The leader algorithm (Horrigan [394]) represents each cluster using a point, known as a leader, and assigns each point to the cluster corresponding to the closest leader, unless this distance is above a user-specified threshold. In that case, the point becomes the leader of a new cluster. What are the advantages and disadvantages of the leader algorithm as compared to K-means? Suggest ways in which the leader algorithm might be improved. Solution (a)K-Means Clustering Advantages and Disadvantages K-Means Advantages: 1) If variables are enormous, then K-Means the majority of the times computationally faster than hierarchical clustering, if we stay k smalls. 2) K-Means create tighter clusters than hierarchical clustering, particularly if the clusters are globular. K-Means Disadvantages: 1) Hard to predict K-Value. 2) With global cluster, it didn\'t work fine. 3) Dissimilar initial partitions can result in dissimilar final clusters. 4) It does not work fine with clusters of dissimilar size and dissimilar density. (b)Definition: The difficulty of leader election is for each processor finally to decide that whether it is a leader or not subject to merely one processor decides that it is the leader. An algorithm resolves the leader election difficulty if: States of processors are alienated into elected and not elected states. Once chosen, it remains as elected. In every implementation, precisely one processor becomes elected and the rest decide that they are not elected. A suitable leader election algorithm must convene the following conditions: Termination: the algorithm should finish eventually within a finite time one leader is selected. In randomized approaches this condition is sometimes weakened Uniqueness: there is exactly one processor that considers itself as leader. Agreement: all other processors know who the leader is. An algorithm for leader election may vary in following aspects: Communication mechanism: the processors are either synchronous in which processes are synchronized by a clock signal or asynchronous where processes run at arbitrary speeds. Process names: whether processes have a unique identity or are indistinguishable. Network topology: for instance, ring, acyclic graph or complete graph. Size of the network: the algorithm may or may not use knowledge of the number of processes in the system. Algorithms: Leader election in rings: Ring network topology A ring network is a connected-graph topology in which each node is exactly connected to two other nodes, i.e., for a graph with n nodes, there are exactly n edges connecting the nodes. A ring can be unidirectional, which means processors only communicate in one direction, or bidirectional, meaning processors may transmit and receive messages in both directions. Anonymous rings: A ring is said to be anonymous if every processor is identical. More formally, the system has the same state machine for every processor. There is no deterministic algorithm to elect a leader in anonym.

1. The leader algorithm (Horrigan [394]) represents each cluster using a point, known as a leader,
and assigns each point to the cluster corresponding to the closest leader, unless this distance is
above a user-specified threshold. In that case, the point becomes the leader of a new cluster.
What are the advantages and disadvantages of the leader algorithm as compared to K-means?
Suggest ways in which the leader algorithm might be improved.
Solution
(a)K-Means Clustering Advantages and Disadvantages
K-Means Advantages:
1) If variables are enormous, then K-Means the majority of the times computationally faster than
hierarchical clustering, if we stay k smalls.
2) K-Means create tighter clusters than hierarchical clustering, particularly if the clusters are
globular.
K-Means Disadvantages:
1) Hard to predict K-Value.
2) With global cluster, it didn't work fine.
3) Dissimilar initial partitions can result in dissimilar final clusters.
4) It does not work fine with clusters of dissimilar size and dissimilar density.
(b)Definition:
The difficulty of leader election is for each processor finally to decide that whether it is a leader
or not subject to merely one processor decides that it is the leader. An algorithm resolves the
leader election difficulty if:
States of processors are alienated into elected and not elected states. Once chosen, it remains as
elected.
In every implementation, precisely one processor becomes elected and the rest decide that they
are not elected.
A suitable leader election algorithm must convene the following conditions:
Termination: the algorithm should finish eventually within a finite time one leader is selected. In
randomized approaches this condition is sometimes weakened
Uniqueness: there is exactly one processor that considers itself as leader.
Agreement: all other processors know who the leader is.
An algorithm for leader election may vary in following aspects:
Communication mechanism: the processors are either synchronous in which processes are
synchronized by a clock signal or asynchronous where processes run at arbitrary speeds.
Process names: whether processes have a unique identity or are indistinguishable.

2. Network topology: for instance, ring, acyclic graph or complete graph.
Size of the network: the algorithm may or may not use knowledge of the number of processes in
the system.
Algorithms:
Leader election in rings:
Ring network topology
A ring network is a connected-graph topology in which each node is exactly connected to two
other nodes, i.e., for a graph with n nodes, there are exactly n edges connecting the nodes. A ring
can be unidirectional, which means processors only communicate in one direction, or
bidirectional, meaning processors may transmit and receive messages in both directions.
Anonymous rings:
A ring is said to be anonymous if every processor is identical. More formally, the system has the
same state machine for every processor. There is no deterministic algorithm to elect a leader in
anonymous rings, even when the size of the network is known to the processes. This is due to the
fact that there is no possibility of breaking symmetry in an anonymous ring if all processes run at
the same speed. The state of processors after some steps only depends on the initial state of
neighbouring nodes. So, because their states are identical and execute the same procedures, in
every round the same messages are sent by each processor. Therefore, each processor state also
changes identically and as a result if one processor is elected as a leader, so are all the others.
Randomized (probabilistic) leader election:
A common approach to solve the problem of leader election in anonymous rings is the use of
probabilistic algorithms. In such approaches, generally processors assume some identities based
on a probabilistic function and communicate it to the rest of the network. At the end, through the
application of an algorithm, a leader is selected (with high probability).
Synchronous ring:
Itai and Rodeh introduced an algorithm for a unidirectional ring with synchronized processes.
They assume the size of the ring (number of nodes) is known to the processes. For a ring of size
n, an processors are active. Each processor decides with probability of a^(-1) whether to become
a candidate. At the end of each phase, each processor calculates the number of candidates c and
if it is equal to 1, it becomes the leader. To determine the value of c, each candidate sends a
token (pebble) at the start of the phase which is passed around the ring, returning after exactly n
time units to its sender. Every processor determines c by counting the number of pebbles which
passed through.
This algorithm achieves leader election with expected message complexity of O(nlogn). A
similar approach is also used in in which a time-out mechanism is employed to detect deadlocks
in the system. There are also algorithms for rings of special sizes such as prime size and odd size.

3. Asynchronous ring:
In the case of asynchronous systems, the problem of leader election becomes more difficult. This
is because of the uncertainty introduced by the arbitrary response time of processors due to the
lack of a global clock. To tackle this problem, there are various approaches introduced. For
instance, Itai and Rodeh extended their algorithm by adding a message buffer and wake up
messages to trigger the computation in processes.
Uniform algorithm:
In typical approaches to leader election, the size of the ring is assumed to be known to the
processes. In the case of anonymous rings, without using an external entity, it is not possible to
elect a leader. Even assuming an algorithm exist, the leader could not estimate the size of the
ring. i.e. in any anonymous ring, there is a positive probability that an algorithm computes a
wrong ring size. To overcome this problem, Fisher and Jiang used a so-called leader oracle ? that
each processor can ask whether there is a unique leader. They show that from some point
upward, it is guaranteed to return the same answer to all processes.
Rings with unique IDs:
In one of the early works, Chang and Roberts proposed a uniform algorithm in which a processor
with the highest ID is selected as the leader. Each processor sends its ID in a clockwise direction.
A process receiving a message and compares it with its own. If it is bigger, it passes it through,
otherwise it will discard the message. They show that this algorithm uses O(n^2) messages and
O(nlogn) in the average case.
Hirschberg and Sinclair improved this algorithm with O(nlogn) message complexity by
introducing a 2 directional message passing scheme allowing the processors to send messages in
both directions.
Leader Election in Mesh:
The mesh is another popular form of network topology specially in parallel systems, redundant
memory systems and interconnection networks.
In a mesh structure, nodes are either corner (only two neighbours), border (only three
neighbours) or interior (with four neighbours). The number of edges in a mesh of size a x b is
m=2ab-a-b.
Unoriented mesh:
A typical algorithm to solve the leader election in an unoriented mesh is to only elect one of the
four corner nodes as the leader. Since the corner nodes might not be aware of the state of other
processes, the algorithm should first wake up the corner nodes. A leader can be elected as
follows.
Wake-up process: in which k nodes initiate the election process. Each initiator sends a wake-up
message to all its neighbouring nodes. If a node is not initiator, it simply forwards the messages

4. to the other nodes. In this stage at most 3n+k messages are sent.
Election process: the election in outer ring takes two stages at most with 6(a+b)-16 messages.
Termination: leader sends a terminating message to all nodes. This requires at most 2n messages.
The message complexity is at most 6(a+b)-16 and if the mesh is square shaped O(n).
Oriented Mesh:
An oriented mesh is a special case where are port numbers are compass labels, i.e. north, south,
east and west. Leader election in an oriented mesh is trivial. We only need to nominate a corner,
e.g. “north” and “east” and make sure that node knows it is a leader.
Torus:
Torus network structure.
A special case of mesh architecture is a torus which is a mesh with “wrap-around”. In this
structure, every node has exactly 4 connecting edges. One approach to elect a leader in such a
structure is known as electoral stages. Similar to procedures in ring structures, this method in
each stage eliminates potential candidates until eventually one candidate node is left. This node
becomes the leader and then notifies all other processes of termination. This approach can be
used to achieve a complexity of O(n). There also more practical approaches introduced for
dealing with presence of faulty links in the network.
Election in Hypercubes:H_4 hypercube network topology.
A Hypercube H_k is a network consisting of n=2^k nodes, each with degree of k and O(n log n)
edges. A similar electoral stages as before can be used to solve the problem of leader election. In
each stage two nodes (called duelists) compete and the winner is promoted to the next stage. This
means in each stage only half of the duelists enter the next stage. This procedure continues until
only one duelist is left, and it becomes the leader. Once selected, it notifies all other processes.
This algorithm requires O(n) messages. In the case of unoriented hypercubes, a similar approach
can be used but with a higher message complexity of O(nloglogn).
Election in complete networks:
Complete network structure.
Complete networks are structures in which all processes are connected to one another, i.e., the
degree of each node is n-1, n being the size of the network. An optimal solution with O(n)
message and space complexity is known. In this algorithm, processes have the following states:
Dummy: nodes that do not participate in the leader election algorithm.
Passive: the initial state of processes before start.
Candidate: the status of nodes after waking up. The candidate nodes will be considered to
become the leader.
To elect a leader, a virtual ring is considered in the network. All processors initially start in a
passive state until they are woken up. Once the nodes are awake, they are candidates to become

5. the leader. Based on a priority scheme, candidate nodes collaborate in the virtual ring. At some
point, candidates become aware of the identity of candidates that precede them in the ring. The
higher priority candidates ask the lower ones about their predecessors. The candidates with lower
priority become dummies after replying to the candidates with higher priority. Based on this
scheme, the highest priority candidate eventually knows that all nodes in the system are dummies
except itself, at which point it knows it is the leader.