Improve explanation of DFS

Author: hollance

Breadth-First Search/README.markdown

@@ -2,6 +2,8 @@
 
 Breadth-first search (BFS) is an algorithm for traversing or searching [tree](../Tree/) or [graph](../Graph/) data structures. It starts at a source node and explores the immediate neighbor nodes first, before moving to the next level neighbors.
 
+Breadth-first search can be used on both directed and undirected graphs.
+
 ## Animated example
 
 Here's how breadth-first search works on a graph:
@@ -83,8 +85,6 @@ The parent of a node is the one that "discovered" that node. The root of the tre
 
 For an unweighted graph, this tree defines a shortest path from the starting node to every other node in the tree. So breadth-first search is one way to find the shortest path between two nodes in a graph.
 
-Breadth-first search can be used on directed and undirected graphs.
-
 ## The code
 
 Simple implementation of breadth-first search using a queue:

Depth-First Search/DepthFirstSearch.playground/Pages/Simple Example.xcplaygroundpage/Contents.swift

@@ -30,7 +30,8 @@ graph.addEdge(nodeB, neighbor: nodeE)
 graph.addEdge(nodeC, neighbor: nodeF)
 graph.addEdge(nodeC, neighbor: nodeG)
 graph.addEdge(nodeE, neighbor: nodeH)
-
+graph.addEdge(nodeE, neighbor: nodeF)
+graph.addEdge(nodeF, neighbor: nodeG)
 
 let nodesExplored = depthFirstSearch(graph, source: nodeA)
 print(nodesExplored)

Depth-First Search/Images/AnimatedExample.gif

@@ -2,7 +2,25 @@
 
 Depth-first search (DFS) is an algorithm for traversing or searching [tree](../Tree/) or [graph](../Graph/) data structures. It starts at a source node and explores as far as possible along each branch before backtracking.
 
-[TODO: this is a work-in-progress]
+Depth-first search can be used on both directed and undirected graphs.
+
+## Animated example
+
+Here's how depth-first search works on a graph:
+
+![Animated example](Images/AnimatedExample.gif)
+
+Let's say we start the search from node `A`. In depth-first search we look at the starting node's first neighbor and visit that. In the example that is node `B`. Then we look at node `B`'s first neighbor and visit it. This is node `D`. Since `D` doesn't have any unvisited neighbors of its own, we backtrack to node `B` and go to its other neighbor `E`. And so on, until we've visited all the nodes in the graph.
+
+Each time we visit the first neighbor and keep going until there's nowhere left to go, and then we backtrack to a point where there are again nodes to visit. When we've backtracked all the way to node `A`, the search is complete.
+
+For the example, the nodes were visited in the order `A`, `B`, `D`, `E`, `H`, `F`, `G`, `C`.
+
+The depth-first search process can also be visualized as a tree:
+
+![Traversal tree](Images/TraversalTree.png)
+
+The parent of a node is the one that "discovered" that node. The root of the tree is the node you started the depth-first search from. Whenever there's a branch, that's where we backtracked.
 
 ## The code
 
@@ -45,20 +63,22 @@ graph.addEdge(nodeB, neighbor: nodeE)
 graph.addEdge(nodeC, neighbor: nodeF)
 graph.addEdge(nodeC, neighbor: nodeG)
 graph.addEdge(nodeE, neighbor: nodeH)
+graph.addEdge(nodeE, neighbor: nodeF)
+graph.addEdge(nodeF, neighbor: nodeG)
 
 let nodesExplored = depthFirstSearch(graph, source: nodeA)
 print(nodesExplored)
 ```
 
-This will output: `["a", "b", "d", "e", "h", "c", "f", "g"]`
+This will output: `["a", "b", "d", "e", "h", "f", "g", "c"]`
 
-## Applications
+## What is DFS good for?
 
 Depth-first search can be used to solve many problems, for example:
 
 * Finding connected components of a sparse graph
-* Topological sorting of nodes in a graph
+* [Topological sorting](../Topological Sort/) of nodes in a graph
 * Finding bridges of a graph (see: [Bridges](https://en.wikipedia.org/wiki/Bridge_(graph_theory)#Bridge-finding_algorithm))
 * And lots of others!
 
-*Written for Swift Algorithm Club by Paulo Tanaka*
+*Written for Swift Algorithm Club by Paulo Tanaka and Matthijs Hollemans*

Depth-First Search/Images/AnimatedExample.graffle

@@ -4,6 +4,7 @@ class DepthFirstSearchTests: XCTestCase {
 
   func testExploringTree() {
     let tree = Graph()
+
     let nodeA = tree.addNode("a")
     let nodeB = tree.addNode("b")
     let nodeC = tree.addNode("c")
@@ -12,6 +13,7 @@ class DepthFirstSearchTests: XCTestCase {
     let nodeF = tree.addNode("f")
     let nodeG = tree.addNode("g")
     let nodeH = tree.addNode("h")
+
     tree.addEdge(nodeA, neighbor: nodeB)
     tree.addEdge(nodeA, neighbor: nodeC)
     tree.addEdge(nodeB, neighbor: nodeD)
@@ -70,7 +72,6 @@ class DepthFirstSearchTests: XCTestCase {
     let nodesExplored = depthFirstSearch(graph, source: nodeA)
 
     XCTAssertEqual(nodesExplored, ["a", "b", "c", "d", "e", "f", "g", "h", "i"])
-
   }
 
   func testExploringGraphWithASingleNode() {
@@ -81,4 +82,4 @@ class DepthFirstSearchTests: XCTestCase {
 
     XCTAssertEqual(nodesExplored, ["a"])
   }
-}
+}

Depth-First Search/Images/AnimatedExample.psd

@@ -16,7 +16,7 @@
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-		83F9C96E1C84449D00B3A87F /* Graph.swift */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.swift; name = Graph.swift; path = ../Graph.swift; sourceTree = SOURCE_ROOT; };
+		83F9C96E1C84449D00B3A87F /* Graph.swift */ = {isa = PBXFileReference; fileEncoding = 4; lastKnownFileType = sourcecode.swift; path = Graph.swift; sourceTree = SOURCE_ROOT; };
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