On the Parallel Complexity of Hierarchical Clustering and

CC-Complete Problems

Dr. Raymond GreenlawSchool of Computing

Armstrong Atlantic State Universityand

Dr. Sanpawat KantabutraDepartment of Computer Science

Chiang Mai University

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 2

Outline• Introduction• Preliminaries• Algorithms for Hierarchical Clustering• Complexity of Hierarchical Clustering• CC-Complete Problems• Conclusions and Open Problems• References• Acknowledgments

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 3

Outline• Introduction• Preliminaries• Algorithms for Hierarchical Clustering• Complexity of Hierarchical Clustering• CC-Complete Problems• Conclusions and Open Problems• References• Acknowledgments

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 4

Introduction• Clustering is a division of data into

groups of ‘similar’ objects, where each group is given a more-compact representation.

• Used to model very large data sets.

• Points are more similar to their own cluster than to points in other clusters.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 5

Introduction• Useful tool in data mining, where

immense data sets which are difficult to store and to manipulate are involved.

• Study the parallel complexity of the hierarchical clustering problem.

• Builds a tree of clusters.• Sibling clusters in this tree partition the

points associated with their parent.• Can explore data using various levels of

granularity.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 6

Introduction• Two widely studied models

– Bottom-Up Starts with single-point clusters and then recursively merges two or more of the most-‘appropriate’ clusters.

– Top-Down Starts with one large cluster consisting of all the data points and then recursively splits the most-‘appropriate’ cluster.

• In both methods, the process continues until a desired stopping condition is met such as a required number of clusters or a diameter bound of the ‘largest’ cluster.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 7

Introduction• A variety of sequential versions of

hierarchical-clustering methods have been studied:– Cure Guha, et al.: Bottom-Up, good for

clusters having arbitrary shapes or outliers– Chameleon Karypis et al.: Bottom-Up,

relies heavily on graph partitioning– Principal Direction Divisive Partitioning

Boley: Top-Down, good for document collections

– Hierarchical Divisive Bisecting k-means Steinbach: Top-Down

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 8

Introduction• Address the parallel complexity of

hierarchical clustering.• Describe known sequential algorithms for

top-down and bottom-up hierarchical clustering.

• Parallelize top-down, when n points are to be clustered, provide an O(log n)-time, n2-processor CREW-PRAM algorithm that computes the same output as the corresponding sequential algorithm.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 9

Introduction• Define a natural decision problem based

on bottom-up hierarchical clustering and add this Hierarchical Clustering Problem (HCP) to the list of CC-complete problems, adding a data mining problem for the first time.

• Show that HCP is one of the computationally most-difficult problems in the Comparator Circuit Value Problem (CCVP) class.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 10

Introduction• Demonstrate that the HCP is very

unlikely to have an NC algorithm.• In sharp contrast, give an NC algorithm

for the top-down sequential approach.• Parallel complexities of top-down and

bottom-up are different, unless CC equals NC.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 11

Outline• Introduction• Preliminaries• Algorithms for Hierarchical Clustering• Complexity of Hierarchical Clustering• CC-Complete Problems• Conclusions and Open Problems• References• Acknowledgments

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 12

Preliminaries• Interested in relating the complexity of

hierarchical clustering to that of a problem involving Boolean circuits containing comparator gates.

• Comparator gates have two output wires, the first outputting the minimum and the second outputting the maximum of its two inputs.

• Each output has a maximum fanout of one.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 13

Preliminaries• Based on the comparator gate• Basis for an entire complexity class

Comparator Circuit Value Problem (CCVP)• Given: An encoding of a Boolean

circuit composed of comparator gates, inputs x1,…,xn, and a designated output y.

• Problem: Is output y of TRUE on input x1,…,xn?

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 14

Preliminaries• Let P denote the class of all languages

decidable in polynomial time.• Let NC denote the class of all languages

decidable in poly-logarithmic time using a polynomial number of processors on a PRAM.

• Let RNC denote the randomized version of NC.• Let NLOG denote the class non-deterministic

logarithmic space.• Let CC denote the class of problems that are

NC many-one reducible to CCVP.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 15

Outline• Introduction• Preliminaries• Algorithms for Hierarchical

Clustering• Complexity of Hierarchical Clustering• CC-Complete Problems• Conclusions and Open Problems• References• Acknowledgments

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 16

Algorithms for Hierarchical Clustering

Sequential Algorithms – Bottom-Up• Input: set of points, distance function,

bound B, and desired number of clusters, k• Output: set of clusters• Pair up all points starting with the two

closest ones, then the next remaining two closest ones, and so on, until all are paired.

• Next, the sets of points X and Y minimizing dmin(X,Y) over all remaining sets are merged, until only k sets remain.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 17

Algorithms for Hierarchical Clustering

Sequential Algorithms – Bottom-Up (cont.)• Assumed that the number of input

points is even.• There are no restrictions placed on the

distance function.• In the first phase of the algorithm points

are clustered whose distance is less than or equal to B.

• Operates in polynomial time.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 18

Algorithms for Hierarchical Clustering

Sequential Algorithms – Top-Down• Function v(G) takes a graph as its

argument and returns a set that consists of the vertices from G.

• Input: set of points, a distance function, and the desired number of clusters k

• Output: set of clusters• All points start in the same cluster.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 19

Algorithms for Hierarchical Clustering

• Compute a minimum-cost spanning tree.

• Form clusters by repeatedly removing the highest-cost edge from what remains of a minimum-cost spanning tree of the graph corresponding to the initial set of points with respect to the distance function, until exactly k sets have been formed.

• Runs in polynomial time.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 20

Algorithms for Hierarchical Clustering

• Top-Down and Bottom-Up have different parallel complexities, unless CC equals NC.

• Prove that the exact same clusters as produced by the Sequential (Top-Down) Hierarchical Clustering Algorithm can be computed in NC.

• A natural decision problem based on the Sequential (Bottom-Up) Hierarchical Clustering Algorithm is CC-complete.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 21

Algorithms for Hierarchical Clustering

• Since a CC-complete problem is very unlikely to have an NC algorithm and a problem with an NC algorithm is very unlikely to be CC-complete, the parallel complexities of these two sequential algorithms are different.

• For a fast parallel algorithm for hierarchical clustering, the algorithm should be based on the Top-Down approach.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 22

Algorithms for Hierarchical Clustering

• Theorem: Let n denote the number of points to be clustered. The Parallel (Top-Down) Hierarchical Clustering Algorithm can be implemented in O(log n) time using n2 processors on the CREW PRAM.

• This algorithm is an NC algorithm, which means that the clusters can be computed very fast in parallel.

• Any reasonable decision problem based on this algorithm will be in NC.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 23

Outline• Introduction• Preliminaries• Algorithms for Hierarchical Clustering• Complexity of Hierarchical

Clustering• CC-Complete Problems• Conclusions and Open Problems• References• Acknowledgments

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 24

Complexity of Hierarchical Clustering

Hierarchical Clustering Problem (HCP)• Given: A set S of n points in Rd, a

distance function dS : S x S N, the number of clusters k ≤ n/2 N, a distance bound B, and two points x, y S.

• Problem: Are x and y with dS(x, y) ≤ B in the same cluster C after the first-phase of the Sequential (Bottom-Up) Hierarchical Clustering Algorithm?

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 25

Complexity of Hierarchical Clustering

• No restrictions placed on the properties the distance function must satisfy, the distances themselves must be natural numbers.

• This version of the problem easily reduces to the version where the weights come from R+.

• Not concerned with the distance between a point and itself, the k is the number of clusters to be formed.

• x and y are required to be no further apart than the distance bound B.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 26

Complexity of Hierarchical Clustering

Lexicographically First Maximal Matching Problem (LFMMP)

• Given: An undirected graph G = (V, E) with an ordering on its edges plus a distinguished edge e E.

• Problem: Is e in the lexicographically first maximal matching of G?

• A matching is maximal if it cannot be extended.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 27

Complexity of Hierarchical Clustering

• LFMMP is CC-complete [Cook 1982, Mayr and Subramanian 1992].

• Theorem: The Hierarchical Clustering Problem is NC many-one reducible to the Lexicographically First Maximal Matching Problem, that is, HCP ≤ LFMMP.

• HCP is in CC.

mNC

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 28

Complexity of Hierarchical Clustering

• Theorem: The Lexicographically First Maximal Matching Problem is NC many-one reducible to the Hierarchical Clustering Problem, that is, LFMMP ≤ HCP.

• Proof Sketch: Let G = (V = {1,…,n},E), ø : E {1,…,|E|} be an ordering on E, and e = {u,v} E be an instance of the LFMMP. Construct instance of HCP, a set S of n points p1,…,pn in Rd, a distance function dS : S x S N, clusters k ≤ n/2 N, bound B, and x,y S.

mNC

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 29

Complexity of Hierarchical Clustering

• Proof (cont.): Let S = {1,…,n,n+1,…,2n}. Let V’ = S – V. Define the distance function between each pair of points in S as follows:

• Let B = |E|, k = n, and take u and v as our points

dS(x,y) = ø({x,y})

if {x,y} E

= 2|E| if x V and y V’ or vice versa

= 3|E| if x V’, y V’, and x ≠ y

= 4|E| if x ≤ n, y ≤ n, x ≠ y, and

{x,y} E

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 30

Complexity of Hierarchical Clustering

• Theorem: The Hierarchical Clustering Problem is CC-complete.

• This expands the list of CC-complete problems and adds the first clustering/data mining problem to the class.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 31

Outline• Introduction• Preliminaries• Algorithms for Hierarchical Clustering• Complexity of Hierarchical Clustering• CC-Complete Problems• Conclusions and Open Problems• References• Acknowledgments

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 32

CC-Complete ProblemsComparator Circuit Value Problem (CCVP)• Given: An encoding of a Boolean circuit

composed of comparator gates, inputs x1,…,xn, and a designated output y.

• Problem: Is output y of TRUE on input x1,…,xn?

• References: [Cook 1982, Mayr and Subramanian 1992]

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 33

CC-Complete ProblemsLexicographically First Maximal Matching

Problem (LFMMP)• Given: An undirected graph G = (V, E) with

an ordering on its edges plus a distinguished edge e E.

• Problem: Is e in the lexicographically first maximal matching of G?

• References: [Cook 1982, Mayr and Subramanian 1992]

• Remarks: Resembles the Lexicographically First Maximal Independent Set Problem which is P-complete.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 34

CC-Complete ProblemsStable Marriage Problem (SMP)• Given: A set of n men and a set of n

women. For each person a ranking of the opposite sex according to their preference for a marriage partner.

• Problem: Does the given instance of the problem have a set of marriages that is stable? The set is stable if there is no unmatched pair {m, w} such that both m and w prefer each other to their current partners.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 35

CC-Complete ProblemsStable Marriage Problem (SMP)• References: [Mayr and Subramanian

1992, Subramanian 1989]• Remarks: If the preference lists are

complete, there is always a solution. Several variations of the SMP are also known to be equivalent to the CCVP. The Male-Optimal Stable Marriage Problem finds a matching in which no man could do any better in a stable marriage.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 36

CC-Complete ProblemsStable Marriage Stable Pair Problem (SMSPP)• Given: A set of n men and n women, for each

person a ranking of the opposite sex according to their preference for a marriage partner, and a designated couple Alice and Bob.

• Problem: Are Alice and Bob a stable pair for the given instance of the problem? That is, is it the case that Alice and Bob are married to each other in some stable marriage?

• References: [Mayr and Subramanian 1992, Subramanian 1989]

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 37

CC-Complete ProblemsStable Marriage Minimum Regret Problem

(SMMRP)• Given: A set of n men and n women, for

each person a ranking of the opposite sex according to their preference for a marriage partner, and a natural number k, 1 ≤ k ≤ n.

• Problem: Is there a stable marriage in which every person has regret at most k? The regret of a person in a stable marriage is the position of her mate on her preference list.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 38

CC-Complete ProblemsStable Marriage Minimum Regret Problem

(SMMRP)• References: [Mayr and Subramanian

1992, Subramanian 1989]• Remarks: The goal in this problem is to

minimize the maximum regret of any person.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 39

CC-Complete ProblemsTelephone Connection Problem (TCP)• Given: A telephone line with a fixed

channel capacity k, a natural number l, and a sequence of calls (s1, f1),…, (sn, fn), where si (fi) denotes the starting (respectively, finishing) time of the i-th call. The i-th call can be serviced at time si if the number of calls being served at that time is less than k. If the call cannot be served, it is discarded. When a call is completed, the channel is freed up.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 40

CC-Complete ProblemsTelephone Connection Problem (TCP)• Problem: Is the l-th call serviced?• References: [Ramachandran and Wang

1991]• Remarks: There is an O(min( ,k) log

n)-time EREW-PRAM algorithm that uses n processors for solving the TCP.

n

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 41

CC-Complete ProblemsInternal Diffusion Limited Aggregation

Predication Problem (IDLAPP)• Given: A time T and a list of moves

(t,i,s), one for each time 0 ≤ t ≤ T indicating that at time t for particle i, if still active, will visit site s, plus a designated site d, and a designated particle p. A particle is active if it is still moving within the cluster, that is, the particle has not yet stuck to the cluster because all of the sites that it has visited so far were occupied already.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 42

CC-Complete ProblemsInternal Diffusion Limited Aggregation

Predication Problem (IDLAPP)• Problem: Is site d occupied and is site p

active at time T?• References: [Moore and Machta 2000]

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 43

CC-Complete ProblemsInternal Diffusion Limited Aggregation

Predication Square Lattice Problem• Given: A time T and a list of moves

(t,i,s) on a square lattice, one for each time 0 ≤ t ≤ T indicating that at time t for particle i, if still active, will visit site s, plus a designated site d, and a designated particle p.

• Problem: Is site d on the square latice occupied and is site p active at time T?

• References: [Moore and Matcha 2000]

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 44

CC-Complete ProblemsHierarchical Clustering Problem (HCP)• Given: A set S of n points in Rd, a distance

function dS : S x S N, the number of clusters k ≤ n/2 N, a distance bound B, and two points x, y S.

• Problem: Are x and y with dS(x, y) ≤ B in the same cluster C after the first-phase of the Sequential (Bottom-Up) Hierarchical Clustering Algorithm?

• Reference: [This work 2006]

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 45

Outline• Introduction• Preliminaries• Algorithms for Hierarchical Clustering• Complexity of Hierarchical Clustering• CC-Complete Problems• Conclusions and Open Problems• References• Acknowledgments

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 46

Conclusions• A natural decision problem based on

bottom-up hierarchical clustering is CC-complete.

• Top-down hierarchical clustering is in NC.• Brings the number of known CC-complete

problems to ten, and shows that the HCP is unlikely to have a NC algorithm.

• Fast parallel algorithms for hierarchical clustering should be based on a top-down approach.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 47

Open Problems• Is Euclidean HCP CC-complete? (It is in

CC.)• Determine the complexity of the

second-phase of the Sequential (Bottom-Up) Hierarchical Clustering Algorithm.

• Add new problems to the class of CC-complete problems.

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 48

Outline• Introduction• Preliminaries• Algorithms for Hierarchical Clustering• Complexity of Hierarchical Clustering• CC-Complete Problems• Conclusions and Open Problems• References• Acknowledgments

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 49

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On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 53

Outline• Introduction• Preliminaries• Algorithms for Hierarchical Clustering• Complexity of Hierarchical Clustering• CC-Complete Problems• Conclusions and Open Problems• References• Acknowledgments

On the Parallel Complexity of Hierarchical Clustering and CC-Complete Problems — Greenlaw and Kantabutra — 54

Acknowledgements• Computer Science Department at

Chiang Mai University, Thailand• Fulbright Commissions of Thailand and

the United States• Jim Hoover and Larry Ruzzo for material

from [Greenlaw, Hoover, and Ruzzo 1995]