### A Parallel GPU Version of the Traveling Salesman Problem

```A Parallel GPU Version of the
Traveling Salesman Problem
Molly A. O’Neil, Dan Tamir, and Martin Burtscher*
Department of Computer Science
The Traveling Salesman Problem
 Common combinatorial optimization problem
 Wire routing, logistics, robot arm movement, etc.
 Given n cities, find shortest Hamiltonian tour
 Must visit all cities exactly once and end in first city
 Usually expressed as a graph problem
 We use complete, undirected, planar, Euclidean graph
 Vertices represent cities
 Edge weights reflect distances
A Parallel GPU Version of the Traveling Salesman Problem
July 2011
TSP Algorithm
 Optimal solution is NP-hard
 Heuristic algorithms used to approximate solution
 We use an iterative hill climbing search algorithm
 Generate k random initial tours (k climbers)
 Iteratively refine them until local minimum reached
 In each iteration, apply best opt-2 move
 Find best pair of edges (a,b) and (c,d)
such that replacing them with (a,d)
and (b,c) minimizes tour length
A Parallel GPU Version of the Traveling Salesman Problem
→
July 2011
GPU Requirements
 Lots of data parallelism
 Need 10,000s of ‘independent’ threads
 Sufficient memory access regularity
Thepcreport.net
 Sets of 32 threads should have ‘nice’ access patterns
 Sufficient code regularity
 Sets of 32 threads should follow the same control flow
 Plenty of data reuse
 At least O(n2) operations on O(n) data
A Parallel GPU Version of the Traveling Salesman Problem
July 2011
TSP_GPU Implementation
 Assuming 100-city problems & 100,000 climbers
 Climbers are independent, can be run in parallel
 Plenty of data parallelism
 Potential load imbalance

Different number of steps required to reach local minimum
 Every step determines best of 4851 opt-2 moves
 Same control flow (but different data)
 Coalesced memory access patterns
 O(n2) operations on O(n) data
A Parallel GPU Version of the Traveling Salesman Problem
July 2011
Code Optimizations
 Key code section: finding best opt-2 move
 Doubly nested loop

Only computes difference in tour length, not absolute length
 Highly optimized to minimize memory accesses
“Caches” rest of data in registers
 Requires only 6 clock cycles per move on a Xeon CPU core

 Local minimum compared to best solution so far

Best solution updated if needed, otherwise tour is discarded
 Other small optimizations (see paper)
A Parallel GPU Version of the Traveling Salesman Problem
July 2011
GPU Optimizations
 Random tours generated in parallel on GPU
 Minimizes data transfer to GPU
 (CPU only generates distance matrix
and prints result)
gamedsforum.ca
 2D distance matrix resident in shared memory
 Ensures hits in software-controlled fast data cache
 Tours copied to local memory in chunks of 1024
 Enables accessing them with coalesced loads & stores
A Parallel GPU Version of the Traveling Salesman Problem
July 2011
Evaluation Method
 Systems
 NVIDIA Tesla C2050 GPU (1.15 GHz 14 SMs w/ 32 PEs)
 Nautilus supercomputer (2.0 GHz 8-core X7550 Xeons)
 Datasets
 Five 100-city inputs from TSPLIB
 Implementations
 CUDA (GPU), Pthreads (CPU), serial C (CPU)
 Use almost identical code for finding best opt-2 move
A Parallel GPU Version of the Traveling Salesman Problem
July 2011
Runtime Comparison (kroE100 Input)
262144
154684
156413 (median)
sequential
Runtimes (in ms)
CUDA GPU
Min
Median
Max
78350
65536
39175
19591
16384
9802
4908
4368
4096
2724
2539
2497
256
CUDA
GPU
1024
seq
CPU
1
2
4
8
16
32
64
128
 GPU is 7.8x faster than CPU with 8 cores
 One GPU chip is as fast as 16 or 32 CPU chips
A Parallel GPU Version of the Traveling Salesman Problem
July 2011
Speedup over Sequential Code
Speedup over Serial (kroE100 Input)
90
80
CUDA GPU
Min
70
Median
60
56.8
Max
60.9
61.9
50
40
31.5
35.4
30
15.8
20
10
(median)
1.0
2.0
3.9
1
2
4
7.9
0
8
16
32
64
128
256
CUDA
GPU
 Pthreads code scales well to 32 threads (4 CPUs)
 CPU performance fluctuates (NUMA), GPU stable
A Parallel GPU Version of the Traveling Salesman Problem
July 2011
Solution Quality
TSPLIB Database
CUDA GPU Solution Quality
Name
Optimal Cost
Min. Tour Cost
Min. Tour #
Runtime (s)
kroA100
21,282
21,282
33,188
2.540
kroB100
22,141
22,141
5,969
2.499
kroC100
20,749
20,749
23,092
2.543
kroD100
21,294
21,294
32,142
2.497
22,084
16,941
2.499
22,068
117,583
4.952
kroE100
22,068
 Optimal tour found in 4 of 5 cases with 100,000 climbers
 200,000 climbers find best solution in fifth case
 Runtime independent of input and linear in climbers
A Parallel GPU Version of the Traveling Salesman Problem
July 2011
Summary
 TSP_GPU source code is freely available at
http://www.cs.txstate.edu/~burtscher/research/TSP_GPU/
 TSP_GPU algorithm
 Highly optimized implementation for GPUs
 Evaluates almost 20 billion tour modifications per
second on a single GPU (as fast as 32 8-core Xeons)
 Produces high-quality results
 May be better suited for GPU than ACO and GA algos.
 Acknowledgments
 NSF TeraGrid (NICS), NVIDIA Corp., and Intel Corp.
A Parallel GPU Version of the Traveling Salesman Problem
July 2011
```