Lecture 8 - Directed and Weighted Graphs
Contents
Directed Graphs
- undirected graphs - an edge relates two vertices equivalently
- directed graphs - only one direction | self loops!?
Keywords
- transitive closure - Directed path from s to t
- shortest path search - Shortest path from s to t
strong connectivity - vertices mutually reachable
in-degree (d^(-1)(v)) - number of directed edges leading into a vertex
out-degree (d(v)) - number of directed edges leading out of a vertex
sink - vertex with out-degree 0 (only in)
source - vertex with in-degree 0 (only out)
reachability - if v->w exists, w is reachable from v
strongly connected - if v->w and w->v exists
strong connectivity - every vertex reachable from every other vertex
strongly-connected component - maximal strongly-connected subgraph
Methods of representation
Adjacency matrix - asymmetric, spare, less space efficient
Adjaceny list - !!!
Edge list - order of edge components matter
Linked data tructures - pointers inherently directional
| - | storage | edge add | has edge | out degree |
|---|---|---|---|---|
| adj mat | O(V+V^w) | O(1) | O(1) | O(V) |
| adj list | O(V+E) | O(d(v)) | O(d(v)) | O(d(v)) |
adj list (y)
digraph ehhhh – sparse
Directed Acyclic Graph (DAG)
Is it a tree? Is it a graph? No it’s a dag!
DAGs are a type of tree/graph, where nodes don’t go back
Weighted Graphs
Some nodes have a higher (or lower) associated weight/value
Theoretical Implementation
Adjacency Matrix - store the weight rather than just true/false
- may need some ‘no edge exists’ value, as zero might be a valid weight
Adjacency List - add weight to each list node
Edge list - add weight to each edge
Linked data structure - links become link/weight pairs
Shortest Path Search
- Find minimum cost path
source-target - shortest from v->w
single-source - from v to all
all-pairs - (v,w) [ minimum for all pairs ]
Single Source Shortest Path Search
O(nlogv) - adjaceny matrix and heap
for an adjacency matrix (O(V^2))
Given a counter; the first time a node is visited is the earliest time it could be visited
Minumum Spanning Trees (MST)
Reminder: A spanning tree is a acyclic graph with all vertices of a graph.
A minimum spanning tree is the smallest representation of this subgraph.
Kruskal's Algorithm
= take all edges, sorted according to their weight; and for each add it to the prototype MST unless it will introduce a cycle
[A, B, C, D, E, F, G, H]
{A, B, C, -, E, -, -, H}
** Cycle checking is expensive - (dfs everything!?)
–> O(ElogE)
Prim's Algorithm
(Prim, Jarnik and Dijkstra)
Start from any vertex with an MST
Choose an edge not already in the MST, that
- does not cause a cycle
- connects to a vertex on the mst (‘on the fringe’)
- minimum possible edge
–> Choose the smallest edge from any of the vertices in the MST that does not cause a cycle
// and takes us somewhere new
// But lol they’re the same! –> Prim == Jarnik == Dijkstra (pick a new edge, minimum, no cycle)