1. List Ranking and Parallel Prefix
Sathish Vadhiyar

2. List Ranking on GPUs
Linked list prefix computations – computations of prefix sum on the elements contained in a linked list
Linked list represented as an array
Irregular memory accesses – successor of each node of a linked list can be contained anywhere
List ranking – special case of list prefix computations in which all the values are identity, i.e., 1.

3. List ranking L is a singly linked list
Each node contains two fields – a data field, and a pointer to the successor
Prefix sums – updating data field with summation of values of its predecessors and itself
L represented by an array X with fields X[i].prefix and X[i].succ

4. Sequential Algorithm Simple and effective Two passes
Pass 1: To identify the head node
Pass 2: Traverses starting from the head, follow the successor nodes accumulating the prefix sums in the traversal order
Works well in practice

6. Parallel Algorithm: Prefix computations on arrays
Array X partitioned into subarrays
Local prefix sums of each subarray calculated in parallel
Prefix sums of last elements of each subarray written to a separate array Y
Prefix sums of elements in Y are calculated.
Each prefix sum of Y is added to corresponding block of X
Divide and conquer strategy

7. Example
1,3,6 4,9,15 7,15,24
6,15,24
6,21,45
1,3,6,10,15,21,28,36,45
Divide
Local prefix sum
Passing last elements to a processor
Computing prefix sum of last elements on the processor
Adding global prefix sum to local prefix sums in each processor

8. Prefix computation on list
The previous strategy cannot be applied here
Division of array X that represents list will lead to subarrays each of which can have many sublist fragments
Head nodes will have to be calculated for each of them

9. Parallel List Ranking (Wyllie’s algorithm)
Involved repeated pointer jumping
Successor pointer of each element is repeatedly updated so that it jumps over its successor until it reaches the end of the list
As each processor traverses and updates the successor, the ranks are updated
A process or thread is assigned to each element of the list

10. Parallel List Ranking (Wyllie’s algorithm)
Will lead to high synchronizations among threads
In CUDA - many kernel invocations

11. Parallel List Ranking (Helman and JaJa)
Randomly select s nodes or splitters. The head node is also a splitter
Form s sublists. In each sublist, start from a splitter as the head node, and traverse till another splitter is reached.
Form prefix sums in each sublist
Form another list, L’, consisting of only these splitters in the order they are traversed. The values in each entry of this list will be the prefix sum calculated in the respective sublists
Calculate prefix sums for this list
Add these sums to the values of the sublists

12. Parallel List Ranking on GPUs: Steps
Step 1: Compute the location of the head of the list
Each of the indices between 0 and n-1, except head node, occur exactly only once in the successors.
Hence head node = n(n-1)/2 – SUM_SUCC
SUM_SUCC = sum of the successor values
Can be done on GPUs using parallel reduction

13. Parallel List Ranking on GPUs: Steps
Step 2: Select s random nodes to split list into s random sublists
For every subarray of X of size X/s, select random location as a splitter.
Highly data parallel, can be done independent of each other

14. Parallel List Ranking on GPUs: Steps
Step 3: Using standard sequential algorithm, compute prefix sums of each sublist separately
The most computationally demanding step
s sublists allocated equally among CUDA blocks, and then allocated equally among threads in a block
Each thread computes prefix sums of each of its sublists, and copy prefix value of last element of sublist i to Sublist[i]

15. Parallel List Ranking on GPUs: Steps
Step 4: Compute prefix sum of splitters, where the successor of a splitter is the next splitter encountered when traversing the list
This list is small
Hence can be done on CPU

16. Parallel List Ranking on GPUs: Steps
Step 5: Update values of prefix sums computed in step 3 using splitter prefix sums of step 4
This can be done using coalesced memory access – access by threads to contiguous locations

17. Choosing s
Large values of s increase the chance of threads dealing with equal number of nodes
However, too large values result in overhead of sublist creation and aggregation

18. Parallel Prefix on GPUs
Using binary tree
An upward reduction phase (reduce phase or up-sweep phase)
Traversing tree from leaves to root forming partial sums at internal nodes
Down-sweep phase
Traversing from root to leaves using partial sums computed in reduction phase

19. Up Sweep

20. Down Sweep

21. Host Code
int main(){
const unsigned int num_threads = num_elements / 2;
/* cudaMalloc d_idata and d_odata */
cudaMemcpy( d_idata, h_data, mem_size, cudaMemcpyHostToDevice) );
dim3 grid(256, 1, 1); dim3 threads(num_threads, 1, 1);
scan<<< grid, threads>>> (d_odata, d_idata);
cudaMemcpy( h_data, d_odata[i], sizeof(float) * num_elements, cudaMemcpyDeviceToHost
/* cudaFree d_idata and d_odata */ }

22. Device Code
__global__ void scan_workefficient(float *g_odata, float *g_idata, int n) { // Dynamically allocated shared memory for scan kernels extern __shared__ float temp[]; int thid = threadIdx.x; int offset = 1; // Cache the computational window in shared memory temp[2*thid] = g_idata[2*thid]; temp[2*thid+1] = g_idata[2*thid+1]; // build the sum in place up the tree for (int d = n>>1; d > 0; d >>= 1) __syncthreads(); if (thid < d) int ai = offset*(2*thid+1)-1; int bi = offset*(2*thid+2)-1; temp[bi] += temp[ai]; } offset *= 2;

23. Device Code // scan back down the tree // clear the last element
// clear the last element
if (thid == 0) temp[n - 1] = 0;  
// traverse down the tree building the scan in place
for (int d = 1; d < n; d *= 2) {
offset >>= 1;
__syncthreads(); 
if (thid < d)
int ai = offset*(2*thid+1)-1;
int bi = offset*(2*thid+2)-1;
float t = temp[ai];
temp[ai] = temp[bi];
temp[bi] += t; }
__syncthreads();
// write results to global memory
g_odata[2*thid] = temp[2*thid]; g_odata[2*thid+1] = temp[2*thid+1];

24. References Fast and Scalable List Ranking on the GPU. ICS 2009.
Optimization of Linked List Prefix Computations on Multithreaded GPUs Using CUDA. IPDPS 2010.