Combined Code and Data Minimization Algorithms

Transcription

1 Problem Statement ombined ode and ata Minimization lgorithms March, 995 Mini onference on Ptolemy Praveen K Murthy (U erkeley, Shuvra S hattacharyya (Hitachi merica Ltd, dward Lee (U erkeley {murthy,shuvra,eal}@eecsberkeleyedu Given an acyclic, multirate SF graph, want a single appearance schedule that minimizes the amount of data needed for buffering uffering Model: uffer on every arc in the graph The size of the buffer is given by the maximum number of tokens queued on the arc in the schedule Total buffering cost given by sum of sizes of individual buffer sizes Want to find a schedule that minimizes this cost Publications on this material are available on WWW: lternative uffering Models lternative #: Naive single appearance schedules with shared buffers uffering requirement can be very bad for some graphs oes not handle delays well Latency is maximized lternative #: Use nested schedules with buffer sharing More awkward to implement ost function is more complicated (50 (0 ( : ost = 5000 ( (5 ( ( ( : ost = 00 Well Ordered Graphs well-ordered graph has only one topological sort (ie, there is a hamiltonian path in the graph Problem of computing minimum buffer schedule boils down to computing an optimum nesting of loops one via dynamic programming in O( n time: b [ i, j ] = MIN i k < j r k O k where c ij [ k] = gcd ( r i,, r j { b[ i, k] + b[ k +, j] + c ij [ k] } Note: r or r( u will mean the repetitions of node u u

2 xample Well Ordered Graph 6 Repetitions vector r = [ 98,,, ] T General cyclic Graphs ny topological sort of an acyclic graph leads to a set of valid single appearance schedules n acyclic graph can have an exponential number of topological sorts in general: a complete n node bipartite graph has ( n! topological sorts The problem is to pick the topological sort that leads to the best nested schedule when nested optimally using dynamic programming algorithm Schedule uffering cost (9(((8 ((((8 7 7 (((((( 0 9 ( ( ( ( (6 : 08 ( ( (9 ( (9 : 0 Two Heuristic Techniques Recursive Partitioning by Min uts We give two heuristic techniques for finding bufferoptimal schedules for acyclic graphs: First technique is a top-down approach using mincuts called Recursive Partitioning by Minimum uts (RPM ffective for irregular topologies Second technique is a bottom-up approach using clustering called cyclic Pairwise Grouping of djacent Nodes (PGN ffective for regular topologies Optimal for a class of graphs Idea: Find a cut of the graph such that a ll arcs cross the cut in the forward direction b The cut results in fairly even-sized sets c mount of data crossing the cut is minimized Recursively schedule the nodes on the left side of the cut before nodes on the right side of the cut V L V R F

3 RPM (cont d Splitting the graph where the minimum amount of data is transferred is a greedy approach and is not optimal in general Finding the minimum cut such that all of the conditions a,b, and c are satisfied is itself a difficult problem: Methods based on max-flow-min-cut theorem do not work Graph partitioning when the size of the partition has to be bounded is NP-complete Therefore, a heuristic solution is needed Heuristic for Legal Min uts Let V R ( u be the set of nodes consisting of u and its descendents Let V L = V \ V R ( u This forms a cut satisfying condition (a Perform a local optimization by moving those nodes from V L that reduce the cost into V R ( u o this for all nodes u in the graph Repeat above steps to generate cuts obtained by letting V L ( u be the set of nodes consisting of u and it ancestors, and letting V R = V \ V L ( u Keep the cut with the lowest cost Runs in time O V V + log ( V XMPL RPM lgorithm F F { } desc (, cost = Find heuristic minimum cut of the graph into sets V L and V R The top level schedule is given by S( V = q S V q S V where q = gcd { r ( v:v V }, i = L, R L L R R i i ontinue recursively until all nodes have been scheduled Post-process resulting schedule by recomputing an optimum nesting of the loops using dynamic programming algorithm with the lexical ordering generated by RPM { } desc ( { }, cost = Runs in time O V for sparse graphs

4 cyclic Pairwise Grouping of Nodes Idea: evelop a loop hierarchy by clustering two adjacent nodes at each step efinition: lustering means combining two or more nodes into one hierarchical node The graph with the hierarchical node instead of the nodes that were clustered is called the clustered graph efinition: clusterizable pair of nodes is a pair of nodes that, when clustered, does not cause deadlock sufficient condition for clusterizability: Two nodes are clusterizable if clustering them does not introduce a cycle in the clustered graph PGN lgorithm luster two nodes that maximize gcd { r (, r( } over all clusterizable pairs {, } ontinue until only one node is left in the clustered graph This is similar to the Huffman coding algorithm fter constructing cluster hierarchy, retrace steps to determine the nested schedule Post-process the schedule using dynamic programming to generate an optimal nesting for the lexical ordering generated by PGN Runs in time O V for sparse graphs PGN in ction Optimality of PGN W yan nodes are clustered at each step W W W 0 W efinition: The buffer memory lower bound for an arc ( uv, is given by ML ( u, v r( uprod ( u, v = gcd { r ( u, r( v } This represents the least amount of buffering needed on this arc in any single appearance schedule ( ( ( ( ( ( 5 W ( ( 5 W ( ( ( 5 W W W efinition: ML schedule for an acyclic SF graph is a single appearance schedule whose buffering cost is equal to the sum of the ML costs for each arc Theorem: The PGN algorithm will find a ML schedule whenever one exists

5 Mobile Satellite Receiver xample Non-uniform Filterbank xample This example is from [Ritz95]: F K G 0 Q 0 L H W 0 0 P N M R 0 S J I 0 V U T ML = 50 PGN = 50 RPM = 80 Ritz * = 00 * Ritz generates a naive single appearance schedule and uses the shared buffer cost a b c d e f g l q o i m p r j n h k ML = 85 RPM = 8 PGN = 7 s t u w x v y z Performance on Practical xamples Performance on Random Graphs Performance of the two heuristics on various acyclic graphs System MU ML PGN RPM verage Random Graph size(nodes/ arcs Fractional decimation /0 Laplacian pyramid / Nonuniform filterbank (/,/ splits, channels /9 Nonuniform filterbank (/,/ splits, 6 channels /7 QMF nonuniform-tree filterbank /5 QMF filterbank (one-sided tree 6 8 0/ QMF analysis only /5 QMF Tree filterbank ( channels / QMF Tree filterbank (8 channels /50 QMF Tree filterbank (6 channels /6 Performance of the two heuristics on random graphs RPM < PGN 6% PGN < RPM 7% RPM < min( random 8% PGN < min( random 68% RPM < min( random 75% PGN < min( random 6% min(rpm,pgn < min( random 87% RPM < PGN by more than % 5% RPM < PGN by more than 0% 5% PGN < RPM by more than % % PGN < RPM by more than 0% %

6 onclusion Have presented algorithms for joint code and data minimization when synthesizing code from SF graphs The problem of jointly minimizing code and data boils down to picking an optimal lexical ordering of the nodes and generating an optimal looping hierarchy for that ordering ynamic programming algorithm generates an optimum looping hierarchy for any given lexical ordering Two heuristics are used to generate lexical orderings: RPM: oes well on some practical examples with irregular topologies and on random graphs PGN: oes well on a lot of practical examples but not as well on random graphs It is optimal for a class of graphs