lecture 23 algorithm design and analysis

Lecture 23: Minimum
Spanning Trees
CSE 373: Data Structures and
Algorithms
CSE 373 SP 18 - KASEY CHAMPION 1

Administriva

Minimum Spanning Trees
It’s the 1920’s. Your friend at the electric company needs to choose where to build wires to
connect all these cities to the plant.
CSE 373 SP 18 – ROBBIE WEBBER 3
She knows how much it would cost to lay electric wires between any pair of locations, and wants the cheapest way to
make sure electricity from the plant to every city.
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Minimum Spanning Trees
What do we need? A set of edges such that:
- Every vertex touches at least one of the edges. (the edges span the graph)
- The graph on just those edges is connected.
- The minimum weight set of edges that meet those conditions.
Assume all edge weights are positive.
Claim: The set of edges we pick never has a cycle. Why?
MST is the exact number of edges to connect all vertices
- taking away 1 edge breaks connectiveness
- adding 1 edge makes a cycle
- contains exactly V – 1 edges
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Notice we do not need a directed graph!
CSE 373 19 WI – KASEY CHAMPION
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Aside: Trees
Our BSTs had:
- A root
- Left and/or right children
- Connected and no cycles
Our heaps had:
- A root
- Varying numbers of children
On graphs our tees:
- Don’t need a root (the vertices aren’t ordered, and we can start BFS from anywhere)
- Varying numbers of children
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An undirected, connected acyclic graph.
Tree (when talking about graphs)
CSE 373 SP 18 – ROBBIE WEBBER
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MST Problem
What do we need? A set of edges such that:
- Every vertex touches at least one of the edges. (the edges span the graph)
- The graph on just those edges is connected.
- The minimum weight set of edges that meet those conditions.
Our goal is a tree!
We’ll go through two different algorithms for this problem today.
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Given: an undirected, weighted graph G
Find: A minimum-weight set of edges such that you can
get from any vertex of G to any other on only those
edges.
Minimum Spanning Tree Problem
CSE 373 SP 18 – ROBBIE WEBBER

Example
Try to find an MST of this graph:
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BFS
1. Pick an arbitrary starting point
2. Queue up unprocessed neighbors
3. Process next neighbor in queue
4. Repeat until all vertices in queue
have been processed
Dijkstra’s
1. Start at source
2. Update distance from current to
unprocessed neighbors
3. Process optimal neighbor
4. Repeat until all vertices have been
marked processed
Graph Algorithm Toolbox
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Example
Try to find an MST of this graph:
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BFS
1. Pick an arbitrary starting point
2. Queue up unprocessed neighbors
3. Process next neighbor in queue
4. Repeat until all vertices in queue
have been processed
Dijkstra’s
1. Start at source
marked processed
Graph Algorithm Toolbox
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Prim’s Algorithm
Dijkstra’s
1. Start at source
marked processed
Dijkstra(Graph G, Vertex source)
initialize distances to ∞
mark source as distance 0
mark all vertices unprocessed
while(there are unprocessed vertices){
let u be the closest unprocessed vertex
foreach(edge (u,v) leaving u){
if(u.dist+weight(u,v) < v.dist){
v.dist = u.dist+weight(u,v)
v.predecessor = u
}
}
mark u as processed
}
Algorithm idea:
1. choose an arbitrary
starting point
2. Investigate edges that
connect unprocessed
vertices
3. Add the lightest edge to
solution (be greedy)
4. Repeat until solution
connects all vertices
Prims(Graph G, Vertex source)
if(weight(u,v) < v.dist){
v.predecessor = u
}
}
mark u as processed
}

Try it Out
PrimMST(Graph G)
foreach(edge (source, v) ) {
v.dist = weight(source,v)
v.bestEdge = (source,v)
}
add u.bestEdge to spanning tree
if(weight(u,v) < v.dist && v unprocessed ){
v.dist = weight(u,v)
v.bestEdge = (u,v)
}
}
mark u as processed
}
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Vertex Distance Best Edge Processed
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Try it Out
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Vertex Distance Best Edge Processed
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(A, B)
(A, C)
(A, D)
X ✓
✓
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(B, F) ✓
(B, E)
(B, G)
PrimMST(Graph G)
foreach(edge (source, v) ) {
v.dist = weight(source,v)
v.bestEdge = (source,v)
}
add u.bestEdge to spanning tree
if(weight(u,v) < v.dist && v unprocessed ){
v.dist = weight(u,v)
v.bestEdge = (u,v)
}
}
mark u as processed
}
✓
---2
---5
--------(C, D)
--------(C, E)
✓
✓
✓

Prim’s Runtime
Dijkstra(Graph G, Vertex source)
if(u.dist+weight(u,v) < v.dist){
v.predecessor = u
}
}
mark u as processed
}
Prims(Graph G, Vertex source)
if(weight(u,v) < v.dist){
v.predecessor = u
}
}
mark u as processed
}
Runtime = VlogV + ElogV
Runtime = VlogV + ElogV

A different Approach
Prim’s Algorithm started from a single vertex and reached more and more
other vertices.
Prim’s thinks vertex by vertex (add the closest vertex to the currently
reachable set).
What if you think edge by edge instead?
Start from the lightest edge; add it if it connects new things to each other
(don’t add it if it would create a cycle)
This is Kruskal’s Algorithm.

Example
Try to find an MST of this graph by adding edges in sorted order
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Kruskal’s Algorithm
KruskalMST(Graph G)
initialize each vertex to be an independent component
sort the edges by weight
foreach(edge (u, v) in sorted order){
if(u and v are in different components){
add (u,v) to the MST
Update u and v to be in the same component
}
}

Try It Out
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KruskalMST(Graph G)
}
}
Edge Include? Reason
(A,C)
(C,E)
(A,B)
(A,D)
(C,D)
Edge (cont.) Inc? Reason
(B,F)
(D,E)
(D,F)
(E,F)
(C,F)

Try It Out
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KruskalMST(Graph G)
}
}
Edge Include? Reason
(A,C) Yes
(C,E) Yes
(A,B) Yes
(A,D) Yes
(C,D) No Cycle A,C,D,A
Edge (cont.) Inc? Reason
(B,F) Yes
(D,E) No Cycle A,C,E,D,A
(D,F) No Cycle A,D,F,B,A
(E,F) No Cycle A,C,E,F,D,A
(C,F) No Cycle C,A,B,F,C

Kruskal’s Algorithm Implementation
KruskalMST(Graph G)
update u and v to be in the same component
}
}
KruskalMST(Graph G)
foreach (V : vertices) {
makeMST(v);
}
sort edges in ascending order by weight
foreach(edge (u, v)){
if(findMST(v) is not in findMST(u)){
union(u, v)
}
}
+V(makeMST)
+ElogE
+E(2findMST + union)
+?
+?
+?
How many times will we call union?
V – 1
-> +Vunion + EfindMST

Appendix: MST Properties, Another
MST Application

Why do all of these MST Algorithms Work?
MSTs satisfy two very useful properties:
Cycle Property: The heaviest edge along a cycle is NEVER part of an MST.
Cut Property: Split the vertices of the graph any way you want into two sets A and B. The lightest
edge with one endpoint in A and the other in B is ALWAYS part of an MST.
Whenever you add an edge to a tree you create exactly one cycle, you can then remove any edge
from that cycle and get another tree out.
This observation, combined with the cycle and cut properties form the basis of all of the greedy
algorithms for MSTs.

One More MST application
Let’s say you’re building a new building.
There are very important building donors coming to visit TOMORROW,
- and the hallways are not finished.
You have n rooms you need to show them, connected by the unfinished hallways.
Thanks to your generous donors you have n-1 construction crews, so you can assign one to each
of that many hallways.
- Sadly the hallways are narrow and you can’t have multiple crews working on the same hallway.
Can you finish enough hallways in time to give them a tour?
Find: A spanning tree such that the weight of the
maximum edge is minimized.
Minimum Bottleneck Spanning Tree Problem

MSTs and MBSTs
Find: A spanning tree such that the weight of the
maximum edge is minimized.
Minimum Bottleneck Spanning Tree Problem
Find: A minimum-weight set of edges such that you
can get from any vertex of G to any other on only
those edges.
Minimum Spanning Tree Problem
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Graph on the right is a minimum bottleneck spanning tree, but not a minimum
spanning tree.

Finding MBSTs
Algorithm Idea: want to use smallest edges. Just start with the smallest edge and add it if it
connects previously unrelated things (and don’t if it makes a cycle).
Hey wait…that’s Kruskal’s Algorithm!
Every MST is an MBST (because Kruskal’s can find any MST when looking for MBSTs)
but not vice versa (see the example on the last slide).
If you need an MBST, any MST algorithm will work.
There are also some specially designed MBST algorithms that are faster (see Wikipedia)
Takeaway: When you’re modeling a problem, be careful to really understand what you’re looking
for. There may be a better algorithm out there.

lecture 23 algorithm design and analysis

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