Lecture 23: Tree and Graph Traversals

10/19/2020

Trees and Traversals

Tree Definition

• A tree consists of:
  • A set of nodes
  • A set of edges that connect those nodes
    • Constraint: There is exactly one path between any two nodes

Rooted Trees Definition

• A rooted tree is a tree where we've chosen one node as the "root"
  • Every node N except the root has exactly one parent, defined as the first node on the path from N to the root
  • A node with no child is called a leaf

Trees

• Trees are a more general concept
  • Organizational charts
  • Family lineages

Example: File Tree System

• Sometimes you want to iterate over a tree. For example, suppose you want to find the total size of all files in a folder called 61b
  • What one might call "tree iteration" is actually called "tree traversal"
  • Unlike lists, there are many orders in which we might visit the nodes
    • Each ordering is useful in different ways

Tree Traversal Orderings

• Level Order
  • Visit top-to-bottom, left-to-right (like reading in English)
• Depth First Traversals
  • 3 types: Preorder, Inorder, Postorder
  • Basic (rough) idea: Traverse "deep nodes" before shallow ones
  • Note: Traversing a node is different than "visiting" a node

Depth First Traversals

preOrder(BSTNode x) {
    if (x == null) return;
    print(x.key)
    preOrder(x.left)
    preOrder(x.right)
}

• Preorder traversal: "Visit" a node, then traverse its children

inOrder(BSTNode x) {
    if (x == null) return;
    inOrder(x.left)
    print(x.key)
    inOrder(x.right)
}

• Inorder traversal: Traverse left child, visit, then traverse right child

postOrder(BSTNode x) {
    if (x == null) return;
    postOrder(x.left)
    postOrder(x.right)
    print(x.key)
}

• Postorder traversal: Traverse left, traverse right, then visit

A Useful Visual Trick (for Humans, not algorithms)

• Preorder traversal: We trave a path around the graph, from the top going counter-clockwise. "Visit" every time we pass the LEFT of a node
• Inorder traversal: "Visit" when you cross the bottom of a node
• Postorder traversal: "Visit" when you cross the right of a node

What Good are all These Traversals

• Example: Preorder Traversal is natural for printing directory listings
• Example: Postorder Traversal for gathering file sizes

Graphs

Trees and Hierarchical Relationships

• Trees are fantastic for representing strict hierarchical relationships
  • But not every relationship is hierarchical
  • Example: Paris Metro map
• The Paris Metro map is not a tree: It contains cycles!

Graph Definition

• A graph consists of:
  • A set of nodes
  • A set of zero or more edges, each of which connects two nodes
  • Note, all trees are graphs
• A simple graph is a graph with:
  • No edges that connect a vertex to itself, i.e. no "loops"
  • No two edges that connect the same vertices, i.e. no "parallel edges"
• In 61B, "graph" refers to "simple graph" unless otherwise stated

Graph Type

• Directed Graph
  • Each edge has a "directionality"
• Undirected Graph
  • Each edge is undirected
• Acyclic Graph:
  • A tree. A graph with no cycles
• Cyclic:
  • A graph that contains a cycle
• With Edge Labels

Graph Terminology

• Graph
  • Set of vertices, aka nodes
  • Set of edges: Pair of vertices
  • Vertices with an edge between them are adjacent
  • Vertices or edges may have labels (or weights)
• A path is a sequence of vertices connected by edges
• A cycle is a path whose first and last vertices are the same
  • A graph with a cycle is "cyclic"
• Two vertices are connected if there is a path between them. If all vertices are connected, we say the graph is connected

Graph Problems

Graph Queries

• There are lots of interesting questions we can ask about a graph:
  • Shortest path between two nodes
  • Longest path between two nodes without cycles
  • Is there a tour that uses each node exactly once?
  • Is there a tour that uses each edge exactly once?

Graph Queries More Theoretically

• Some well known graph problems and their common names:
  • s-t Path. Is there a path between vertices s and t?
  • Connectivity. Is the graph connected, i.e. is there a path between all vertices?
  • Biconnectivity. Is there a vertex whose removal disconnects the graph?
  • Shortest s-t Path. What is the shortest path between vertices s and t?
  • Cycle Detection. Does the graph contain any cycles?
  • Euler Tour. Is there a cycle that uses every edge exactly once?
  • Hamilton Tour. Is there a cycle that uses every vertex exactly once?
  • Planarity. Can you draw the graph on paper with no crossing edges?
  • Isomorphism. Are two graphs isomorphic?
• Often can't tell how difficult a graph problem is without very deep consideration

Graph Problem Difficulty

• Some well known graph problems
  • Euler Tour
  • Hamilton Tour
• Difficulty can be deceiving
  • An efficient Euler tour algorithm O(# edges) was found as early as 1873
  • Despite decades of intense study, no efficient algorithm for a Hamilton tour exists. Best algorithms are exponential time

Depth-First Traversal

s-t Connectivity

• Let's solve a classic graph problem called the s-t connectivity problem
  • Given source vertex s and a target vertex t, is there a path between s and t?
• Requires us to traverse the graph somehow
• One possible recursive algorithm for connected(s, t)
  • Does s == t? If so, return true
  • Otherwise, if connected(v, t) for any neighbor v of s, return true
  • Return false
• What is wrong with it? Can get caught in an infinite loop
• How do we fix it?
  • Mark s
  • Does s == t? If so, return true
  • Otherwise, if connected(v, t) for any unmarked neighbor v of s, return true
  • Return false
• Basic idea is same as before, but visit each vertex at most once
  • Marking nodes prevents multiple visits

Depth First Traversal

• This idea of exploring a neighbor's entire subgraph before moving on to the next neighbor is known as Depth First Traversal
  • Called "depth first" because you go as deep as possible

The Power of Depth First Search

• DFS is a very powerful technique that can be used for many types of graph problems
• Another example:
  • Let's discuss an algorithm that computes a path to every vertex
  • Let's call this algorithm DepthFirstPaths

DepthFirstPaths

• Gaol: Find a path from s to every other reachable vertex, visiting each vertex at most once. dfs(v) is as follows:
  • Mark v.
  • For each unmarked adjacent vertex w:
    • set edgeTo[w] = v
    • dfs(w)
  • Now you're left with an artifact to compute paths from s to every other reachable vertex

Tree Vs. Graph Traversals

Tree Traversals

• There are many tree traversals
  • Preorder
  • Inorder
  • Postorder
  • Level order (non depth-first traversal)

Graph Traversal

• What we just did in DepthFirstPaths is called "DFS Preorder"
  • DFS Preorder: Action is before DFS calls to neighbors
    • Our action was setting edgeTo
    • Preorders are equivalent to the order of dfs calls
• Could also do actions in DFS Postorder
  • DFS Postorder: Action is after DFS calls to neighbors
    • Equivalent to the order of dfs returns
• There are also many graph traversals, given some source:
  • DFS Preorder
  • DFS PostOrder
  • BFS order: (Breadth first search) Act in order of distance from s
    • Analogous to "level order". Search is wide, not deep

Summary

Summary

• Graphs are a more general idea than a tree
  • A tree is a graph where there are no cycles and every vertex is connected
  • Key graph terms: Directed, Undirected, Cyclic, Acyclic, Path, Cycle
• Graph problems vary widely in difficulty
  • Common tool for solving almost all graph problems is traversal
  • A traversal is an order in which you visit / act upon vertices
  • Tree traversals:
    • Preorder, inorder, postorder, level order
  • Graph traversals:
    • DFS preorder, DFS postorder, BFS
  • By performing actions / setting instance variables during a graph (or tree) traversal, you can solve problems like s-t connectivity or path finding