EE/CSCI 451 Spring 2018 Homework 2 Assigned: February 7, 2018 Due: February 14, 2018, before 11:59 pm Total Points: 100

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1 EE/CSCI 45 Spring 08 Homework Assigned: February 7, 08 Due: February 4, 08, before :59 pm Total Points: 00 [0 points] Explain the following terms:. Diameter of a network. Bisection width of a network. Multistage network 4. Rearrangeable network 5. Non blocking network [0 points] Figure shows a 4 input/output CLOS network with control setting for each switch box. What permutation does this network realize? Figure : 4 input/output CLOS network Permutation: 0

2 [0 points] In the class, we discussed an algorithm to perform routing on a Shuffle-Exchange Network which requires k routing steps (shuffles and exchanges), where k is the number of bits in the binary representation of source and destination. For a binary number x k x k... x 0, we define suffix as the most significant bits x k x k... of the number and prefix as the least significant bits... x x 0 of the number. Figure : Prefix 0 of s is equal to suffix 0 of d Let l denote the length of the longest prefix of the source (in binary representation) which is a suffix of the destination (again in binary representation). See Figure for example.. Consider a 6 input and output shuffle-exchange network, a source node s = 00 and a destination node d = 00. For this pair of source and destination, find a route which has 4 links ((k l) = (4 ) = 4). Write all the intermediate nodes and the operation (Shuffle or Exchange) performed at each step. (5 points) Hint: Longest prefix of s which is equal to the suffix of d is 0.. Modify the algorithm discussed in the lecture so that it runs in at most (k l) steps. 4 [5 points] A mesh of trees is a network that imposes a tree interconnection on a grid of processing nodes. A p p mesh of trees is constructed as follows. Starting with a p p grid, a complete binary tree is imposed on each row of the grid. Then a complete binary tree is imposed on each column of the grid. Figure illustrates the construction of a 4 4 mesh of trees. Assume that the nodes at intermediate levels are switching nodes. Determine the diameter of a p p mesh of trees. 5 [5 points] In the class, we defined an n input and n output CLOS network as a -stage network with n n n crossbar switches in each stage.. Draw the network for n = 6.. We can extend the -stage CLOS network as follows: A p input and p output CLOS network can be defined as a -stage network where Stage 0 and Stage consist of p switches while Stage consists of two p p switches. The definition can be applied recursively to Stage until the network consists only of switches. Using the above definition, each n n crossbar switch of the network in part a can be implemented using only switches. Draw a 6 6 crossbar switch and define the connectivity requirement from Stage 0 to Stage.

3 complete binary tree is imposed on each column of the grid. Figure.6 illustrates the construction of a 4 x 4 mesh of trees. Assume that the nodes at intermediate levels are switching nodes. Determine the bisection width, diameter, and total number of switching nodes in a mesh. Figure.6. The construction of a 4 x 4 mesh of trees: (a) a 4 x 4 grid, (b) complete binary trees imposed over individual rows, (c) complete binary trees imposed over each column, and (d) the complete 4 x 4 mesh of trees..8 [Lei9] Extend the two-dimensional mesh of trees (Problem.7) to d dimensions to construct Figurea : p /d The x pconstruction /d x x p /d of mesh a 4 of 4 mesh trees. of We trees: can do (a) this a 4 by 4fixing grid, grid (b) complete positions in all dimensions binary treesto imposed different over values individual and imposing rows, (c) a complete complete binary binary trees tree imposed on the over oneeach dimension column, that and is (d) being the complete varied. 4 4 mesh of trees. Derive the total number of switching nodes in a p /d x p /d x x p /d mesh of trees. Calculate the diameter, bisection width, and wiring cost in terms of the total number of wires. What are the advantages and disadvantages of a mesh of trees as compared to a wraparound mesh?.9 [Lei9] A network related to the mesh of trees is the d-dimensional pyramidal mesh. A d-dimensional pyramidal mesh imposes a pyramid on the underlying grid of processing nodes (as opposed to a complete tree in the mesh of trees). The generalization

4 . In general, derive an expression for the total number of switches and the total delay from an input to an output for an n input and n output CLOS network with n n n crossbar switches each of which is implemented using the definition of CLOS network used in Part. (Assume the delay of each switch is equal to unit) 6 [0 points] In this problem, we will design a network for n n = k nodes. The nodes will be laid out in dimensions with n rows and n columns. Each row is a Shuffle-Exchange network of size n = k. Similarly, each column is a Shuffle-Exchange network of size n = k. To perform routing on this network, we split the k bits into two chunks of k bits each. The most significant k bits (left chunk) will be used to perform Shuffle-Exchange Routing in vertical direction (across columns) and the least significant k bits (right chunk) will be used to perform Shuffle-Exchange Routing in horizontal direction (along a row).. Fill the connections for k = 4 in the network shown in Figure 4.. Show the intermediate nodes while routing from source s = 00 to destination d = 00 in the network shown in Figure Figure 4: The network for k = 4 4

5 7 [0 points] Suppose we want to perform matrix transpose for an n n matrix using an n n -D mesh with a simple routing algorithm that routes Matrix(i, j) to PE(j, i) (0 i, j < n). Matrix(i, j) is initially stored in PE(i, j) of the -D mesh. The matrix transpose is performed as follows: Matrix(i, j) is first routed along row i to PE(i, i), then routed along column i to PE(j, i). Using the simple definition of congestion, what is the congestion in the network? Specify all nodes where the congestion is maximum. 8 Submission Instructions Upload your file to Google drive. Create a share link and give edit permission to grader (shubhamb@usc.edu). Fill in your name and link in Google sheet d/9fststzqexz-neel_bqpulxf9xgayzkrqhexnrems4/edit?usp=sharing. Send the files to me (zhuoyw@gmail.com) in . (This is for record only) 5