Graph Algorithms
Shortest paths trees, dynamic programming, and minimum spanning trees.
Shortest Paths Trees
- Trace Dijkstra’s algorithm for shortest paths trees.
- Explain why Dijkstra’s algorithm might not work with negative edge weights.
- Explain the runtime for Dijkstra’s algorithm in terms of priority queue operations.
Breadth-first search (BFS) can be used to solve the unweighted shortest paths problem, of which we’ve studied 3 variations:
- Single-pair shortest paths
- Given a source vertex s and target t, what is the shortest path from s to t?
- Single-source shortest paths
- Given a source vertex s what are the shortest paths from s to all vertices in the graph?
- Multiple-source shortest paths
- Given a set of source vertices S, what are the shortest paths from any vertex in S to all vertices in the graph.
In the context of BFS and unweighted shortest paths, the metric by which we define “shortest” is the number of edges on the path. But, in Husky Maps, we need a metric that is sensitive to the fact that road segments are not all the same length. In Husky Maps, the weight of an edge represents the physical distance of a road segment. Longer road segments have larger weight values than shorter road segments.
- Weighted shortest paths problem
- For the single pair variant: Given a source vertex s and target vertex t what is the shortest path from s to t minimizing the sum total weight of edges?
Dijkstra’s algorithm
Dijkstra’s algorithm is the most well-known algorithm for finding a weighted SPT in a graph.
Dijkstra’s algorithm gradually builds a shortest paths tree on each iteration of the while loop. The algorithm selects the next unvisited vertex based on the sum total cost of its shortest path.
public class DijkstraSolver<V> implements ShortestPathSolver<V> {
private final Map<V, Edge<V>> edgeTo;
// Maps each vertex to the weight of the best-known shortest path.
private final Map<V, Double> distTo;
/**
* Constructs a new instance by executing Dijkstra's algorithm on the graph from the start.
*
* @param graph the input graph.
* @param start the start vertex.
*/
public DijkstraSolver(Graph<V> graph, V start) {
edgeTo = new HashMap<>();
distTo = new HashMap<>();
MinPQ<V> perimeter = new DoubleMapMinPQ<>();
perimeter.add(start, 0.0);
// The shortest path from the start to the start requires no edges (0 cost).
edgeTo.put(start, null);
distTo.put(start, 0.0);
Set<V> visited = new HashSet<>();
while (!perimeter.isEmpty()) {
V from = perimeter.removeMin();
visited.add(from);
for (Edge<V> edge : graph.neighbors(from)) {
V to = edge.to;
// oldDist is the weight of the best-known path not using this edge.
double oldDist = distTo.getOrDefault(to, Double.POSITIVE_INFINITY);
// newDist is the weight of the shortest path using this edge.
double newDist = distTo.get(from) + edge.weight;
// Check that we haven't added the vertex to the SPT already...
// AND the path using this edge is better than the best-known path.
if (!visited.contains(to) && newDist < oldDist) {
edgeTo.put(to, edge);
distTo.put(to, newDist);
perimeter.addOrChangePriority(to, newDist);
}
// This entire if block is called "relaxing" an edge.
}
}
}
/** Returns a single-pair weighted shortest path from start to goal. */
public List<V> solution(V goal) {
List<V> path = new ArrayList<>();
V curr = goal;
path.add(curr);
while (edgeTo.get(curr) != null) {
curr = edgeTo.get(curr).from;
path.add(curr);
}
Collections.reverse(path);
return path;
}
}
This code works fine, but in practice, you’ll often see a similar version of this code that is basically the same except it makes no mention of the visited set. If we were to remove the visited set from BFS, BFS can get stuck in a loop or cause other kinds of problems. But Dijkstra’s algorithm actually runs the same with or without the visited set. In class, we’ll learn more about why this optimization works without causing any infinite loops.
Dynamic Programming
- Identify whether a recursive algorithm can be rewritten using DP.
- Explain how unique paths can be counted using recursion and DP.
- Explain how a min-cost seam can be found using recursion and DP.
Graph data structures (like adjacency lists) and graph algorithms (like Dijkstra’s algorithm) provided reusable abstractions for addressing many kinds of problems. Reduction refers to exactly this problem solving approach: modifying the input data to a problem so that an existing algorithm can solve it.
- Reduction
- A problem-solving approach that works by reframing problem A in terms of another problem B:
- Preprocess. Take the original input to A and modify it so that it works as an input to B.
- Run a standard algorithm for problem B on this input, returning some output.
- Postprocess. Modify this output so that it can be understood as the expected output for A.
We’ve actually already seen several examples of reduction algorithms that we just didn’t call “reductions” at the time. For example, in the project, we learned that seam finding reduces to single-pair shortest paths:
- First, we constructed a
PixelGraphfrom your inputPicture. - Then, we ran a shortest paths algorithm, returning the shortest path as a list of nodes.
- Finally, we extracted the y-index of each node representing the pixels in the seam to remove.
Reduction algorithms are a very common approach to solving problems in computer science. They also work without graphs as input or context. Remember dup1 and dup2 from the first week of the course? dup1 detects duplicates in an array by checking every pair of items in quadratic time. dup2 detects duplicates in a sorted array in just linear time. The problem of duplicate detection reduces to sorting because we can just sort the array and then run dup2, which can be much faster than dup1. Inventing faster sorting algorithms doesn’t just mean faster sorting, but faster everything because so many problems reduce to or otherwise depend on sorting.
Eric Allender summarizes the importance of this insight:1
If A is efficiently reducible to B, and B is efficiently reducible to A, then in a very meaningful sense, A and B are “equivalent”—they are merely two different ways of looking at the same problem. Thus instead of infinitely many different apparently unrelated computational problems, one can deal instead with a much smaller number of classes of equivalent problems […] Nothing had prepared the computing community for the shocking insight that there are really just a handful of fundamentally different computational problems that people want to solve.
Although reductions provide a powerful way to define classes of equivalent computational problems, they don’t necessarily lead to the most experimentally-efficient solutions. The seam finding reduction to single-pair shortest paths will be limited by the runtime for shortest paths algorithms like Dijkstra’s algorithm. In practice, algorithm designers apply algorithm design paradigms like dynamic programming to develop more efficient solutions when greater efficiency is necessary.
Fibonacci sequence case study
The Fibonacci Sequence is a series of numbers in which each number is the sum of the two preceding numbers, starting with 0 and 1. We can represent this rule as a recurrence: F(N) = F(N - 1) + F(N - 2) with the base cases F(0) = 0 and F(1) = 1.
By converting this recurrence into a program, we can compute the Nth Fibonacci number using multiple recursion.
public static long fib(int N) {
if (N == 0 || N == 1) {
return N;
}
return fib(N - 1) + fib(N - 2);
}
We can draw a recursion tree diagram to visualize the process for computing fib(N). Each node in the diagram represents a call to fib, starting from F(N), which calls F(N - 1) and F(N - 2). F(N - 1) then calls F(N - 2) and F(N - 3), and so forth. Repeated subproblems are highlighted in different colors.
Top-down dynamic programming
To save time, we can reuse the solutions to repeated subproblems. Top-down dynamic programming is a kind of dynamic programming that augments the multiple recursion algorithm with a data structure to store or cache the result of each subproblem. Recursion is only used to solve each unique subproblem exactly once, leading to a linear time algorithm for computing the Nth Fibonacci number.
private static long[] F = new long[92];
public static long fib(int N) {
if (N == 0 || N == 1) {
return N;
}
if (F[N] == 0) {
F[N] = fib(N - 1) + fib(N - 2);
}
return F[N];
}
Top-down dynamic programming is also known as memoization because it provides a way to turn multiple recursion with repeated subproblems into dynamic programming by remembering past results.
| index | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| F(N) | 0 | 1 | 1 | 2 | 3 | 5 | 8 |
The data structure needs to map each input to its respective result. In the case of fib, the input is an int and the result is a long. Because the integers can be directly used as indices into an array, we chose to use a long[] as the data structure for caching results. But, in other programs, we may instead want to use some other kind of data structure like a Map to keep track of each input and result.
Bottom-up dynamic programming
Like top-down dynamic programming, bottom-up dynamic programming is another kind of dynamic programming that also uses the same data structure to store results. But bottom-up dynamic programming populates the data structure using iteration rather than recursion, requiring the algorithm designer to carefully order computations.
public static long fib(int N) {
long[] F = new long[N + 1];
F[0] = 0;
F[1] = 1;
for (int i = 2; i <= N; i += 1) {
F[i] = F[i - 1] + F[i - 2];
}
return F[N];
}
After learning both top-down and bottom-up dynamic programming, we can see that there’s a common property between the two approaches.
- Dynamic programmming
- An algorithm design paradigm for speeding-up algorithms that makes multiple recursive calls by identifying and reusing solutions to repeated recursive subproblems.
Minimum Spanning Trees
- Trace Prim’s algorithm to find a minimum spanning tree in a graph.
- Compare and contrast Prim’s algorithm and breadth-first search.
- Apply the super-source node pattern to simplify graph problems.
The minimum spanning tree (MST) problem is about finding a spanning tree of minimum total weight in a connected and weighted undirected graph.
- Spanning tree
- A tree that connects all the vertices in an undirected graph.
- Tree (graph theory)
- A selection of vertices and edges in an undirected graph where there is only a single path between any two vertices.
Prim’s algorithm
Prim’s algorithm for finding an MST builds on the foundation of Dijkstra’s for finding an SPT. Just like Dijkstra’s, Prim’s algorithm starts from a given vertex and gradually builds the tree structure on each iteration of the while loop. But there is 1 major difference:
- Earlier we learned that Dijkstra’s algorithm gradually builds a shortest paths tree on each iteration of the while loop. Prim’s algorithm does this similarily but instead selects the next unvisited vertex based on edge weight alone for the MST, whereas Dijkstra’s algorithm selects the next unvisited vertex based on the sum total cost of its shortest path.
In short, Prim’s algorithm builds a MST by repeatedly choosing the next-smallest edge to an unvisited vertex. The algorithm is finished once all the reachable vertices are visited.
Review the comments in this code snippet to identify how each difference appears in Prim’s algorithm.
public class PrimMST<V> implements MSTSolver<V> {
// Same edgeTo map as in Dijkstra's for shortest paths trees.
private final Map<V, Edge<V>> edgeTo;
// Same distTo map as in Dijkstra's for shortest paths trees.
private final Map<V, Double> distTo;
public PrimMST(Graph<V> graph) {
edgeTo = new HashMap<>();
distTo = new HashMap<>();
// The MST problem does not specify a start vertex.
// But, like Dijkstra's, Prim's algorithm requires a start vertex, so pick any vertex.
V start = graph.randomVertex();
// Unvisited vertices are considered in order of edge weight.
MinPQ<V> perimeter = new DoubleMapMinPQ<>();
perimeter.add(start, 0.0);
Set<V> visited = new HashSet<>();
while (!perimeter.isEmpty()) {
// Remove the next-smallest weight vertex from the perimeter.
V from = perimeter.removeMin();
// Mark a vertex as visited only after it's removed from the perimeter.
visited.add(from);
for (Edge<V> edge : graph.neighbors(from)) {
V to = edge.to;
double oldWeight = distTo.getOrDefault(to, Double.POSITIVE_INFINITY);
// Check that we haven't added the vertex to the MST already...
// Diff 1. AND this edge weight is better than the previous best edge weight to this vertex
// (infinity if this vertex has not been encountered before).
if (!visited.contains(to) && edge.weight < oldWeight) {
edgeTo.put(to, edge);
// Diff 1. Store the edge weight rather than distance from start.
distTo.put(to, edge.weight);
perimeter.addOrChangePriority(to, edge.weight);
}
// This entire if block is called "relaxing" an edge.
}
}
}
/** Returns a collection of edges representing a minimum spanning tree in the graph. */
public Collection<Edge<V>> mst() {
return edgeTo.values();
}
}
Cut property
How do we know Prim’s algorithm works?
- Prim’s algorithm stores all the next potential vertices to visit sorted by lowest-cost edge weight. In each iteration of the
whileloop, Prim’s algorithm grows the MST-in-progress by 1 vertex.
When Prim’s algorithm “grows the MST-in-progress” by selecting the lowest-cost edge to an unvisited vertex, it’s actually applying the cut property.
There are three key terms to define about the cut property.
- Cut
- A partitioning of the vertices of a graph into two non-empty sets.
- Crossing edge
- An edge that has one endpoint in each set of a cut.
- Cut property
- The idea that, for any cut, a minimum-weight crossing edge must be in an MST.
Prim’s algorithm can be understood as a repeated application of the cut property.
- Start with a single vertex in one set and every other unvisited vertex in the other set.
- On each iteration of the
whileloop, apply the cut property to choose the minimum-weight crossing edge. This expands the visited set by taking 1 vertex from the unvisited vertices. - Once the
whileloop is done and all reachable vertices have been visited, return the MST.
Eric Allender. 2009. A Status Report on the P versus NP Question. ↩