A Parallel GPU Version of the Traveling Salesman Problem

By Molly A. O’Neil, Dan Tamir and Martin Burtscher

Presented ByRukshan Siriwardhane (148208V)Vimukthi Wickramasinghe (148245F)

Outline

● The Travelling Salesman Problem● The TSP algorithm used● Using a GPU to solve TSP● Optimizations used● Evaluation method● Results

The Traveling Salesman Problem

Defn - Given n cities, find the shortest Hamiltonian tour between the cities● Combinatorial optimization problem

○ Eg: Finding effective drilling arm movement, best routing, logistics etc.

● A brute force search in the solution space is not feasible● Usually expressed as a graph problem

○ Complete, undirected, planar, Euclidean graph is used

○ Vertices represent cities

○ Edge weights reflect distances or costs

● Optimal solution is NP-hard○ Heuristic algorithms used to find an approximate solution.

● Here an iterative hill climbing search algorithm is used○ 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

The TSP Algorithm used

The TSP Algorithm used

Using a GPU to solve TSP

Parallelism Memory access regularity

Code regularity Data reuse

More than 10,000 threads

Sets of 32 threads needs to have good access to memory

Sets of 32 threads need to follow the same control flow

At least O(n2) operations on O(n) data

Using a GPU to solve TSP

▪ Assuming 100-city problems & 100,000 climbers

▪ Climbers are independent, can be run in parallel▪ Pro - Plenty of data parallelism

▪ Con - 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

Optimizations - code

● Main 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

Optimizations - GPU

● Random tours generated in parallel on GPU○ Minimizes data transfer to GPU

● 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

Evaluation Method

● Hardware○ NVIDIA Tesla C2050 GPU○ (1.15 GHz 14 SMs w/ 32 PEs, 3GB global memory)○ Nautilus supercomputer (2.0 GHz 8-core X7550 Xeons, sharing 4TB main

memory)

● Data○ Five 100-city inputs from TSPLIB

● Implementations○ CUDA (GPU), Pthreads (CPU), serial C (CPU)○ Use almost identical code for finding best opt-2 move

Results - Runtime Comparison

● GPU is 7.8x faster than CPU with 8 cores

● One GPU chip is as fast as 16 or 32 CPU chips

Speedup over Serial

● Pthreads code scales well up to 32 threads (4 CPUs)

● CPU performance fluctuates (NUMA), GPU stable

Results - Solution Quality

● 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

Summary▪ 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 Ant Colony Optimization and GAs.

Any Questions?

Thank You..