Randomized Graph Algorithms - NTUAopapadig/archive/slidesrga.pdf · Minimum Spanning Trees The problem The problem " Given a connected, undirected graph, a spanning tree of that graph

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......Randomized Graph Algorithms

Vasileios-Orestis Papadigenopoulos

School of Electrical and Computer Engineering - N.T.U.A

papadigenopoulos [email protected]

July 22, 2014

Vasileios-Orestis Papadigenopoulos (NTUA) Randomized Graph Algorithms July 22, 2014 1 / 37

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Overview

...1 The Min-Cut problemThe problemThe s-t min-cut problemMin-cut and s-t min-cut problemThe Contract algorithmThe Contraction algorithmThe FastCut algorithm

...2 Minimum Spanning TreesThe problemDeterministic approachesBoruvka’s algorithmHeavy Edges and MST VerificationRandom Sampling for MSTsThe linear-time MST algorithm


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The Min-Cut problem

Overview




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The Min-Cut problem The problem

Min-cut


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The problem

”In graph theory, a minimum cut of a graph is a cut (a partition of thevertices of a graph into two disjoint subsets that are joined by at least one

edge) whose cut set has the smallest number of edges... ”wikipedia


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The problem

.Definition..

......

A cut in G is a partition of the vertices V = (C , C ) into two non-emptysets.

.Definition........The value or size of a cut C is the number of edges crossing the cut.

.Definition..

......

The min-cut problem is that of finding a cut with minimum size in aninput graph G.


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Also...

Multiple edges.

Connected graph assumption or else trivially solved with findingconnected components.

Weighted graphs generalization.

Assuming non-negative edges, or else the problem is NP-complete.The max-cut is a special case of it.


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The Min-Cut problem The s-t min-cut problem

The s-t min-cut problem

.Definition..

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The s-t min-cut problem is the min-cut problem with the restriction thattwo distinguished vertices s and t have the property s ∈ C and t ∈ C .

.Lemma..

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The value of a maximum s-t flow in a network equals the value of a s-tmin-cut.

.Maximum flow duality..

......

The s-t min-cut problem can be solved via the maximum flow problem inO(mn log (n2/m)).


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The Min-Cut problem Min-cut and s-t min-cut problem

Deterministic approach of min-cut via s-t min-cut

.Lemma..

......

The min-cut problem for a graph G can be solved via polynomial numberof invocations of an s-t algorithm applied to the same graph.

.Deterministic complexity..

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It can be shown that n-1 flow computations (s-t min-cut calculations)suffice. Since the best deterministic maximum flow algorithm runs inO(mn log (n2/m)), this approach will require Ω(mn2). Fortunately, the n-1maximum flow computations can be implemented in time proportional tothe cost of a single maximum flow computation. So we can compute amin-cut in time O(mn log (n2/m)).


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The Min-Cut problem The Contract algorithm

The Contract algorithm

.Definition..

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Given an edge (x,y) in a multigraph G(V,E), a contraction of the edge(x,y) corresponds to replacing the vertices x and y by a new vertex z, andfor each u /∈ x , y replacing the edge (x,u) and (y,u) by the edge (z,u).Any multiple edges created are to be retained. The graph obtained by thiscontraction is denoted by G/(x,y).

.Definition..

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Given a collection of edges F ⊆ E , the contracted graph is denoted byG/F. The vertex set and the edge set of the graph G/F are denoted byV/F and E/F.


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The Contract algorithm


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The Contract algorithm - Analysis

.Lemma..

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It is possible to keep track of the correspondence between the originalgraph and the contracted one in O(n) time per contraction.

.Lemma..

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The effect of contracting the edges in F is independent of the order ofcontraction.

.Lemma..

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The selection of an edge (x,y) uniformly at random from the set of alledges in the graph G can be done in O(n).


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The Contract algorithm - Analysis

.Theorem..

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The Contract algorithm can be implemented to run in O(n2) time on anyn-vertex multigraph G.


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The Min-Cut problem The Contraction algorithm

The Contraction algorithm

.Lemma..

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A cut in C is produced as output by the algorithm Contract iff none ofthe edges crossing this cut is contracted by the algorithm.

Let K be the a min-cut in G with size of k..Lemma..

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In a n-vertex multigraph G with min-cut value k, no vertex has a degreesmaller than k. Further, the total number of edges in the graph satisfiesm ≥ nk/2.

.Lemma..

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Given an edge (x,y) in a graph G, the min-cut value in G/(x,y) is at leastas large as the min-cut value in G.


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The number of vertices in H decreases by one during each iteration ofContract algorithm. At the i-th iteration, there are ni = n − i + 1 verticesin H. Since the number of edges in each step is at least nik/2(handshaking lemma), the probability that any edge of K is contracted isat most 2/ni . So:


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.Theorem..

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Any specific min-cut K is output by the algorithm Contract withprobability Ω(n−2)

.The algorithm..

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The above theorem gives as a Monte Carlo algorithm running inO(n4 log n), by repeating the algorithm contract O(n2 log n) times.

.Lemma..

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Suppose the algorithm Contract is terminated when the number of verticesremaining in the contracted graph is exactly t. Then any specific min-cutK survives in the resulting contracted graph with probability at least:(t2

)/(n2

)= Ω(

(tn

)2)


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The Min-Cut problem The FastCut algorithm

The FastCut algorithm

At this point, we will modify the implementation of the contractionalgorithm to reduce its running time to O(n2 log n). As the complexity ofour general module algorithm contract cannot be less than O(n2), the onlyway to succeed it is by increasing the probability that a min-cut isproduced by the algorithm contract. The crucial insight is to use twoindependent invocations of the contract algorithm itself on the contractedgraph with n/

√2 vertices.


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The FastCut algorithm


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The FastCut algorithm - Analysis

.Theorem..

......Algorithm FastCut runs in O(n2 log n) time and O(n2) space.

.Theorem..

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Algorithm FastCut succeeds in finding a min-cut with probabilityΩ(1/ log n).


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Minimum Spanning Trees

Overview




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Minimum Spanning Trees The problem

Minimum Spanning Trees


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The problem

”Given a connected, undirected graph, a spanning tree of that graph is asubgraph that is a tree and connects all the vertices together. A single

graph can have many different spanning trees. We can also assign a weightto each edge, which is a number representing how unfavourable it is, and

use this to assign a weight to a spanning tree by computing the sum of theweights of the edges in that spanning tree. A minimum spanning tree(MST) or minimum weight spanning tree is then a spanning tree with

weight less than or equal to the weight of every other spanning tree.”wikipedia


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Also...

Minimum Spanning Forest.

Distinct edge weights assumption.

The exact weight is irrelevant in the unit-cost RAM model.


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Minimum Spanning Trees Deterministic approaches

Deterministic approaches

Prim’s algorithm, greedy with Fibonacci heap O(E + V logV ).

Kruskal’s algorithm, greedy with union-find O(E log E ).

Boruvka’s algorithm, O(E logV ).

Best-known algorithm runs in O(E log β(E ,V )), whereβ(E ,V ) = min i | log (i)V ≤ E/V . Unfortunately, the algorithmhas a large constant due to its complicated data structures. Notapplicable in practice.


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Minimum Spanning Trees Boruvka’s algorithm

Boruvka’s algorithm

At this point, we are going to present the idea behind Boruvka’s algorithm,as it will be useful for the randomized algorithm we are going to create.Sequential implementation in O(m log n):

Data: A graph GResult: The MST of Ginitialization;in each phase in time O(m) with two passes of the edges;1st pass: find the less-weighted edge of each component. Consecutive”find”, the edges of the same component are noted;2nd pass: include the less-weighted edges in the MST and update thecomponents using ”union”;

Algorithm 1: Boruvka’s algorithm


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The reasoning of the algorithm is based on the following lemmas:.Lemma..

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Let v ∈ V be any vertex in G. The MST for G must contain the edge(v,w) that is the minimum-weight edge incident on v.

.Lemma..

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Given a graph G with n vertices and m edges, a Boruvka phase can beimplemented in time O(n +m) .


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.Lemma..

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The set of edges marked for contraction during a Boruvka phase induces aforest in G.

.Lemma..

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Let G’ be the graph obtained from G after a Boruvka phase. The MST ofG is the union of the edges marked for contraction with the edges in theMST of G’


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Conclusions:.Reduction..

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Boruvka’s algorithm reduces the MST problem in a n-vertex graph with medges to the MST problem in an (n/2)-vertex graph with at most m edges.

.Boruvka phase complexity........In each Boruvka phase the time required for the reduction is O(n +m).

.Boruvka’s running time..

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It follows that the Boruvka’s algorithm worst-case running time isO(m log n)


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Minimum Spanning Trees Heavy Edges and MST Verification

Heavy Edges and MST Verification

Now, we are going to touch the subject of MST-Verification problemcomplexity:.The problem........Take as input a tree T in a graph G. Check if T is an MST of G.

.Definition..

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Fix a forest F in G and consider any pair of vertices u, v ∈ V . Let wF (u,v)denote the maximum weight of an edge on the (unique) path P(u,v) in F.We set wF (u,v)=∞ when u and v are disconnected in G.

.Definition..

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An edge (u, v) ∈ E is said to be F-heavy if w(u,v)> wF (u,v). The edge(u,v) is said to be F-light if w(u,v)≤ wF (u,v).


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Minimum Spanning Trees Heavy Edges and MST Verification

Heavy Edges and MST Verification

.Lemma........All edges in a forest F must be F-light.

.Lemma..

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Let F be any forest in G. If an edge (u,v) is F-heavy, then it does not lie inthe MST for G.

.Existence of MST-verification algorithms..

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There exist linear-time verification algorithms that can also identify allF-heavy and F-light edges with respect to any forest F.

.Theorem..

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Given a graph G and a forest F, all F-heavy edges in G can be identified intime O(n +m).


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Minimum Spanning Trees Random Sampling for MSTs

Random Sampling for MSTs

In the MST algorithm we are constructing, the use of randomization is inthe use of random sampling to identify and eliminate edges that areguaranteed not to belong to the MST.


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Let G a graph..Definition..

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We define as G(p) the random graph obtained by independently includingeach edge of G in G(p) with probability p.

Obviously, G(p) is expected to have n vertices and mp edges. There is noguarantee, although, that G(p) is connected..

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Let F be the minimum spanning forest for G(p). Then, for reasonably largevalues of p, F should be a very good approximation to the MST of G.

We remind the lemma suggesting that F-heavy edges does not lie in theMST for G. So we expect very few edges in G to be F-light.


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.Lemma..

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Let F be the minimum spanning forest in the random graph G(p) obtainedby independently including each edge of G with probability p. Then thenumber of F-light edges in G is stochastically dominated by a randomvariable X that has a negative binomial distribution with parameters n andp. In particular, the expected number of F-light edges in G is at most n/p.


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Minimum Spanning Trees The linear-time MST algorithm

The expected linear-time MST algorithm


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Minimum Spanning Trees The linear-time MST algorithm

Properties

Correctness. In each recursion phase the Boruvka step includes edgesthat are in the MST. The verification step excludes only F-heavyedges of G that cannot be part of the MST.

Complexity. Recursive relationT (n,m) ≤ T (n/8,m/2) + T (n/8, n/4) + c(n +m) leading toexpected O(n +m)

Worst case analysis. Worst runtime is equivalent to the runtime ofBoruvka’s algorithm (”Las Vegas” algorithm).


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References

References

Rajeev Motwani & Prabhakar Raghavan

Randomized Algorithms

Karger David, (1993)

”Global Min-cuts in RNC and Other Ramifications of a Simple Mincut Algorithm”

Karger, David R.; Klein, Philip N.; Tarjan, Robert E. (1995)

”A randomized linear-time algorithm to find minimum spanning trees”

Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein

Introduction to Algorithms

V.K Balakrishnan

Schaum’s Outline of Theory and Problems of Combinatorics


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References

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Randomized Graph Algorithms - NTUAopapadig/archive/slidesrga.pdf · Minimum Spanning Trees The problem The problem " Given a connected, undirected graph, a spanning tree of that graph