CS 261 Reachability Problems

The classic Reachability algorithm findReachable (graph g, vertex start) { create a set of reachable vertices, initially empty. call this r. create a container for vertices known to be reachable. call this c add start vertex to container c while the container c is not empty { remove first entry from the container c, assign to v if v is not already in the set of reachable vertices r { add v to the reachable set r add the neighbors of v to the container c } return r }

Different containers different algorithms What is interesting about this algorithm is that if you use different containers You get different algorithms

Example Solving a Maze

Represented as a graph

Use a Stack: { 1 }

Use a Stack: { 2, 6 }

Use a Stack: { 7, 3, 6 }

Use a Stack: { 3, 6 } dead end

Try alternative: { 4, 8, 6 }

Keep plowing on: { 5, 9, 8, 6 }

Keep on: { 10, 9, 8, 6 }

Keep on: { 15, 9, 8, 6 }

Keep on: { 14, 20, 9, 8, 6 }

Note duplicate: { 9, 20, 9, 8, 6 }

Opps, 4 again: { 4, 20, 9, 8, 6 }

Pop, try alternative: { 20, 9, 8, 6 }

Keep going: { 19, 9, 8, 6 }

And going: { 24, 9, 8, 6 }

Finished: { 25, 9, 8, 6 }

Depth first search Keep going in one direction as long as possible When you get stuck, back up to last choice, try alternative Did anybody do a Corn Maze last fall? Probably used this technique.

What if we use a queue?

Initial queue: {1}

First neighbors: {2, 6}

Neighbors of 2: {6, 3, 7}

Neighbors of 6: {3, 7, 11}

Neighbors of 3: {7, 11, 4, 8}

Neighbors of 7: {11, 4, 8}

Neighbors of 11: {4, 8, 12, 16}

Neighbors of 4: {8, 12, 16, 5, 9}

Breadth-first search Notice how this search jumps all over the place It’s - you go that way, I’ll go this way (but with a very large group of friends) Or think about spilling ink at the start, and watching it seep through the entire maze

Questions you can ask Which is faster? Which is guaranteed to find a solution? What if the graph is infinite, but there is a finite solution - which is guaranteed to find a solution?

But there is more! If we have a weighted graph, and use a priority queue - then the same technique is called Dijkstra’s algorithm Idea: queue keeps shortest distance from some place you have been to someplace you might not have been

Essence of Dijkstra’s algorithm! Idea: queue keeps shortest distance from some place you have been to someplace you might not have been When you pull an item from the queue, if it is someplace you have seen, toss it out, If it is new, add to destinations, put neighbors into queue

What about Weighted Graphs? Pendleton Pierre Pensacola Princeton Pittsburgh Peoria Pueblo Phoenix Dijkstra’s algorithm. Use a priority queue instead of a stack. Return a map of city, distance pairs. Pqueue orders values on shortest distance Dijkstra (String startCity, Map[String, Map[String, double]] distances) Make empty map of distances into variable reachable Put (StartingCity, 0) into Pqueue While Pqueue not empty pull new city from queue, if not ready in reachable, add to reachable add neighbors to queue, adding weight to distance from starting city When done with loop, return reachable map

Example: What is the distance from Pierre Pendleton Pierre Pensacola Princeton Pittsburgh Peoria Pueblo Phoenix Pierre: Dijkstra (String startCity, Map[String, Map[String, double]] distances) Make empty map of distances into variable reachable Put (StartingCity, 0) into Pqueue While Pqueue not empty pull new city from queue, if not ready in reachable, add to reachable add neighbors to queue, adding weight to distance from starting city When done with loop, return reachable map

Example: What is the distance from Pierre Pendleton Pierre Pensacola Princeton Pittsburgh Peoria Pueblo Phoenix Pierre: Pendeleton: Dijkstra (String startCity, Map[String, Map[String, double]] distances) Make empty map of distances into variable reachable Put (StartingCity, 0) into Pqueue While Pqueue not empty pull new city from queue, if not ready in reachable, add to reachable add neighbors to queue, adding weight to distance from starting city When done with loop, return reachable map

Example: What is the distance from Pierre Pendleton Pierre Pensacola Princeton Pittsburgh Peoria Pueblo Phoenix Pierre: 0, Pendleton: Phoenix: 6, Pueblo: 10 Notice how the distances have been added Dijkstra (String startCity, Map[String, Map[String, double]] distances) Make empty map of distances into variable reachable Put (StartingCity, 0) into Pqueue While Pqueue not empty pull new city from queue, if not ready in reachable, add to reachable add neighbors to queue, adding weight to distance from starting city When done with loop, return reachable map

Example: What is the distance from Pierre Pendleton Pierre Pensacola Princeton Pittsburgh Peoria Pueblo Phoenix Pierre: 0, Pendleton: 2, Phoenix: Pueblo: 9, Peoria: 10, Pueblo: 10, Pittsburgh: 16 Notice how values are stored in the Pqueue in distance order Dijkstra (String startCity, Map[String, Map[String, double]] distances) Make empty map of distances into variable reachable Put (StartingCity, 0) into Pqueue While Pqueue not empty pull new city from queue, if not ready in reachable, add to reachable add neighbors to queue, adding weight to distance from starting city When done with loop, return reachable map

Example: What is the distance from Pierre Pendleton Pierre Pensacola Princeton Pittsburgh Peoria Pueblo Phoenix Pierre: 0, Pendleton: 2, Phoenix: 6, Pueblo: Peoria: 10, Pueblo: 10, Pierre: 13, Pittsburgh: 16 Pierre gets put in queue, although it is known to be reachable Dijkstra (String startCity, Map[String, Map[String, double]] distances) Make empty map of distances into variable reachable Put (StartingCity, 0) into Pqueue While Pqueue not empty pull new city from queue, if not ready in reachable, add to reachable add neighbors to queue, adding weight to distance from starting city When done with loop, return reachable map

Example: What is the distance from Pierre Pendleton Pierre Pensacola Princeton Pittsburgh Peoria Pueblo Phoenix Pierre: 0, Pendleton: 2, Phoenix: 6, Pueblo: 9, Peoria: Pueblo: 10, Pierre: 13, Pueblo: 13, Pittsburgh: 15, Pittsburgh: 16 Duplicates only removed when pulled out of queue Dijkstra (String startCity, Map[String, Map[String, double]] distances) Make empty map of distances into variable reachable Put (StartingCity, 0) into Pqueue While Pqueue not empty pull new city from queue, if not ready in reachable, add to reachable add neighbors to queue, adding weight to distance from starting city When done with loop, return reachable map

Depth first reachability, Breadth first Reachability, Dijkstras Algorithm All three are essentially the same algorithm The only difference is the type of container they use

Bottom line So the key idea from this course is Selecting appropriate data structures is the key to programming You need to know what tools are available to make the best choices And with that, we end this course