com/introductiontopologicalsortingisacommonoperationperformed-221122093203-5bab4e12/85/IntroductionTopological-sorting-is-a-common-operation-performed-docx-6-320.jpg)
![should use to test your code. To run our main function, use the
following command:
$ java -cp <colon separated path(s) to classes> Project5Test
[Graph_File_Path]
If you want to specify the number of stations for task 4, use the
following command:
$ java -cp <colon separated path(s) to classes> Project5Test
[Graph_File_Path] -s [Number_Of_Stations]
Analysis Questions
Your report must answer each question briefly and clearly.
1 State and explain the asymptotic performance of your
algorithm determining the topological numbers.
2 State and explain the asymptotic performance of your
algorithm determining the longest path in the dag.
3 State and explain the asymptotic performance of your
algorithm determining a schedule achieving minimum
production span.
4 State and explain the asymptotic performance of your
algorithm determining a schedule using no more than k stations.
5 Give an example of a dag consisting of at least 15 vertices on
which your k-station algorithm fails to find a schedule
minimizing the production span for k>1. Explain at what point
your algorithm makes the wrong decision.
6 Give an example of a dag consisting of at least 15 vertices on
which your k-station algorithm generates a schedule achieving
minimum production span, k>1.
7 For the dag you used in question 6: Does your algorithm
produce a schedule achieving minimum production span when
the topological ordering used is the one generated by executing
a DFS on the dag? Explain and illustrate your answer.
For the dag in file dagQ8.txt(400 vertices, 850 edges) create a
plot having k on the x-axis (starting with k=1) and the resulting
production span on the y-axis. Note that the dag has a longest
path length of 17 and thus the minimum production span
possible is 18 (it requires 107 stations). Discuss your plot.](https://image.slidesharecdn.com/introductiontopologicalsortingisacommonoperationperformed-221122093203-5bab4e12/85/IntroductionTopological-sorting-is-a-common-operation-performed-docx-7-320.jpg)
IntroductionTopological sorting is a common operation performed .docx
- 1. Introduction Topological sorting is a common operation performed on directed acyclic graphs (dags). It arises in numerous applications including task scheduling, serialization and dependency determinations, and graph drawing. The goal of this project is to manipulate graph representations, perform topological sorting on a dag, and use the topological numbers in a longest path computation as well as in a heuristic solving a scheduling problem. Degree-based Topological Sort Given is a dag G=(V,E), V={0, 1, … ,n-1}, having n vertices and m edges. The graph is represented in the form of adjacency lists. For each vertex u, Aout[u] points to a list containing all vertices v such that <u,v> is an edge from u to v and Ain[u] points to a list containing all vertices v such that <v,u> is an edge from v to u. The length of list Aout[u] is the out-degree of node u and the length of list Ain[u] is the in-degree of node u. In any dag there exists at least one vertex having in-degree 0 (called the sources) and at least one vertex with out-degree 0 (called the sinks). The dags used in the project can have an arbitrary number of sources, but will always have one sink. The dag shown in Figure 1 has two sources (vertices A and G) and one sink (vertex F). Figure 1: A 7-vertex dag with two sources and one sink. Topological sorting assigns to each vertex an integer (also called its topological number). See Chapter 4.2 in the text for details. The topological numbers represent a linear ordering of the vertices such that for every edge <u,v> vertex u appears before v in the ordering. For the graph in Figure 1, the ordering A G B D C E F is a topological ordering and the topological numbers are A=0, B=2, C=4, D=3, E=5, F=6, G=1. Note that topological orderings are not unique. We note that the textbook presents topological sorting as an application of DFS (Algorithm 4.5 in Chapter 4.2). Another
- 2. approach is a non-recursive implementation based on a degree- driven graph exploration. In this project you need to implement the degree-based topological sorting approach described below. All source vertices are ready to be placed. For Figure 1, vertices A and G can be placed right away. Once a vertex u has all the vertices incident to its incoming edges placed, u can be placed. This observation forms the basis of degree-based topological sorting algorithm. The algorithm uses a queue TSQ. Whether one uses a FIFO or a LIFO (i.e., stack) queue does not matter for determining a topological ordering (different queues will produce different orderings). The topological numbers are maintained in an array, say array T. The high-level description of the degree-driven algorithm is as follows: 1 Make all initializations; includes a variable count set to 0 and placing all sources in queue TSQ. 2 u = dequeue() and we set T[u]=count 3 All of u’s out-going edges are examined. For each edge <u,v>, reduce the number of edges into v by 1. Once all incoming edges of vertex v have received topological numbers, this count would be zero and v is placed into TSQ. 4 Repeat steps 2 to 3 until TSQ is empty. The needed book-keeping details are yours to work out. In particular, you need to decide how to manage the counts into vertices used in step 3. Keep in mind that the running time of step 3 needs to be proportional to the number of v's outgoing edges. You can assume that all sinks can be reached from at least one source. The algorithm implementation needs to run in O(n+m) time. Longest Path Computation and Scheduling Assume the dag describes a manufacturing process and each vertex corresponds to one production step. Production step u cannot start until all production steps associated with the incoming edges incident to u have been completed. A
- 3. topological ordering now represents one possible production sequence. Manufacturing times at different production steps vary in applications, but in this project we assume that every production step takes the same amount of time. When there is no limit on the number of production steps that can be carried out simultaneously, what is the minimum time in which all production steps can be completed? We refer to the time needed to complete a manufacturing process as the production span. The production span cannot be shorter than the length of the longest path from any source to the sink, plus 1 (because the longest path counts the number of edges). The longest paths is also called the critical path length (critical as reducing the production span needs to shorten the length of the longest paths in the dag.) You need to determine the length of the longest path in the dag and a manufacturing schedule which achieving minimum production span. The number of production steps that can execute at the same time is not limited. Figure 2: A 8-vertex dag with a longest path of 3. The DAG in Figure 2 has a longest path of length of 3. This means that it is possible to schedule the eight production steps in a production span of 4. Here is a possible schedule: A, (B, C, D), (E, F, G), H. At time 0, production step A is the only one executing; at time 1, three production steps are executing simultaneously, etc. At any point, no more than 3 stations are used simultaneously (and thus no more than 3 production steps execute simultaneously). Algorithm 4.10 in Section 4.4 of the text presents an O(n+m) time shortest path algorithm for a DAG. By changing the vertex relaxation, the algorithm can be changed into the needed longest path algorithm. See your class notes for more detail. Scheduling with k Stations Assume now that at most k manufacturing stations can be used at any time. You have already developed solutions for two cases: when k=1 and when the value of k can be as large as
- 4. needed. Assume k>1. It turns out that generating a schedule using at most k stations simultaneously and minimizing the production span is not an easy problem. You will implement an algorithm that is a natural generalization of the two earlier algorithms. The algorithm will never use more than k manufacturing stations, but the resulting production span may not be the smallest possible (for k stations). This means that the algorithm will make decisions on which production steps to assign to stations at times steps that may to not result in the best possible production span. However, your algorithm will be be fast - run in O(n+m) time - and will never use more than k stations. You are asked to develop an O(n+m) time algorithm for scheduling the n manufacturing steps when at any time at most k stations are available. Thus, k tasks can execute simultaneously at any time. The value of k is given. Your algorithm should assign to each vertex u a level number L(u) such that · every level contains at most k vertices, and · for every edge <u,v>, we have L(u)<L(v). The largest level represents the achieved production span. You need to work out a number of implementation details in your scheduling algorithm. Make sure it runs in O(n+m) time. Your algorithm should be based on the following approach: 1 The vertices are considered in topological order. For each vertex v, assign v to a level p if (i) level p is at least one level higher than the largest level assigned to any of v’s incoming edges, and (ii) level p does not already have k vertices assigned to it. If more than one vertex qualifies for placement, an arbitrary one is chosen. 2 Repeat the process until all vertices have been scheduled. 3 Report the schedule and the achieved production span. Summary of Tasks This project asks you to implement the following tasks. Each task needs to have an O(n+m) time implementation. 1 For a given dag G, determine the topological number of each
- 5. vertex using the degree-based topological sorting algorithm. 2 For a given dag G, determine for the length of the longest path. 3 For a given dag G, use the topological numbers and the longest paths entries to generate a schedule that completes all manufacturing steps in the shortest production span. Return the schedule. 4 For a given dag G and an integer k, 1< k < n, use the topological numbers to determine a schedule using at most k stations. Return the schedule. For those interested in using graph visualization tools, we provide the following links to resources: · Network Workbench - http://nwb.cns.iu.edu/index.html · graphviz - http://www.linuxdevcenter.com/pub/a/linux/2004/05/06/graphvi z_dot.html · Cytoscape - http://www.cytoscape.org/ Programming Assignment The skeleton code contains a number example dags. Each dag file has the following format: <number of vertices> vertices, <number of edges> edges 0: In: <incoming edges for vertice> Out: <outgoing edges for vertice> 1: In: <incoming edges for vertice> Out: <outgoing edges for vertice> ... n: In: <incoming edges for vertice> Out: <outgoing edges for vertice> For example, for a 4-vertex dag the input can look like: 4 vertices, 6 edges 0: In: 2 Out: 1 3
- 6. 1: In: 0 2 Out: 3 2: In: Out: 3 0 1 3: In: 2 0 1 Out: Represent each dag so that it maintains for each vertex the list of out-edges as well as in-edges. You need to define a class Digraph that you can adapt from the one given from Princeton library. Add additional entries as needed. For tasks 3 and 4, the generated schedule is represented as described in Schedule.java. The data structure resembles the structure of adjacency lists: it is an array of lists. The size of the array equals the production span and the list at the j-th entry of the array contains the steps executing simultaneously at time j (listed in arbitrary order). Do not modify Schedule.java. You can use methods in Schedule.java to construct a schedule. You must implement the following methods of the DagUtilities class: · public static int[] topologicalSort(Digraph G) · perform topological sort on dag G and assign topological numbers. · public static int longestPath(Digraph G) · return the length of the longest path in the dag G. · public static Schedule minProdSpanSchedule(Digraph G) · for dag G, return a schedule minimizing the production span. · public static Schedule spanKStations(Digraph G, int k) · for dag G and integer k, return a schedule using at most k stations. · public static Digraph readGraphFromFile(File file) · Read a dag from input file and return the dag Testing You have been provided with several example trees which you
- 7. should use to test your code. To run our main function, use the following command: $ java -cp <colon separated path(s) to classes> Project5Test [Graph_File_Path] If you want to specify the number of stations for task 4, use the following command: $ java -cp <colon separated path(s) to classes> Project5Test [Graph_File_Path] -s [Number_Of_Stations] Analysis Questions Your report must answer each question briefly and clearly. 1 State and explain the asymptotic performance of your algorithm determining the topological numbers. 2 State and explain the asymptotic performance of your algorithm determining the longest path in the dag. 3 State and explain the asymptotic performance of your algorithm determining a schedule achieving minimum production span. 4 State and explain the asymptotic performance of your algorithm determining a schedule using no more than k stations. 5 Give an example of a dag consisting of at least 15 vertices on which your k-station algorithm fails to find a schedule minimizing the production span for k>1. Explain at what point your algorithm makes the wrong decision. 6 Give an example of a dag consisting of at least 15 vertices on which your k-station algorithm generates a schedule achieving minimum production span, k>1. 7 For the dag you used in question 6: Does your algorithm produce a schedule achieving minimum production span when the topological ordering used is the one generated by executing a DFS on the dag? Explain and illustrate your answer. For the dag in file dagQ8.txt(400 vertices, 850 edges) create a plot having k on the x-axis (starting with k=1) and the resulting production span on the y-axis. Note that the dag has a longest path length of 17 and thus the minimum production span possible is 18 (it requires 107 stations). Discuss your plot.
- 8. Project 5: Implement four tasks on dags, each having O(n+m) time performance. 1.For a given dag G, determine the topological number of each vertex using the degree-based topological sorting algorithm. • Do not use the DFS approach, but the described degree-based approach 2.For a given dag G, determine for the length of the longest path. • Use the topological numbers and edge relaxation(on max) 17 · Longest path of length is3 · Shortest production span is4 · Possible schedule: A, (B, C, D), (E, F, G), H · Possible schedules if only two stations are available: A, (B,C), (D,E), (F,G), H A, (B,D), (C,F), (E,G, H 18 Project 5: implement four tasks on dags, each having O(n+m) time performance. 3. For a given dag G, use the topological numbers and the longest paths entries to generate a schedule that completes all manufacturing steps in the shortest production span. Return the schedule. 4. For a given dag G and an integer k, 1< k < n, use the topological numbers to determine a schedule using at most k stations. Return the schedule. · The schedule for k=1 is the topological order · No benefit when having more stations than long · About the dags and schedules Take a careful look at the input format Dags can have multiple sources, but have only one sink
- 9. Determine what is needed in class Digraph on entries and methods Class schedule cannot be changed There are other representations of schedule, but you need to use the one given 20 Analysis question 5 Give an example of a dag consisting of at least 15 vertices on which your k-station algorithm fails to find a schedule minimizing the production span for k>1. Explain at what point your algorithm makes the wrong decision. feasible and whether it generates minimum production span. whether your schedule is feasible. However, determining whether it produces minimum production span is not easy. algorithm makes a mistake.You have to do more than find a counterexample. The example has to fool your algorithm! 21 About the schedules The graph were generated by a dag library and they seem to be “schedule friendly” For k>1, two students may generate schedules having a different production span. For every production span, there generally exist a number of valid schedules. Finding an example where your algorithm fails corresponds