2016/1/6Part I1 A Taste of Parallel Algorithms
2016/1/6Part I2 We examine five simple building-block parallel operations and look at the corresponding algorithms on four simple parallel architectures: linear array, binary tree, 2D mesh, and a simple sharedvariable computer.
2016/1/6Part I3 Semigroup Computation
2016/1/6Part I4 Parallel Prefix Computation
2016/1/6Part I5 Packet Routing A packet of information resides at Processor i and must be sent to Processor j. The problem is to route the packet through intermediate processors, if needed, such that it gets to the destination as quickly as possible. The problem becomes more challenging when multiple packets reside at different processors, each with its own destination. When each processor has at most one packet to send and one packet to receive, the packet routing problem is called one-to- one communication or 1-1 routing.
2016/1/6Part I6 Broadcasting Given a value a known at a certain processor i, disseminate it to all p processors as quickly as possible, so that at the end, every processor has access to, or "knows," the value. This is sometimes referred to as one-to-all communication. one-to-many communication, is known as multicasting.
2016/1/6Part I7 Sorting Rather than sorting a set of records, each with a key and data elements, we focus on sorting a set of keys for simplicity.
2016/1/6Part I8 Linear Array D=p-1 d=2 Ring?
2016/1/6Part I9 Binary Tree If all leaf levels are identical and every nonleaf processor has two children, the binary tree is said to be complete. D= d=3
2016/1/6Part I10 2D Mesh D= d=4 Torus?
2016/1/6Part I11 Shared memory A shared-memory multiprocessor can be modeled as a complete graph, in which every node is connected to every other node. D=1 d=p-1
2016/1/6Part I12 Algorithms for a Linear Array (1) Semigroup Computation –Let us consider first a special case of semigroup computation, namely, that of maximum finding. Each of the p processors holds a value initially and our goal is for every processor to know the largest of these values.
2016/1/6Part I13 Algorithms for a Linear Array (2) Parallel Prefix Computation (Case1)
2016/1/6Part I14 Algorithms for a Linear Array (3) Parallel Prefix Computation (Case2, more than one value)
2016/1/6Part I15 Algorithms for a Linear Array (4) Packet Routing
2016/1/6Part I16 Algorithms for a Linear Array (5) Broadcasting –If Processor i wants to broadcast a value a to all processors, it sends an rbcast(a) (read r-broadcast) message to its right neighbor and an lbcast(a) message to its left neighbor.
2016/1/6Part I17 Algorithms for a Linear Array (6) Sorting (Case 1)
2016/1/6Part I18 Algorithms for a Linear Array (7) Sorting (Case 2, odd-even transposition) (efficiency?)
2016/1/6Part I19 Algorithms for a Binary Tree (1) In algorithms for a binary tree of processors, we will assume that the data elements are initially held by the leaf processors only. The nonleaf (inner) processors participate in the computation, but do not hold data elements of their own.
2016/1/6Part I20 Algorithms for a Binary Tree (2) Semigroup Computation –Each inner node receives two values from its children, applies the operator to them, and passes the result upward to its parent.
2016/1/6Part I21 Algorithms for a Binary Tree (3) Parallel Prefix Computation
2016/1/6Part I22 Algorithms for a Binary Tree (4) Packet Routing –depends on the processor numbering scheme used. –Preorder
2016/1/6Part I23 Algorithms for a Binary Tree (5) Broadcasting –Processor i sends the desired data upwards to the root processor, which then broadcasts the data downwards to all processors.
2016/1/6Part I24 Algorithms for a Binary Tree (6) Sorting
2016/1/6Part I25 Algorithms for 2D Mesh (1) In all of the 2D mesh algorithms presented in this section, we use the linear-array algorithms of Section 2.3 as building blocks. This leads to simple algorithms, but not necessarily the most efficient ones. Mesh-based architectures and their algorithms will be discussed in great detail in Part III.
2016/1/6Part I26 Algorithms for 2D Mesh (2) Semigroup Computation –For example, in finding the maximum of a set of p values, stored one per processor, the row maximums are computed first and made available to every processor in the row. Then column maximums are identified.
2016/1/6Part I27 Algorithms for 2D Mesh (3) Parallel Prefix Computation –(1) do a parallel prefix computation on each row, –(2) do a diminished parallel prefix computation in the rightmost column, and –(3) broadcast the results in the rightmost column to all of the elements in the respective rows and combine with the initially computed row prefix value.
2016/1/6Part I28 Algorithms for 2D Mesh (4) Packet Routing –To route a data packet from the processor in Row r, Column c, to the processor in Row r', Column c', we first route it within Row r to Column c'. Then, we route it in Column c' from Row r to Row r'. (row-first routing)
2016/1/6Part I29 Algorithms for 2D Mesh (5) Broadcasting –(1) broadcast the packet to every processor in the source node's row and –(2) broadcast in all columns.
2016/1/6Part I30 Algorithms for 2D Mesh (6) Sorting
2016/1/6Part I31 Algorithms for Shared Variables Semigroup Computation Parallel Prefix computation Packet Routing (Trivial in view of the direct communication path between any pair of processors) Broadcasting (Trivial, as each processor can send a data item to all processors directly) Sorting