Shortest Paths

Designing and analyzing shortest paths.

Seam finding interfaces
Graph interfaces
Reference implementation
1. Node interface
Design and implement
Analyze and compare
1. Experimental analysis

In addition to implementing navigation directions in Husky Maps, shortest paths is also useful for seam carving: a technique for image resizing where the size of an image is reduced by one pixel in height (by removing a horizontal seam) or width (by removing a vertical seam) at a time. Rather than cropping pixels from the edges or scaling the entire image, seam carving is considered content-aware image resizing because it attempts to identify and preserve the most important content in an image.

In this project, we will compare 2 graph representations, 2 graph algorithms, and 1 dynamic programming algorithm for seam carving. By the end of this project, students will be able to:

Design and implement graph representations and algorithms for seam finding.
Analyze and compare implementation runtime and adaptibility to new metrics.

Can I work with someone else on this project?

Although this project requires an individual submission, you may find it useful to discuss overarching ideas with your classmates. Our primary rule is that we ask that you do not claim to be responsible for work that is not yours. If you get some help from someone else or from an online resource, cite it. I believe that there is a lot of value in learning from others so long as you do not deprive yourself (or others) of the opportunity to learn.

We are comfortable doing this because each submission in this class comes in the form of a video that you record. Your video is a demonstration of everything that you learned throughout the process of working on an assignment. Our goal is for students to support each other and find community through this course. The real advantage of taking a course on-campus at a university is to be able to connect with others who share common interests in learning.

What am I submitting at the end of this project?

Satisfactory completion of the project requires one Canvas submission including: a video-recorded individual presentation, a link to your filled-out slide template for Shortest Paths and a link to your src/main/java folder on GitLab from your projects repo that addresses all deliverables (in green).

The project instructions contain a lot of details to provide context, clarify common confusions, and help students get started. Your Canvas submissions only need to address tasks that are described in green callouts like this one.

Your video presentation should meet the following requirements:

Your presentation should not be much longer than 7 minutes (8 minutes maximum) and should include your voiceover. (Your video is appreciated but not necessary.)
Your video presentation must include or use our provided slide template for Shortest Paths. This is done intentionally to help guide you toward the correct solution.
After submitting to Canvas, add a submission comment linking to your slides or document and a link to your src/main/java folder on GitLab. Remember to commit and push to Gitlab and check the associated status for your latest Pipeline is a green checkmark! Please be sure to check access permissions on your slides and video, so any TA can view your work.

Students are expected to follow Washington state law on the Student Conduct Code for the University of Washington. In this course, students must:

Indicate on assignment submissions any assistance received, including materials distributed in this course.
Not receive, generate, or otherwise acquire any substantial portion or walkthrough for an assignment.
Not aid, assist, attempt, or tolerate prohibited academic conduct in others.

Within the provided slide template, you must cite any external sources you used, either through collaboration or online resources. If you consulted any Generative AI on this project, you must include all prompts and outputs. Online sources should be cited with a simple link. Collaboration with peers must include names and the extent of that collaboration. The more detail, the better!

Submitted work that is not consistent with sources will require makeups in Office Hours.

Seam finding interfaces

The SeamCarver class depends on algorithms that can find a least-noticeable horizontal or vertical seam. The interfaces for seam finding are defined in the src/main/java/seamfinding folder.

SeamFinder: An interface specifying a single method, findHorizontal, for finding a least-noticeable horizontal seam in a given Picture according to the EnergyFunction. The horizontal seam is returned as a list of integer indices representing the y-value (vertical) pixel indices for each pixel in the width of the picture.
Picture: A class representing a digital image where the color of each pixel is an int. In image processing, pixel (x, y) refers to the pixel in column x and row y where pixel (0, 0) is the upper-left corner and the lower-right corner is the pixel with the largest coordinates.

This is opposite to linear algebra, where (i, j) is row i column j and (0, 0) is the lower-left corner.

EnergyFunction: An interface specifying a single method, apply, for computing the importance of a given pixel in the picture. The higher the energy of a pixel, the more noticeable it is in the picture.

Seam finder implementations work by applying the EnergyFunction to each pixel in the given Picture. Then, we can use a shortest paths algorithm to find the least-noticeable horizontal seam from the left side of the picture to the right side of the picture.

Graph interfaces

The graph interfaces and algorithms are defined in the src/main/java/graphs folder.

Graph: An interface representing a directed weighted graph with a single method that returns a list of the neighbors of a given vertex. The directed Edge class provides 3 fields: the origin vertex from, the destination vertex to, and the edge weight.
ShortestPathSolver: An interface for finding a shortest paths tree in a Graph. Implementations of this interface must provide a public constructor that accepts two parameters: a Graph and a start vertex. The solution method returns the list of vertices representing a shortest path from the start vertex to the given goal vertex.

The generic type V is used throughout the graphs package to indicate the vertex data type. For seam carving, all vertices will be of the interface type Node (described below).

Reference implementation

AdjacencyListSeamFinder implements SeamFinder by building an adjacency list graph representation of the picture and then running a single-source shortest paths algorithm to find a lowest-cost horizontal seam.

public List<Integer> findHorizontal(Picture picture, EnergyFunction f) {
    PixelGraph graph = new PixelGraph(picture, f);
    List<Node> seam = sps.run(graph, graph.source).solution(graph.sink);
    seam = seam.subList(1, seam.size() - 1); // Skip the source and sink nodes
    List<Integer> result = new ArrayList<>(seam.size());
    for (Node node : seam) {
        // All remaining nodes must be Pixels
        PixelGraph.Pixel pixel = (PixelGraph.Pixel) node;
        result.add(pixel.y);
    }
    return result;
}

Here’s an explanation of each line of code in this method. Given a Picture and an EnergyFunction that defines the way that we want to measure the importance of each pixel:

PixelGraph graph = new PixelGraph(picture, f): The first line creates a PixelGraph where each vertex represents a pixel and each edge represents the energy cost for the neighboring pixel. The PixelGraph constructor creates a new Pixel (node) for each pixel in the image. It also creates a source node and a sink node.
List<Node> seam = sps.run(graph, graph.source).solution(graph.sink): sps.run calls the ShortestPathSolver (such as DijkstraSolver) to find the shortest path in the PixelGraph from the source and immediately asks for a shortest path to the sink. The seam stores the solution to the shortest path problem.
seam = seam.subList(1, seam.size() - 1): The seam includes the source and sink, which we don’t need in our final solution.
for (Node node : seam) { ... }: Since the remaining nodes in the seam must be Pixel nodes, add each pixel’s y index to the result list and return the result.

Look inside the AdjacencyListSeamFinder class to find a static nested class called PixelGraph. PixelGraph implements Graph<Node> by constructing a 2-dimensional grid storing each Node in the seam carving graph. This class also includes two fields called source and sink.

Node interface

Node is an interface that adapts the Graph.neighbors method for use with different types of nodes in the AdjacencyListSeamFinder. This is helpful because not all nodes are the same. Although most nodes represent pixels that have x and y coordinates, the graph also contains source and sink nodes that don’t have x or y coordinates! Using the Node interface allows for these different kinds of nodes to all work together in a single program.

Inside of the PixelGraph class, you’ll find the three types of nodes implemented as follows.

source: A field that implements Node.neighbors by returning a list of picture.height() outgoing edges to each Pixel in the first column of the picture. The weight for each outgoing edge represents the energy of the corresponding pixel in the leftmost column.
Pixel: An inner class representing an (x, y) pixel in the picture with directed edges to its right-up, right-middle, and right-down neighbors. Most pixels have 3 adjacent neighbors except for pixels at the boundary of the picture that only have 2 adjacent neighbors. The weight of an edge represents the energy of the neighboring to-side pixel.
sink: A field that implements Node.neighbors by returning an empty list. It has one incoming edge from each Pixel in the rightmost column of the picture.

The source and sink are defined using a feature in Java called anonymous classes. Anonymous classes are great for objects that implement an interface but only need to be instantiated once.

Design and implement

Design and implement 1 graph representation, 1 graph algorithm, and 1 dynamic programming algorithm for seam finding.

Make sure to pass all the tests for all 5 implementations, push your code to Gitlab, and check that the associated status for your latest Pipeline is passing. Include a link to the passing pipeline of your projects repository in the provided slide template.

GenerativeSeamFinder

A graph representation that implements SeamFinder. Similar to AdjacencyListSeamFinder but rather than creating the neighbors for every node in the PixelGraph constructor ahead of time, this approach only creates vertices and edges when Pixel.neighbors is called.

To approach this implementation, consider the following steps:

Study the AdjacencyListSeamFinder.PixelGraph constructor, which constructs the entire graph and represents it as a Pixel[][].
Adapt the ideas to implement GenerativeSeamFinder.PixelGraph.Pixel.neighbors, which should return the list of neighbors for a given Pixel. Create a new Pixel for each neighbor.
- For this method, you should identify where the code in AdjacencyListSeamFinder connects a given Pixel to its right-up, right-middle, and right-down neighbors.
- You should also be wary of any potential edge cases, based on the restrictions of the picture’s height and width, as well as the source and sink nodes.
Then, define the source and sink nodes. You should implement their respective Node.neighbors methods.
- Recall that the source node serves as the starting point for establishing the leftmost column of pixels. What are the bounds of your picture? Which edges must be created?
- Recall that the sink node serves as the ending point from the rightmost column of pixels. Accordingly, how should it implement the method?

The slides below show how a PixelGraph is constructed in AdjacencyListSeamFinder and explain how this differs from GenerativeSeamFinder. Click on the image below to advance to the next slide.

Walk through how you adapted the code from AdjacencyListSeamFinder.PixelGraph to implement GenerativeSeamFinder.PixelGraph.Pixel.neighbors. What are the key similarities and differences between these two implementations?

ToposortDAGSolver

A graph algorithm that implements ShortestPathSolver. Finds a shortest paths tree in a directed acyclic graph (DAG) using topological sorting.

You may find it useful to consult the Shortest Path Trees lesson and review the Topological Sorting algorithm.

Initialize edgeTo and distTo data structures just as in DijkstraSolver.
- Recall that edgeTo is a Map used to store the previous edge used to reach some given to vertex. We trace through the values of this Map from goal until we reach some desired start node (as backpointers), then reverse the output to determine the path from start to goal.
- Recall that distTois a Map used to store the summed shortest distance from start up to some given to vertex. Based on the SPT we create for some start node, we can extract any value to determine the length of the path to reach that goal node.
List all reachable vertices in depth-first search postorder. Then, Collections.reverse the list.
- Recall that DFS postorder adds vertices to the result list once you reach the dead end of a path, at a node whose neighbors have all been visited. This differs from preorder, where nodes are added to the result list as soon as they are reached.
For each node in reverse DFS postorder, relax each neighboring edge.
- This means that the process of creating your SPT and determining the shortest possible edges to use should start from the end of your result list that you create using postorder DFS. This is to ensure you respect all dependencies in the graph and visit nodes in order from “parent” to “child” in a hierarchical manner.

Edge relaxation: If the new distance to the neighboring node using this edge is better than the best-known distTo the node, update distTo and edgeTo accordingly.

DynamicProgrammingSeamFinder

A dynamic programming algorithm that implements SeamFinder. The dynamic programming approach processes pixels in a topological order: start from the leftmost column and work your way right, using the previous columns to help identify the best shortest paths. The difference is that dynamic programming does not create a graph representation (vertices and edges) nor does it use a graph algorithm.

How does dynamic programming solve the seam finding problem? We need to first generate a dynamic programming table storing the accumulated path costs, where each entry represents the total energy cost of the least-noticeable path from the left edge to the given pixel.

Initialize a 2-d double[picture.width()][picture.height()] array.
- Recall that (x, y) refers to column x and row y.
Fill out the leftmost column in the 2-d array with the energy for each pixel.
- You may find inspiration from your implementation of Node.neighbors in GenerativeSeamFinder for the source node.
For each pixel in each of the remaining columns, determine the lowest-energy predecessor to the pixel: the minimum of its left-up, left-middle, and left-down neighbors. Compute the total energy cost to the current pixel by adding its energy to the total cost for the least-noticeable predecessor.
- Consider how the loops in GenerativeSeamFinder and AdjacencyListSeamFinder overlap with this process of computing energy pixel-by-pixel for a given column by looking at its previous 3 (or 2) neighbors.
- Be sure to include the energy cost of the current pixel as well.
- It is preferred that you create a separate method for this table filling process entirely.

The slides below show how the DynamicProgrammingSeamFinder implementation works and highlight key ideas to account for in your code. Click on the image below to advance to the next slide.

Once we’ve generated this table, we can use it to find the shortest path.

Add the y value of the least-noticeable pixel in the rightmost column to the result.
- We can simply determine the minimum value in this column first.
Follow the path back to the left by adding the y-value of each predecessor to the result.
- To trace through each predecessor and its respective neighbors, consider what extra field you should store besides the value of the minimum energy cost. Recall what needs to be returned for the findHorizontal method in a SeamFinder.
- It is preferred that you create a separate method for this minimum seam process entirely.
Finally, to get the coordinates from left to right, Collections.reverse the result.

Walk through your implementation for DynamicProgrammingSeamFinder. First, discuss the portion of your code that fills the table with energy costs. How does it select the minimum from its neighbors? How does the code handle pixels at the edges of the image? Next, discuss the portion of your code that determines the shortest path and finds the seam with the lowest energy cost. How is the first minimum found in the rightmost column? Afterwards, how do we continue to select previous neighbors with the minimum energy cost? What is ultimately returned from your findHorizontal method?

Analyze and compare

Experimental analysis

Run the provided RuntimeExperiments to compare the real-world runtime of each implementation. For each implementation, RuntimeExperiments constructs an empty instance and records the number of seconds to findHorizontal through randomly-generated square-sized pictures of increasing resolution.

The first column denotes N, the image dimensions (or resolution) in pixels.
The second column denotes the average runtime for findHorizontal in seconds.

RuntimeExperiments are disabled by default. Comment-out the @Disabled tag above the RuntimeExperiments class header to enable it.

Copy-paste the text into plotting software such as Desmos. Plot the runtimes of all 5 approaches on the same graph.

AdjacencyListSeamFinder(DijkstraSolver::new)
AdjacencyListSeamFinder(ToposortDAGSolver::new)
GenerativeSeamFinder(DijkstraSolver::new)
GenerativeSeamFinder(ToposortDAGSolver::new)
DynamicProgrammingSeamFinder()

Once you are done running RuntimeExperiments, please uncomment the @Disabled line to ensure that our pipeline to check your tests will work!

Display a legible and labelled plot to compare runtimes across all 5 implementations. Which implementation is the fastest, and why?

Determine how differences in SeamFinders and ShortestPathSolvers impact their respective runtimes. How do the two SeamFinders differ in computing neighbors? How do the two ShortestPathSolvers differ in the number of times they traverse the PixelGraph? Based on these properties, discuss how combinations of SeamFinders and ShortestPathSolvers cause certain pairings to be favorable, and others to be unfavorable.

Through this analysis, you should speak to the results of all 4 pairings.

Consider the following questions to facilitate your discussion:

What differences between ToposortDAGSolver and DijkstraSolver might affect runtimes? How many times does each implementation traverse the PixelGraph?
- Consider the difference in the number of steps required between these approaches.
- Consider the total number of vertices explored in the two algorithms.
What differences between GenerativeSeamFinder and AdjacencyListSeamFinder might affect runtimes? How might the runtimes of their respective neighbors() methods compare?
- Based on the number of traversals, consider which SeamFinder may be faster.
- Based on the number of vertices explored, consider which SeamFinder may be faster.
Although we don’t require asymptotic analysis for this project, does the experimental analysis agree with your expected asymptotic analysis?