Math 118 concepts exam 1
how do you find the possible number of spanning trees in a network?
multiple (the # of edges of one circuit) x ( the # of edges of another circuit) -if they share an edge of minus 1
How many degrees does a circle add?
two degrees because it connects twice to the edge
How do you know it's a tree?
-no circuit but has a network -every edge is a bridge -only one path connecting a given pair of vertices -doesn't have to visit every vertices
Processor
"workers" who carry out the work N = number of processors
Kruskal's Algorithm
(Similar to Cheapest Link Algorithm) 1. Pick cheapest edge available (mark it) 2. Continue with next cheapest, mark it, until you are done ** do not create a circuit** -always gives an optimal MinST solution
How do you find the number of hamilton CIRCUTS?
(n-1)!
What does it mean to do closed unicursal tracing?
-euler circut -eulerization
What does it mean to have a open unicursal tracing?
-euler path -semi eulerization
Whats the difference between Hamilton and Euler graphs?
Hamilton visits ALL the VERTICES once. Euler visits ALL the EDGES once.
Repeated Nearest Neighbor Algorithm
-refined version of NNA 1. Do NNA starting at all vertices 2. Choose the one at the least cost - if necessary rewrite tour so it starts at the designated starting vertex
How to Eulerize a graph
1. Identify odd vertices 2. Duplicate copy of edges - > make it even -find the optimal route! fewest edges!
A connected graph has exactly 20 vertices of odd degree 1. in order to draw the graph using a closed unicursal tracing you must lift your pencil how many times?
10 times. -need eulerization -> 10 more make it even
A connected graph has exactly 20 vertices of odd degree 1. in order to draw the graph using a open unicursal tracing you must lift your pencil how many times?
9 times one less than needed b/c it will leave 2 odds for an euler path
Euler Circuit
A circuit that uses every edge exactly once -does not have odd vertices -starts and ends at the same vertex
Directed Graph (Digraph)
A graph in which each edge is directed from one vertex to another (or the same) vertex -a asymmetric relationships
Fleury's Algorithm
A way to find euler path/circuit 1. Make sure graph is connected, EC. no odd vertices EP. just 2 odd Vertices 2. EC. Start at any point EP. Start at one of the 2 odd vertices 3. Don't choose a bridge unless you have to 4. Finish at EC. Starting vertex, EP. Opposite odd vertex circuit is complete
How do you find the critical time in the backflow algorithm
Add the processing time of X plus the LARGEST critical time among the vertices that X is incident to
Which algorithm will always yield the optimal tour for a weighted complete graph?
Brute force
How do you find the relative error of a tour
E = (cost found - optimal cost )/divided by optimal cost x 100 = ___ % also known as Y-X/X
N-1 Edges property
Every tree with n vertices, has N-1 edges
Which path/circuit are the brute force, NNA, RNNA and Cheapest Link Algorithms used for?
Hamilton paths/circuits!! to find the lowest (optimal) weight!
Euler's Sum of Degrees Theorem
the sum of the degrees of all the vertices = 2 times the number of edges
How do you find the number of hamilton PATHS?
N!
What does P mean in scheduling?
P = time
Decreasing Time Priority List
Prioritize bigger jobs first to complete then leave small jobs last
Whats the difference between spanning tree and tree?
Spanning tree must touch every vertices. Tree doesn't have to touch every vertices -both are NOT closed circuits
Backflow Algorithm
Step 1. Working backwards (from END to START) find the critical time, keep time of each vertex in [brackets] 1. A Look at the vertices that X is incident to and choose among them the one with the largest critical time 2. Once you have the critical time for every vertex, follow the path along the largest critical times -> move to adjacent ones (pick largest one and carry on)
Hamilton path
a circuit that visits all the vertices once and only once The route is NOT required to start and end at the same vertex
Hamilton Circuit
a circuit that visits all the vertices once and only once -required to start and end at same vertex
How does a hamilton circuit differ from a euler circuit
a hamilton circuit doesn't have to touch every edge and can have odd degrees.
Euler Path
a path that travels along each edge of a graph once and only once - has ONLY two odd vertices -starts and ends at the odd vertices
Critical-Time Priority List
a priority list where the tasks are listed in decreasing order of critical times (longest critical time first, shortest last)
Traveling Salesman Problem
a problem whose objective is to find the shortest or cheapest possible route that visits each vertices exactly once and returns to the starting vertices
Brute Force Algorithm
an algorithm that checks the cost of EVERY POSSIBLE Hamilton circuit and chooses the optimal one
Cheapest Link Algorithm
at each step chooses the cheapest link available that does not violate the partial-circuit rule or the three-edge rule
If there are 8 odd vertices how many edges must we add at least?
at least 4 duplicate edges (it connects the 8 with eachother)
Degree of seperation
distance from one point (vertices) to another
How do you find the optimal eulerization?
go through the least amount of edges - shortest circuit, but can't skip vertices
optimal tour/solution
hamilton CIRCUIT at the least total cost (circuit b/c start and end at same place)
Reduncancy of a network- how do you know if there is a tree or not?
if R = 0 the network is a tree if R > 0 the network is not a tree. It becomes a circuit
Can a graph contain a euler path and circuit
no. it can only have one or the other
outdegrees
number of arcs going away from the vertices
indegrees
number of arcs going in toward the vertices
How many times can hamilton paths/circuits go to each of the vertices?
once and only once can't visit a node twice like euler path/circuits can
Eulerization
the process of duplicating edges in a graph to make it have all even vertices
Semi-Eulerization
the process of duplicating edges in a graph to make it have almost all even edges but it's one less than a semi eulerization b/c it will create a euler path (odd vertices)
Nearest Neighbor Algorithm
start at one vertex and move to the nearest vertex that has the smallest weight so long it doesn't make a circuit. Continue until all of the vertices have been visited and return to the starting vertex.
Priority List
tasks executed by PRIORITY/READINESS, if INELIGIBLE tasks are skipped over until they become ready
Degree of a vertex
the amount of edges that meet at a vertex
What are arcs?
the edge for diagraphs
What does n represent?
the number of vertices
How do you find hamilton paths?
you start and end at different spots