Parallel Prefix Sum Algorithm Explained
Learn about the Parallel Prefix Sum algorithm, a technique used to efficiently compute prefix sums of an array in parallel. Discover how the recursive doubling method enables this algorithm to overcome inherent sequential challenges, showcasing a beautiful parallel computing approach.
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Given array A[0..N-1], produce B[N], such that B[k] is the sum of all elements of A upto A[k] Prefix Sum Problem 0 1 2 3 4 5 6 7 A 3 1 4 2 5 2 7 4 + + + + + B 3 4 8 10 15 17 24 28 B[3] is the sum of A[0], A[1], A[2], A[3] But B[3] can also be calculated as B[2]+ A[3] 1 Charm Tutorial
Parallel Prefix Data dependency from iteration to iteration. How can this be parallelized at all? B[0] = A[0]; for (i=1; i<N; i++) B[i] = B[i-1] + A[i]; It looks like the problem is inherently sequential, but theoreticians came up with a beautiful algorithm called recursive doubling or just parallel prefix 2 Charm Tutorial
Parallel prefix : recursive doubling 0 1 2 3 4 5 6 7 N Data Items P Processors N=P 5 3 7 2 1 3 1 2 5 8 10 9 3 4 4 3 Log P Phases P additions in each phase P log P operations Completes in O(logP) time 5 8 15 17 13 13 7 7 5 8 15 17 18 21 22 24 3 Charm Tutorial
Parallel Prefix Example: prefix.ci mainmodule prefix { readonly CProxy_Main mainProxy; readonly CProxy_Prefix prefixArray; readonlyint numElements; mainchare Main { entry Main(CkArgMsg* msg); entry [reductiontarget] void done(); }; array [1D] Prefix { entry Prefix(); entryvoid step(); entryvoid passValue(int value); } } Charm Tutorial 4
#include "prefix.decl.h" #include <math.h> /*readonly*/ CProxy_Main mainProxy; /*readonly*/ CProxy_Prefix prefixArray; /*readonly*/ int numElements; class Main : public CBase_Main { public: Main(CkArgMsg *msg) { mainProxy = thisProxy; numElements = (msg->argc > 1) ? atoi(msg->argv[1]) : 8; delete msg; prefixArray = CProxy_Prefix::ckNew(numElements); } void done() { CkExit(); } }; Charm Tutorial 5
class Prefix : public CBase_Prefix { int value, distance; public: Prefix() : distance(1) { srand(time(NULL)); value = rand() % 10; step(); } ... Charm Tutorial 6
void step() { if (distance >= numElements) { CkPrintf("Prefix[%d].value = %d\n", thisIndex, value); CkCallback cb(CkReductionTarget(Main, done), mainProxy); contribute(sizeof(int), &value, CkReduction::sum_int, cb); } else { if (thisIndex+distance < numElements) thisProxy[thisIndex + distance].passValue(value); } } } void passValue(int incoming_value) { ... } }; #include "prefix.def.h" void passValue(int incoming_value) { value += incoming_value; distance *= 2; step(); } Charm Tutorial 7
void step() { if (distance >= numElements) { CkPrintf("Prefix[%d].value = %d\n", thisIndex, value); CkCallback cb(CkReductionTarget(Main, done), mainProxy); contribute(sizeof(int), &value, CkReduction::sum_int, cb); } else { if (thisIndex+distance < numElements) thisProxy[thisIndex + distance].passValue(value); } //if you no longer receive, but need to continue sending if (thisIndex - distance < 0) { distance = distance*2; step(); } } } void passValue(int incoming_value) { ... } }; #include "prefix.def.h" void passValue(int incoming_value) { value += incoming_value; distance *= 2; step(); } Charm Tutorial 8
void step() { if (distance >= numElements) { CkPrintf("Prefix[%d].value = %d\n", thisIndex, value); CkCallback cb(CkReductionTarget(Main, done), mainProxy); contribute(sizeof(int), &value, CkReduction::sum_int, cb); } else { if (thisIndex+distance < numElements) thisProxy[thisIndex + distance].passValue(value); } //if you no longer receive, but need to continue sending if (thisIndex - distance < 0) { distance = distance*2; step(); } } } void passValue(int incoming_value) { ... } }; #include "prefix.def.h" Still wrong Parallel Prefix: Why? void passValue(int incoming_value) { value += incoming_value; distance *= 2; step(); } Charm Tutorial 9
Parallel Prefix Example, Correct Version: prefix.ci mainmodule prefix { readonly CProxy_Main mainProxy; readonly CProxy_Prefix prefixArray; readonlyint numElements; readonly int numStages; mainchare Main { entry Main(CkArgMsg* msg); entry [reductiontarget] void done(); }; array [1D] Prefix { entry Prefix(); entryvoid step(); entryvoid passValue(int stage, int value); } } Charm Tutorial 10
#include "prefix.decl.h" #include <math.h> /*readonly*/ CProxy_Main mainProxy; /*readonly*/ CProxy_Prefix prefixArray; /*readonly*/ int numElements; /*readonly*/ int numStages; class Main : public CBase_Main { public: Main(CkArgMsg *msg) { mainProxy = thisProxy; numElements = (msg->argc > 1) ? atoi(msg->argv[1]) : 8; numStages = (int) ceil(log2(numElements)); delete msg; prefixArray = CProxy_Prefix::ckNew(numElements); } void done() { CkExit(); } }; Charm Tutorial 11
class Prefix : public CBase_Prefix { int *flagBuf, *valueBuf, value, stage; public: Prefix() : stage(0) { srand(time(NULL)); value = rand() % 10; valueBuf = newint[numStages]; flagBuf = newint[numStages]; step(); } ... Charm Tutorial 12
void step() { if (stage >= numStages) { CkPrintf("Prefix[%d].value = %d\n", thisIndex, value); CkCallback cb(CkReductionTarget(Main, done), mainProxy); contribute(sizeof(int), &value, CkReduction::sum_int, cb); } else { int sendIndex = thisIndex + (1 << stage); // thisIndex + distance if (sendIndex < numElements) thisProxy[sendIndex].passValue(stage, value); void passValue(int stg, int val) { flagBuf[stg] = 1; valueBuf[stg] = val; if (flagBuf[stg] == 1) updateValue(); } if (flagBuf[stage] == 1) updateValue(); elseif (thisIndex - (1 << stage) < 0) { stage++; step(); } } } ... void updateValue() { value += valueBuf[stage]; flagBuf[stage] = 0; stage++; step(); } Charm Tutorial 13
Parallel Prefix with SDAG: prefix.ci mainmodule prefix { readonly CProxy_Main mainProxy; readonly CProxy_Prefix prefixArray; readonlyint numElements; readonlyint numStages; mainchare Main { entry Main(CkArgMsg* msg); entry [reductiontarget] void done(); }; array [1D] Prefix { entry Prefix(); entryvoid passValue(int incoming_stage, int incoming_val); entry void step_through() { ... }; } } Charm Tutorial 14
entryvoid step_through() { for (stage = 0; stage < numStages; stage++) { serial "send_value" { int sendIndex = thisIndex + (1 << stage); if (sendIndex < numElements) thisProxy[sendIndex].passValue(stage, value); } if (thisIndex - (1 << stage) >= 0) { when passValue[stage](int stg, int val) { serial { value += val; } } } } serial "done" { CkPrintf("Prefix[%d].value = %d\n", thisIndex, value); CkCallback cb(CkReductionTarget(Main, done), mainProxy); contribute(sizeof(int), &value, CkReduction::sum_int, cb); } }; Charm Tutorial 15
serial "done" { CkPrintf("Prefix[%d].value = %d\n", thisIndex, value); CkCallback cb(CkReductionTarget(Main, done), mainProxy); contribute(sizeof(int), &value, CkReduction::sum_int, cb); } }; Charm Tutorial 16
