graphs

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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