The leader election problem is a fundamental problem in distributed computing.The classical leader election problem can be considered as findingthe processor with the maximum key in a distributed networkin which each processor has one key and a total order is defined on the keys.In this paper we define a generalized leader election problem that findsall the processors with the maximal keys on the basis of a partial order on the keys.We propose two distributed algorithms for the generalized leader election problem.The first algorithm solves the problem on a network by using a spanning tree of the network.The message complexity of the algorithm is $O(mn)$ where $m$ is the number of different keys and $n$ is the number of processors.The time complexity of the algorithm is $O(n)$.The second algorithm solves the problem using a coterie of the $n$ processors.The number of messages exchanged on the coterie is $O(?max?{rn n^{1.5}?})$ where $r$ is the number of the maximal keys.When the physical network for connecting the $n$ processors is considered the message and time complexities ofthe second algorithm are $O(?max ?{drn dn^{1.5}?})$ and $O(d)$ respectively where $d$ is the diameter of the network.

The leader election problem is a fundamental problem in distributed computing.The classical leader election problem can be considered as findingthe processor with the maximum key in a distributed networkin which each processor has one key and a total order is defined on the keys.In this paper, we define a generalized leader election problem that findsall the processors with the maximal keys on the basis of a partial order on the keys.We propose two distributed algorithms for the generalized leader election problem.The first algorithm solves the problem on a network by using a spanning tree of the network.The message complexity of the algorithm is $O(mn)$,where $m$ is the number of different keys and $n$ is the number of processors.The time complexity of the algorithm is $O(n)$.The second algorithm solves the problem using a coterie of the $n$ processors.The number of messages exchanged on the coterie is $O(\max\{rn,n^{1.5}\})$,where $r$ is the number of the maximal keys.When the physical network for connecting the $n$ processors is considered,the message and time complexities ofthe second algorithm are $O(\max \{drn,dn^{1.5}\})$ and $O(d)$, respectively,where $d$ is the diameter of the network.