Shortest Paths (Optional)

Designing and analyzing shortest paths.

1. Seam-finding interfaces
2. Graph interfaces
3. Reference implementation
   1. Node interface
4. Design and implement
   1. GenerativeSeamFinder
   2. ToposortDAGSolver
   3. DynamicProgrammingSeamFinder
5. Analyze and compare
   1. Affordance analysis
   2. Experimental analysis
6. Above and beyond
   1. Minimum Cost to Make at least One Valid Path in a Grid

Shortest paths is not only essential for navigation directions in Husky Maps, but also essential for image processing. Seam carving is 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 cropping pixels from the edges or scaling the entire image, seam carving is considered content-aware 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 graph algorithms for image processing.
• Analyze and compare the runtimes and affordances for graph algorithms.

Seam-finding interfaces

Seam carving depends on algorithms that can find a least-noticeable horizontal seam. The seam-finding interfaces are defined in the src/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/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 (introduced 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 how each line of code in this method works starting from the method signature.

findHorizontal(Picture picture, EnergyFunction f)

The Picture represents the image we want to process. The EnergyFunction 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.

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.

1. Study the AdjacencyListSeamFinder.PixelGraph constructor, which constructs the entire graph and represents it as a Pixel[][].
2. 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.
3. Then, define the source and sink nodes.

ToposortDAGSolver

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

1. Initialize edgeTo and distTo data structures just as in DijkstraSolver.
2. List all reachable vertices in depth-first search postorder. Then, Collections.reverse the list.
3. For each node in reverse DFS postorder, relax each neighboring edge.

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.

1. Initialize a 2-d double[picture.width()][picture.height()] array.
2. Fill out the leftmost column in the 2-d array with the energy for each pixel.
3. 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.

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

1. Add the y value of the least-noticeable pixel in the rightmost column to the result.
2. Follow the path back to the left by adding the y-value of each predecessor to the result.
3. Finally, to get the coordinates from left to right, Collections.reverse the result.

Analyze and compare

Affordance analysis

How does the choice of energy function affect the effectiveness of the seam carving algorithm? The DualGradientEnergyFunction defines the energy of a pixel as a function of red, green, and blue color differences in adjacent horizontal and vertical pixels.¹ (The exact details of how the energy calculated isn’t important for the purposes of your analysis.)

However, some images don’t work well with seam carving for content-aware image resizing.

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 pictures of increasing resolution.

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

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

Above and beyond

Optionally, apply what you’ve learned by working on these project ideas.

Minimum Cost to Make at least One Valid Path in a Grid

LeetCode 1368. Minimum Cost to Make at least One Valid Path in a Grid

Given an m x n grid. Each cell of the grid has a sign pointing to the next cell you should visit if you are currently in this cell. The sign of grid[i][j] can be:

• 1 which means go to the cell to the right. (i.e go from grid[i][j] to grid[i][j + 1])
• 2 which means go to the cell to the left. (i.e go from grid[i][j] to grid[i][j - 1])
• 3 which means go to the lower cell. (i.e go from grid[i][j] to grid[i + 1][j])
• 4 which means go to the upper cell. (i.e go from grid[i][j] to grid[i - 1][j])

Notice that there could be some signs on the cells of the grid that point outside the grid.

You will initially start at the upper left cell (0, 0). A valid path in the grid is a path that starts from the upper left cell (0, 0) and ends at the bottom-right cell (m - 1, n - 1) following the signs on the grid. The valid path does not have to be the shortest.

You can modify the sign on a cell with cost = 1. You can modify the sign on a cell one time only.

Return the minimum cost to make the grid have at least one valid path.