2.7. graphs

Some notes on computer science graphs.

• Edges are of the form (v1, v2)

  • Can be ordered pair or unordered pair

• Definitions

  • A weight or cost can be associated with each edge - this is determined by the application

  • w is adjacent to v iff (v, w) \(\in\) E

  • path: sequence of vertices w1, w2, w3, …, wn such that (wi, wi+1) ∈ E for 1 ≤ i < n

  • length of a path: number of edges in the path

  • simple path: all vertices are distinct

  • cycle:

    • directed graph: path of length \(\geq\) 1 such that w1 = wn

    • undirected graph: same, except all edges are distinct

  • connected: there is a path from every vertex to every other vertex

  • loop: (v, v) \(\in\) E

  • complete graph: there is an edge between every pair of vertices

• digraph

  • directed acyclic graph: no cycles; often called a “DAG”

  • strongly connected: there is a path from every vertex to every other vertex

  • weakly connected: the underlying undirected graph is connected

• For Google Maps, an adjacency matrix would be infeasible - almost all zeros (sparse)

  • an adjacency list would work much better

  • an adjacency matrix would work for airline routes

• detect cycle

  • dfs from every vertex and keep track of visited, if repeat then cycle

• Topological Sort

  • Given a directed acyclic graph, construct an ordering of the vertices such that if there is a path from vi to vj, then vj appears after vi in the ordering

  • The result is a linear list of vertices

  • indegree of v: number of edges (u, v) – meaning the number of incoming edges

  • Algorithm

    • start with something of indegree 0

    • take it out, and take out the edges that start from it

    • keep doing this as we take out more and more edges

  • can have multiple possible topological sorts

• Shortest Path

  • single-source - start somewhere, get shortest path to everywhere

  • unweighted shortest path - breadth first search

  • Weighted Shortest Path

    • We assume no negative weight edges

    • Djikstra’s algorithm: uses similar ideas as the unweighted case

    • Greedy algorithms: do what seems to be best at every decision point

    • Djikstra: v^2

      • Initialize each vertex’s distance as infinity

      • Start at a given vertex s

        • Update s’s distance to be 0

      • Repeat

        • Pick the next unknown vertex with the shortest distance to be the next v

        • If no more vertices are unknown, terminate loop

      • Mark v as known

      • For each edge from v to adjacent unknown vertices w

        • If the total distance to w is less than the current distance to w

        • Update w’s distance and the path to w

      • It picks the unvisited vertex with the lowest-distance, calculates the distance through it to each unvisited neighbor, and updates the neighbor’s distance if smaller. Mark visited (set to red) when done with neighbors.

  • Shortest path from a start node to a finish node

    • 1. We can just run Djikstra until we get to the finish node

    • 1. Have different kinds of nodes

      • Assume you are starting on a “side road”

      • Transition to a “main road”

      • Transition to a “highway”

      • Get as close as you can to your destination via the highway system

      • Transition to a “main road”, and get as close as you can to your destination

      • Transition to a “side road”, and go to your destination

• Traveling Salesman

  • Given a number of cities and the costs of traveling from any city to any other city, what is the least-cost round-trip route that visits each city exactly once and then returns to the starting city?

  • Hamiltonian path: a path in a connected graph that visits each vertex exactly once

  • Hamiltonian cycle: a Hamiltonian path that ends where it started

  • The traveling salesperson problem is thus to find the least weight Hamiltonian path (cycle) in a connected, weighted graph

• Minimum Spanning Tree

  • Want fully connected

  • Want to minimize number of links used

    • We won’t have cycles

  • Any solution is a tree

  • Slow algorithm: Construct a spanning tree:

    • Start with the graph

    • Remove an edge from each cycle

    • What remains has the same set of vertices but is a tree

    • Spanning Trees

• Minimal-weight spanning tree: spanning tree with the minimal total weight

  • Generic Minimum Spanning Tree Algorithm

    • KnownVertices <- {}

    • while KnownVertices does not form a spanning tree, loop:

      • find edge (u,v) that is “safe” for KnownVertices

      • KnownVertices <- KnownVertices U {(u,v)}

    • end loop

  • Prim’s algorithm

    • Idea: Grow a tree by adding an edge to the “known” vertices from the “unknown” vertices. Pick the edge with the smallest weight.

    • Pick one node as the root,

    • Incrementally add edges that connect a “new” vertex to the tree.

    • Pick the edge (u,v) where:

    • u is in the tree, v is not, AND

    • where the edge weight is the smallest of all edges (where u is in the tree and v is not)

    • Running time: Same as Dijkstra’s: Θ(e log v)

  • Kruskal’s algorithm

    • Idea: Grow a forest out of edges that do not create a cycle. Pick an edge with the smallest weight.

    • When optimized, it has the same running time as Prim’s and Dijkstra’s: Θ(e log v)

    • unoptomized: v^2