Trees Featuring Minimum Spanning Trees HKOI Training 2005 Advanced Group Presented by Liu Chi Man (cx)

TreesFeaturing Minimum Spanning Trees

HKOI Training 2005Advanced Group

Presented by Liu Chi Man (cx)

2

Prerequisites

Asymptotic complexity Set theory Elementary graph theory Priority queues

3

Assumptions

In the coming slides, we only talk about simple graphs Extensions to multigraphs and

pseudographs should be straightforward if you understand the materials well

4

Notations

A graph G = (V, E) V is the set of vertices E is the set of edges

|V| is the number of vertices in V Sometimes we use V for

convenience |E| is the number of edges in E

Sometimes we use E for convenience

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Uh… a long way to go

What is a Tree? Disjoint Sets Minimum Spanning Trees Various Tree Topics

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What is a Tree?

A tree in Pokfulam, Hong Kong. (Photographer: cx)

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What is a Tree?

In graph theory, a tree is an acyclic, connected graph Acyclic means “with no cycles”

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What is a Tree?

Properties of a tree: |E| = |V| - 1

|E| = (|V|)

Adding an edge between a pair of non-adjacent vertices creates exactly one cycle

Removing any one edge from the tree makes it disconnected

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What is a Tree?

The following four conditions are equivalent: G is connected and acyclic G is connected and |E| = |V| - 1 G is acyclic and |E| = |V| - 1 Between any pair of vertices in G,

there exists a unique path G is a tree if at least one of the

above conditions is satisfied

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Other Properties of Trees

A tree is bipartite A tree with at least two vertices

has at least two leaves (vertices of degree 1)

A tree is planar

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ZzZZzzz… bored to death?


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The Union-Find Problem

Initially there are N objects, each in its own singleton set Let’s label the objects by 1, 2, …, N So the sets are {1}, {2}, …, {N}

There are two kinds of operations: Merge two sets (Union) Find the set to which an object

belongs Let there be a sequence of M

operations

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The Union-Find Problem

An example, with N = 4, M = 5 Initial: {1}, {2}, {3}, {4} Union {1}, {3} {1, 3}, {2},

{4} Find 3. Answer: {1, 3} Union {4}, {1,3} {1, 3, 4},

{2} Find 2. Answer: {2} Find 1. Answer {1, 3, 4}

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Disjoint Sets

We usually use disjoint-set data structures to solve the union-find problem

A disjoint set is simply a data structure that supports the Union and Find operations

How should a set be named? Each set has its own

representative element

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Disjoint Sets

Implementation: Naïve arrays Set[x] stores the representative of

the set containing x, for 1 ≤ x ≤ N <per Union, per Find>: <O(N),

O(1)> Slight modifications give <O(U),

O(1)> U is the size of the union

Worst case is O(MN) for all M operations

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Disjoint Sets

An example, with N = 4

x 1 2 3 4

Set[x] 1 2 3 4Initial:

x 1 2 3 4

Set[x] 1 2 1 4Union 1 3:

x 1 2 3 4

Set[x] 1 2 1 2Union 2 4:

x 1 2 3 4

Set[x] 1 1 1 1Union 1 2:

Find 4: Answer = 2

Find 4: Answer = 1

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Disjoint Sets

Why is it slow? Consider Union 1 2 on the table:

x 1 2 3 4 5 6 7 8 9

Set[x] 1 2 2 2 2 2 2 2 2

x 1 2 3 4 5 6 7 8 9

Set[x] 1 1 1 1 1 1 1 1 1

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Disjoint Sets

Idea for improvement: When doing a Union, always

update the set of smaller size The overall running time can be

shown to be O(M lgN)

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Disjoint Sets

Implementation: Forest A forest is a set of trees Each set is represented by a

rooted tree, with the root being the representative element

1

5 3

6

4

2 7

Two sets: {1, 3, 5} and {2, 4, 6, 7}.

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Disjoint Sets

Ideas Find

Traverses from the vertex up to the root

Union Merges two trees in the forest That is, the root of one tree becomes

a child of the other tree

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Disjoint Sets

An example, with N = 4

Initial:

Union 1 3:

Union 2 4:

Find 4:

1 3 42

1

3

42

1

3 4

2

1

3 4

2

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Disjoint Sets

An example, with N = 4 (cont.)

Union 1 2:

Find 4:

1

3

4

2

1

3

4

2

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Disjoint Sets

How to represent the trees? Leftmost-Child-Right-Sibling

(LCRS)? Too complex! We just need to traverse from bottom

to top – parent-finding should be supported

An array Parent[] is enough Parent[x] stores the parent vertex of x

if x is not the root of a tree Parent[x] stores x if x is the root of a

tree

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Disjoint Sets

The worst case is still O(MN ) overall

To improve the worst case time complexity, we employ two techniques Union-by-rank Path compression

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Disjoint Sets

Union-by-rank Recall that in our previous naïve

array implementation, the worst case time bound can be improved by always updating the smaller set in a Union

Here, we make the “smaller” tree a subtree of the “larger” tree To keep the resulting tree “small”

To compare the “sizes” of trees, we assign a rank to every tree in the forest

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Disjoint Sets

Union-by-rank (cont.) The rank of the tree is the height

of the tree when it is created A tree is created either at the

beginning, or after a Union

That means we need to keep track of the trees’ ranks during the Unions

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Disjoint Sets

Path compression See also the solution for Symbolic

Links (HKOI2005 Senior Final) When we carry out a Find, we

traverse from a vertex to the root Why don’t we move the vertices

closer to the root at the same time? Closer to the root means shorter

query time

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Disjoint Sets

An example, with N = 7 Now we Find 4

3

5 1

6

4

2 7

3

5 1

6

4

2 7

The root is 3

The root is 3

The root is 3

3

5 164

2 7

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Disjoint Sets

We ignore the effect of path compression on tree ranks

Union-by-rank alone gives a running time of O(M lgN)

Together with path compression, the improved running time is O(M(N)) is the inverse of the Ackermann's

function (n) is almost constant

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When can I have my lunch?


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Minimum Spanning Trees

Given a connected graph G = (V, E), a spanning tree of G is a graph T such that T is a subgraph of G T is a tree T contains every vertex of G

A connected graph must have at least one spanning tree

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Given a weighted connected graph G, a minimum spanning tree T* of G is a spanning tree of G with minimum total edge weight

Application: Minimizing the total length of wires needed to connect up a collection of computers

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Two algorithms Kruskal’s algorithm Prim’s algorithm

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Kruskal’s Algorithm

Algorithm Initially T is an empty set While T has less than |V|-1 edges

Do Choose an edge with minimum weight

such that it does not form a cycle with the edges in T, and add it to T

The edges in T form a MST

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Algorithm (Detailed) T is an empty set Sort the edges in G by their

weights For (in ascending weight) each

edge e Do If T {e} does not contain a cycle

Then Add e to T

Return T

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How to check if a cycle is formed? Depth-first search (DFS)

Time complexity is O(E2)

Alternatively, a disjoint set! The sets are connected components

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Algorithm (disjoint-set) T is an empty set Create sets {1}, {2}, …, {V} Sort the edges in G by their

weights For (in ascending weight) each

edge e Do Let e connect vertices x and y If Find(x) Find(y) Then

Add e to T, then Union(Find(x), Find(y))

Return T

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The improved time complexity is O(ElgV)

In fact, the bottleneck is sorting!

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Prim’s Algorithm

Algorithm Initially T is a set containing any

one vertex in V While T has less than |V|-1 edges

Do Among those edges connecting a

vertex in T and a vertex in V-T, choose one with minimum weight, and add it (with its two end vertices) to T

T is a minimum spanning tree

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Prim’s Algorithm Algorithm (Detailed)

T Cost[x] for all x V Marked[x] false for all x V Pred[x] for all x V Pick any u V, and set Cost[u] to 0 Repeat |V| times

Let w be a vertex of minimum Cost among all unMarked vertices

Marked[w] true T T Pred[x] For each unMarked vertex v such that (w, v) E Do

If weight(w, v) < Cost[v] Then Cost[v] weight(w, v) Pred[x] { (w, v) }

Return T

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Prim’s Algorithm

Time complexity: O(V2) If a (binary) heap is used to

accelerate the Find-Min operations, the time complexity is reduced to O(ElgV) O(VlgV+E) if a Fibonacci heap is

used

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The fastest known algorithm runs in O(E(V,E))

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MST Extensions

Second-best MST We don’t want the best!

Online MST See IOI2003 Path Maintenance

Minimum Bottleneck Spanning Tree The bottleneck of a spanning tree is

the weight of its maximum weight edge

An algorithm that runs in O(V+E) exists

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MST Extensions (NP-Hard)

Minimum Steiner Tree No need to connect all vertices,

but at least a given subset B V Degree-bounded MST

Every vertex of the spanning tree must have degree not greater than a given value K

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Are you with me?


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Various Tree Topics

Center and diameter Tree isomorphism

Canonical representation Prüfer code Lowest common ancestor (LCA)

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Supplementary Readings

Advanced: Disjoint set forest (Lecture slides) Prim’s algorithm Kruskal’s algorithm Center and diameter

Post-advanced (so-called Beginners): Lowest common ancestor Maximum branching

Documents

Trees Featuring Minimum Spanning Trees HKOI Training 2005 Advanced Group Presented by Liu Chi Man (cx)