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CSC 301.01, Class 32: Sets and union-find

Overview

  • Preliminaries
    • Notes and news
    • Upcoming work
    • Extra credit
    • Questions
  • A set ADT
  • Data structure design, revisited
  • The union-find structure
  • Analyzing union find
  • Improving union-find

News / Etc.

  • I expect that most upper-level CS classes will fill this semester. You help the department better consider stress points and how to address them if you register earlier, rather than later.
  • Prof. French noted that Kruskal’s algorithm serves as second purpose: It allows you to count the number of conntected components in a graph.
  • The due date of Exam 2 has been pushed to Friday.

Upcoming work

  • Exam 2 due Friday at 4:00 p.m.
  • Read Skiena 6.1-6.4 for Wednesday (or Friday).

Extra Credit (Academic/Artistic)

  • CS Table, Tomorrow
  • Convocation Thursday at 11 a.m. in JRC 101. “Work’s Provocative Future: Which Graduates Will Thrive?”

Extra credit (Peer)

  • Pub-Free Quiz: Math/Stats, Wednesday
  • Swim meet, Saturday
  • Orchestra Saturday at 2pm in Sebring Lewis

Extra Credit (Misc)

Other good things

  • “The First Time I Walked on the Moon”. Thursday 7:30, Friday 7:30, Saturday 2:00, Saturday 7:30, Sunday 2:00.
  • Fresh Flutes Thursday
  • Voice recitals Friday at 4:15 (Henderson) and 7:00 (Manuel)
  • Women’s Basketball vs. Emmaus Wed at 5:00 p.m.
  • Men’s Basketball vs. Emmaus Wed at 7:00 p.m.

Questions

How do I do part c of problem 3? (That is, how do I determine the final value based on the initial values, given that things are random?)
A good invariant will help.
Should we prove the invarant correct
A good argument suffices.
Does pseudocode suffice for the online algorithm?
Psuedocode is fine. But you do need to argue that it’s O(n).
A description is also fine. But you do need to argue that it’s O(n).
For problem 4: Are v and u already in the graph?
Maybe. At least one is. You may suppose wlog that v is in the graph.
“wlog” = “without loss of generality”
Does that mean we need to check if it’s already in the graph?
It seems so.
Can we use existing algorithms and their running time?
Certainly.
Can we use the Web site that builds red-black trees for you?
Yes, if you cite it.
But you might want to build them yourself, first.
Do we have to prove that our solution to #4 is correct?
No.
Do we have an in-class final?
Yes.
I have two in-class finals that day. Will you find another time?
Yes.
When is our final?
Wednesday at 9am!
Next Wednesday?
No. Wednesday of finals week.

A set ADT

Suppose we have a program that requires us to deal with sets.

  • What goes in the set interface? (Alternately: What operations do you want as you deal with sets?)

  • set.add!(value) or set.add(value) => set
  • set.remove!(value) or set.remove(value) => set
  • set.contains?(value) => boolean
  • set.union!(set) or set.union(set) => set
  • set.intersect!(set) or set.intersect(set) => set
  • set.subtract!(set) or set.subtract(set) => set
  • set.disjunction!(set) or set.disjunction(set) => set
  • set.filter(proc) => set
  • set.subsetOf?(set) => boolean
  • set.reduce(binproc) => val
  • set.map(proc) => set
  • set.product(set) => set
  • set.iterate(proc) or set.iterator() => Iterator.

Data structure design, revisited

  • Sometimes it’s better to focus on the particular things you need for a particular task, rather than the “kitchen sink” approach.
  • We will look at a small set of operations and consider how to implement them efficiently.
  • SetOfSets.set-of-one(val)
  • SetOfSets.union!(set,set)
  • SetOfSets.find-set(val)
  • Assume that each value can only be in one set.
  • Note: We will use find-set primarily to determine if two values are in the same set.

What strategies might you use?

  • “hash tables”
  • “trees”

Examples

  • Values a, b, c, d, e,f are all in separate sets.
  • We union the {a} and {b} sets, giving us {a,b}
  • We union the {a,b} and {c} sets, giving us {a,b,c}
  • We union the {d} and {e} sets, giving us {d,e}
  • We union the {a,b,c} and {d,e} sets, giving us {a,b,c,d,e}
  • We union the {a,b,c,d,e} and {f} sets, giving us {a,b,c,d,e,f}

The union-find structure

  • Represent each set as a directed graph: An upside-down tree. Children point to parents.
  • The root is the representative element (what we get from find-set)
  • How do we keep the tree shallow?
    • When unioning two trees, have the root of the smaller one point to the root of the bigger one.
    • Our height will never be bigger than log(n).
  • So we now have union is O(1) and find is O(logn)
    • union could be O(logn) if we need to find the root
  • Keep the size in each tree so that we know which is smaller
  • Note: This is really helpful for Kruskal’s, which repeatedly asks if an edge joins two different components

Analyzing union find

Improving union-find

  • Whenver you run find, update the links from every element along the way.

Kruskal’s counts connected components