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**Parallel Graph Algorithms**

Kamesh Madduri Lawrence Berkeley National Laboratory CS267, Spring 2011 April 7, 2011

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**Lecture Outline Applications**

Designing parallel graph algorithms, performance on current systems Case studies: Graph traversal-based problems, parallel algorithms Breadth-First Search Single-source Shortest paths Betweenness Centrality

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**Routing in transportation networks**

Road networks, Point-to-point shortest paths: 15 seconds (naïve) 10 microseconds H. Bast et al., “Fast Routing in Road Networks with Transit Nodes”, Science 27, 2007.

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Internet and the WWW The world-wide web can be represented as a directed graph Web search and crawl: traversal Link analysis, ranking: Page rank and HITS Document classification and clustering Internet topologies (router networks) are naturally modeled as graphs

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**Scientific Computing Reorderings for sparse solvers**

Fill reducing orderings Partitioning, eigenvectors Heavy diagonal to reduce pivoting (matching) Data structures for efficient exploitation of sparsity Derivative computations for optimization graph colorings, spanning trees Preconditioning Incomplete Factorizations Partitioning for domain decomposition Graph techniques in algebraic multigrid Independent sets, matchings, etc. Support Theory Spanning trees & graph embedding techniques Image source: Yifan Hu, “A gallery of large graphs” B. Hendrickson, “Graphs and HPC: Lessons for Future Architectures”, Image source: Tim Davis, UF Sparse Matrix Collection.

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**Large-scale data analysis**

Graph abstractions are very useful to analyze complex data sets. Sources of data: petascale simulations, experimental devices, the Internet, sensor networks Challenges: data size, heterogeneity, uncertainty, data quality Astrophysics: massive datasets, temporal variations Bioinformatics: data quality, heterogeneity Social Informatics: new analytics challenges, data uncertainty Image sources: (1) (2,3)

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**Data Analysis and Graph Algorithms in Systems Biology**

Study of the interactions between various components in a biological system Graph-theoretic formulations are pervasive: Predicting new interactions: modeling Functional annotation of novel proteins: matching, clustering Identifying metabolic pathways: paths, clustering Identifying new protein complexes: clustering, centrality Image Source: Giot et al., “A Protein Interaction Map of Drosophila melanogaster”, Science 302, , 2003.

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**Graph –theoretic problems in social networks**

Targeted advertising: clustering and centrality Studying the spread of information Image Source: Nexus (Facebook application)

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**Network Analysis for Intelligence and Survelliance**

[Krebs ’04] Post 9/11 Terrorist Network Analysis from public domain information Plot masterminds correctly identified from interaction patterns: centrality A global view of entities is often more insightful Detect anomalous activities by exact/approximate subgraph isomorphism. Image Source: Image Source: T. Coffman, S. Greenblatt, S. Marcus, Graph-based technologies for intelligence analysis, CACM, 47 (3, March 2004): pp 45-47

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**Research in Parallel Graph Algorithms**

Application Areas Methods/ Problems Graph Algorithms Architectures GPUs FPGAs x86 multicore servers Massively multithreaded architectures Multicore Clusters Clouds Traversal Shortest Paths Connectivity Max Flow … Social Network Analysis WWW Computational Biology Scientific Computing Engineering Find central entities Community detection Network dynamics Data size Marketing Social Search Gene regulation Metabolic pathways Genomics Problem Complexity Graph partitioning Matching Coloring VLSI CAD Route planning

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**Characterizing Graph-theoretic computations**

Factors that influence choice of algorithm Input data Graph kernel graph sparsity (m/n ratio) static/dynamic nature weighted/unweighted, weight distribution vertex degree distribution directed/undirected simple/multi/hyper graph problem size granularity of computation at nodes/edges domain-specific characteristics traversal shortest path algorithms flow algorithms spanning tree algorithms topological sort ….. Problem: Find *** paths clusters partitions matchings patterns orderings Graph problems are often recast as sparse linear algebra (e.g., partitioning) or linear programming (e.g., matching) computations

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**Lecture Outline Applications**

Designing parallel graph algorithms, performance on current systems Case studies: Graph traversal-based problems, parallel algorithms Breadth-First Search Single-source Shortest paths Betweenness Centrality

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History 1735: “Seven Bridges of Königsberg” problem, resolved by Euler, one of the first graph theory results. … 1966: Flynn’s Taxonomy. 1968: Batcher’s “sorting networks” 1969: Harary’s “Graph Theory” 1972: Tarjan’s “Depth-first search and linear graph algorithms” 1975: Reghbati and Corneil, Parallel Connected Components 1982: Misra and Chandy, distributed graph algorithms. 1984: Quinn and Deo’s survey paper on “parallel graph algorithms”

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**The PRAM model Idealized parallel shared memory system model**

Unbounded number of synchronous processors; no synchronization, communication cost; no parallel overhead EREW (Exclusive Read Exclusive Write), CREW (Concurrent Read Exclusive Write) Measuring performance: space and time complexity; total number of operations (work)

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**PRAM Pros and Cons Pros Cons Simple and clean semantics.**

The majority of theoretical parallel algorithms are designed using the PRAM model. Independent of the communication network topology. Cons Not realistic, too powerful communication model. Algorithm designer is misled to use IPC without hesitation. Synchronized processors. No local memory. Big-O notation is often misleading.

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**PRAM Algorithms for Connected Components**

Reghbati and Corneil [1975], O(log2n) time and O(n3) processors Wyllie [1979], O(log2n) time and O(m) processors Shiloach and Vishkin [1982], O(log n) time and O(m) processors Reif and Spirakis [1982], O(log log n) time and O(n) processors (expected)

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**Building blocks of classical PRAM graph algorithms**

Prefix sums Symmetry breaking Pointer jumping List ranking Euler tours Vertex collapse Tree contraction

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**Data structures: graph representation**

Static case Dense graphs (m = O(n2)): adjacency matrix commonly used. Sparse graphs: adjacency lists Dynamic representation depends on common-case query Edge insertions or deletions? Vertex insertions or deletions? Edge weight updates? Graph update rate Queries: connectivity, paths, flow, etc. Optimizing for locality a key design consideration.

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**Graph representation Compressed Sparse Row-like 5 8 1 7 3 4 6 9 2**

Vertex Degree Adjacencies Compressed Sparse Row-like 1 2 3 4 5 6 7 8 9 4 1 2 3 2 5 7 6 3 4 8 1 9 5 8 1 7 3 4 6 9 Flatten adjacency arrays 2 Index into adjacency array 4 5 7 … 28 Size: n+1 Adjacencies 2 5 7 6 3 4 …. Size: 2*m

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**Distributed Graph representation**

Each processor stores the entire graph (“full replication”) Each processor stores n/p vertices and all adjacencies out of these vertices (“1D partitioning”) How to create these “p” vertex partitions? Graph partitioning algorithms: recursively optimize for conductance (edge cut/size of smaller partition) Randomly shuffling the vertex identifiers ensures that edge count/processor are roughly the same

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2D graph partitioning Consider a logical 2D processor grid (pr * pc = p) and the matrix representation of the graph Assign each processor a sub-matrix (i.e, the edges within the sub-matrix) 9 vertices, 9 processors, 3x3 processor grid x 5 8 1 7 3 4 6 Flatten Sparse matrices 2 Per-processor local graph representation

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**Data structures in (parallel) graph algorithms**

A wide range seen in graph algorithms: array, list, queue, stack, set, multiset, tree Implementations are typically array-based for performance considerations. Key data structure considerations in parallel graph algorithm design Practical parallel priority queues Space-efficiency Parallel set/multiset operations, e.g., union, intersection, etc.

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**Graph Algorithms on current systems**

Concurrency Simulating PRAM algorithms: hardware limits of memory bandwidth, number of outstanding memory references; synchronization Locality Try to improve cache locality, but avoid “too much” superfluous computation Work-efficiency Is (Parallel time) * (# of processors) = (Serial work)?

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The locality challenge “Large memory footprint, low spatial and temporal locality impede performance” Serial Performance of “approximate betweenness centrality” on a 2.67 GHz Intel Xeon 5560 (12 GB RAM, 8MB L3 cache) Input: Synthetic R-MAT graphs (# of edges m = 8n) No Last-level Cache (LLC) misses ~ 5X drop in performance O(m) LLC misses

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**Tuning implementation to minimize parallel overhead is non-trivial**

The parallel scaling challenge “Classical parallel graph algorithms perform poorly on current parallel systems” Graph topology assumptions in classical algorithms do not match real-world datasets Parallelization strategies at loggerheads with techniques for enhancing memory locality Classical “work-efficient” graph algorithms may not fully exploit new architectural features Increasing complexity of memory hierarchy, processor heterogeneity, wide SIMD. Tuning implementation to minimize parallel overhead is non-trivial Shared memory: minimizing overhead of locks, barriers. Distributed memory: bounding message buffer sizes, bundling messages, overlapping communication w/ computation.

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**Lecture Outline Applications**

Designing parallel graph algorithms, performance on current systems Case studies: Graph traversal-based problems, parallel algorithms Breadth-First Search Single-source Shortest paths Betweenness Centrality

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**Graph traversal (BFS) problem definition**

1 2 distance from source vertex Output: Input: 5 8 4 1 1 2 3 3 4 7 3 4 6 9 source vertex 1 2 Memory requirements (# of machine words): Sparse graph representation: m+n Stack of visited vertices: n Distance array: n

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Optimizing BFS on cache-based multicore platforms, for networks with “power-law” degree distributions Problem Spec. Assumptions No. of vertices/edges 106 ~ 109 Edge/vertex ratio 1 ~ 100 Static/dynamic? Static Diameter O(1) ~ O(log n) Weighted/Unweighted Unweighted Vertex degree distribution Unbalanced (“power law”) Directed/undirected? Both Simple/multi/hypergraph? Multigraph Granularity of computation at vertices/edges? Minimal Exploiting domain-specific characteristics? Partially Test data (Data: Mislove et al., IMC 2007.) Synthetic R-MAT networks

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**Parallel BFS Strategies**

1. Expand current frontier (level-synchronous approach, suited for low diameter graphs) 5 8 1 O(D) parallel steps Adjacencies of all vertices in current frontier are visited in parallel 7 3 4 6 9 source vertex 2 2. Stitch multiple concurrent traversals (Ullman-Yannakakis approach, suited for high-diameter graphs) 5 8 path-limited searches from “super vertices” APSP between “super vertices” 1 source vertex 7 3 4 6 9 2

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**A deeper dive into the “level synchronous” strategy**

Locality (where are the random accesses originating from?) 1. Ordering of vertices in the “current frontier” array, i.e., accesses to adjacency indexing array, cumulative accesses O(n). 2. Ordering of adjacency list of each vertex, cumulative O(m). 3. Sifting through adjacencies to check whether visited or not, cumulative accesses O(m). 53 84 93 31 44 74 63 26 11 1. Access Pattern: idx array -- 53, 31, 74, 26 2,3. Access Pattern: d array -- 0, 84, 0, 84, 93, 44, 63, 0, 0, 11

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**Performance Observations**

Youtube social network Flickr social network Graph expansion Edge filtering

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**Improving locality: Vertex relabeling**

Well-studied problem, slight differences in problem formulations Linear algebra: sparse matrix column reordering to reduce bandwidth, reveal dense blocks. Databases/data mining: reordering bitmap indices for better compression; permuting vertices of WWW snapshots, online social networks for compression NP-hard problem, several known heuristics We require fast, linear-work approaches Existing ones: BFS or DFS-based, Cuthill-McKee, Reverse Cuthill-McKee, exploit overlap in adjacency lists, dimensionality reduction

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**Improving locality: Optimizations**

Recall: Potential O(m) non-contiguous memory references in edge traversal (to check if vertex is visited). e.g., access order: 53, 31, 31, 26, 74, 84, 0, … Objective: Reduce TLB misses, private cache misses, exploit shared cache. Optimizations: Sort the adjacency lists of each vertex – helps order memory accesses, reduce TLB misses. Permute vertex labels – enhance spatial locality. Cache-blocked edge visits – exploit temporal locality. 84 53 93 31 44 74 63 26 11

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**Improving locality: Cache blocking**

Metadata denoting blocking pattern Process high-degree vertices separately Adjacencies (d) Adjacencies (d) 1 2 3 x x x x x x x x x x x x x x x x x x x x x x x x frontier x x frontier x x Tune to L2 cache size x x x x x x x x x x linear processing New: cache-blocked approach Instead of processing adjacencies of each vertex serially, exploit sorted adjacency list structure w/ blocked accesses Requires multiple passes through the frontier array, tuning for optimal block size. Note: frontier array size may be O(n)

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**Vertex relabeling heuristic**

Similar to older heuristics, but tuned for small-world networks: High percentage of vertices with (out) degrees 0, 1, and 2 in social and information networks => store adjacencies explicitly (in indexing data structure). Augment the adjacency indexing data structure (with two additional words) and frontier array (with one bit) Process “high-degree vertices” adjacencies in linear order, but other vertices with d-array cache blocking. Form dense blocks around high-degree vertices Reverse Cuthill-McKee, removing degree 1 and degree 2 vertices

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**Architecture-specific Optimizations**

1. Software prefetching on the Intel Core i7 (supports 32 loads and 20 stores in flight) Speculative loads of index array and adjacencies of frontier vertices will reduce compulsory cache misses. 2. Aligning adjacency lists to optimize memory accesses 16-byte aligned loads and stores are faster. Alignment helps reduce cache misses due to fragmentation 16-byte aligned non-temporal stores (during creation of new frontier) are fast. 3. SIMD SSE integer intrinsics to process “high-degree vertex” adjacencies. 4. Fast atomics (BFS is lock-free w/ low contention, and CAS-based intrinsics have very low overhead) 5. Hugepage support (significant TLB miss reduction) 6. NUMA-aware memory allocation exploiting first-touch policy

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**Experimental Setup Network n m Max. out-degree**

% of vertices w/ out-degree 0,1,2 Orkut 3.07M 223M 32K 5 LiveJournal 5.28M 77.4M 9K 40 Flickr 1.86M 22.6M 26K 73 Youtube 1.15M 4.94M 28K 76 R-MAT 8M-64M 8n n0.6 Intel Xeon 5560 (Core i7, “Nehalem”) 2 sockets x 4 cores x 2-way SMT 12 GB DRAM, 8 MB shared L3 51.2 GBytes/sec peak bandwidth 2.66 GHz proc. Performance averaged over 10 different source vertices, 3 runs each.

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**Impact of optimization strategies**

Generality Impact* Tuning required? (Preproc.) Sort adjacency lists High -- No (Preproc.) Permute vertex labels Medium Yes Preproc. + binning frontier vertices + cache blocking M 2.5x Lock-free parallelization 2.0x Low-degree vertex filtering Low 1.3x Software Prefetching 1.10x Aligning adjacencies, streaming stores 1.15x Fast atomic intrinsics H 2.2x * Optimization speedup (performance on 4 cores) w.r.t baseline parallel approach, on a synthetic R-MAT graph (n=223,m=226)

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**Cache locality improvement**

Performance count: # of non-contiguous memory accesses (assuming cache line size of 16 words) Theoretical count of the number of non-contiguous memory accesses: m+3n

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**Parallel performance (Orkut graph)**

Execution time: 0.28 seconds (8 threads) Parallel speedup: 4.9 Speedup over baseline: 2.9 Graph: 3.07 million vertices, 220 million edges Single socket of Intel Xeon 5560 (Core i7)

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Performance Analysis How well does my implementation match theoretical bounds? When can I stop optimizing my code? Begin with asymptotic analysis Express work performed in terms of machine-independent performance counts Add input graph characteristics in analysis Use simple kernels or micro-benchmarks to provide an estimate of achievable peak performance Relate observed performance to machine characteristics

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**Graph 500 “Search” Benchmark (graph500.org)**

BFS (from a single vertex) on a static, undirected R-MAT network with average vertex degree 16. Evaluation criteria: largest problem size that can be solved on a system, minimum execution time. Reference MPI, shared memory implementations provided. NERSC Franklin system is ranked #2 on current list (Nov 2010). BFS using 500 nodes of Franklin

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**Lecture Outline Applications**

Designing parallel graph algorithms, performance on current systems Case studies: Graph traversal-based problems, parallel algorithms Breadth-First Search Single-source Shortest paths Betweenness Centrality

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**Parallel Single-source Shortest Paths (SSSP) algorithms**

Edge weights: concurrency primary challenge! No known PRAM algorithm that runs in sub-linear time and O(m+nlog n) work Parallel priority queues: relaxed heaps [DGST88], [BTZ98] Ullman-Yannakakis randomized approach [UY90] Meyer and Sanders, ∆ - stepping algorithm [MS03] Distributed memory implementations based on graph partitioning Heuristics for load balancing and termination detection K. Madduri, D.A. Bader, J.W. Berry, and J.R. Crobak, “An Experimental Study of A Parallel Shortest Path Algorithm for Solving Large-Scale Graph Instances,” Workshop on Algorithm Engineering and Experiments (ALENEX), New Orleans, LA, January 6, 2007.

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**∆ - stepping algorithm [MS03]**

Label-correcting algorithm: Can relax edges from unsettled vertices also ∆ - stepping: “approximate bucket implementation of Dijkstra’s algorithm” ∆: bucket width Vertices are ordered using buckets representing priority range of size ∆ Each bucket may be processed in parallel

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**∆ - stepping algorithm: illustration**

∆ = 0.1 (say) 0.05 One parallel phase while (bucket is non-empty) Inspect light edges Construct a set of “requests” (R) Clear the current bucket Remember deleted vertices (S) Relax request pairs in R Relax heavy request pairs (from S) Go on to the next bucket 0.56 3 6 0.07 0.01 0.15 0.23 4 2 0.02 0.13 0.18 1 5 d array ∞ ∞ ∞ ∞ ∞ ∞ ∞ Buckets

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**∆ - stepping algorithm: illustration**

0.05 One parallel phase while (bucket is non-empty) Inspect light edges Construct a set of “requests” (R) Clear the current bucket Remember deleted vertices (S) Relax request pairs in R Relax heavy request pairs (from S) Go on to the next bucket 0.56 3 6 0.07 0.01 0.15 0.23 4 2 0.02 0.13 0.18 1 5 d array ∞ ∞ ∞ ∞ ∞ ∞ Buckets Initialization: Insert s into bucket, d(s) = 0

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**∆ - stepping algorithm: illustration**

One parallel phase while (bucket is non-empty) Inspect light edges Construct a set of “requests” (R) Clear the current bucket Remember deleted vertices (S) Relax request pairs in R Relax heavy request pairs (from S) Go on to the next bucket 0.05 0.56 3 6 0.07 0.01 0.15 0.23 4 2 0.02 0.13 0.18 1 5 d array ∞ ∞ ∞ ∞ ∞ ∞ Buckets R 2 .01 S

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**∆ - stepping algorithm: illustration**

0.05 One parallel phase while (bucket is non-empty) Inspect light edges Construct a set of “requests” (R) Clear the current bucket Remember deleted vertices (S) Relax request pairs in R Relax heavy request pairs (from S) Go on to the next bucket 0.56 3 6 0.07 0.01 0.15 0.23 4 2 0.02 0.13 0.18 1 5 d array ∞ ∞ ∞ ∞ ∞ ∞ Buckets R 2 .01 S

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**∆ - stepping algorithm: illustration**

0.05 One parallel phase while (bucket is non-empty) Inspect light edges Construct a set of “requests” (R) Clear the current bucket Remember deleted vertices (S) Relax request pairs in R Relax heavy request pairs (from S) Go on to the next bucket 0.56 3 6 0.07 0.01 0.15 0.23 4 2 0.02 0.13 0.18 1 5 d array ∞ ∞ ∞ ∞ ∞ .01 Buckets R 2 S

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**∆ - stepping algorithm: illustration**

0.05 One parallel phase while (bucket is non-empty) Inspect light edges Construct a set of “requests” (R) Clear the current bucket Remember deleted vertices (S) Relax request pairs in R Relax heavy request pairs (from S) Go on to the next bucket 0.56 3 6 0.07 0.01 0.15 0.23 4 2 0.02 0.13 0.18 1 5 d array ∞ ∞ ∞ ∞ ∞ .01 Buckets R 1 3 2 .03 .06 S

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**∆ - stepping algorithm: illustration**

0.05 One parallel phase while (bucket is non-empty) Inspect light edges Construct a set of “requests” (R) Clear the current bucket Remember deleted vertices (S) Relax request pairs in R Relax heavy request pairs (from S) Go on to the next bucket 0.56 3 6 0.07 0.01 0.15 0.23 4 2 0.02 0.13 0.18 1 5 d array ∞ ∞ ∞ ∞ ∞ .01 Buckets R 1 3 .03 .06 S 2

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**∆ - stepping algorithm: illustration**

0.05 One parallel phase while (bucket is non-empty) Inspect light edges Construct a set of “requests” (R) Clear the current bucket Remember deleted vertices (S) Relax request pairs in R Relax heavy request pairs (from S) Go on to the next bucket 0.56 3 6 0.07 0.01 0.15 0.23 2 4 0.02 0.13 0.18 1 5 d array ∞ ∞ ∞ .03 .01 .06 Buckets R 1 3 S 2

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**∆ - stepping algorithm: illustration**

0.05 One parallel phase while (bucket is non-empty) Inspect light edges Construct a set of “requests” (R) Clear the current bucket Remember deleted vertices (S) Relax request pairs in R Relax heavy request pairs (from S) Go on to the next bucket 0.56 3 6 0.07 0.01 0.15 0.23 4 2 0.02 0.13 0.18 1 5 d array .03 .01 .06 .16 .29 .62 Buckets R 1 4 2 5 S 2 1 3 6 6

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**Classify edges as “heavy” and “light”**

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**Relax light edges (phase) Repeat until B[i] Is empty**

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**Relax heavy edges. No reinsertions in this step.**

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**No. of phases (machine-independent performance count)**

high diameter low diameter

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Average shortest path weight for various graph families ~ 220 vertices, 222 edges, directed graph, edge weights normalized to [0,1]

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**Last non-empty bucket (machine-independent performance count)**

Fewer buckets, more parallelism

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**Number of bucket insertions (machine-independent performance count)**

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**Lecture Outline Applications**

Designing parallel graph algorithms, performance on current systems Case studies: Graph traversal-based problems, parallel algorithms Breadth-First Search Single-source Shortest paths Betweenness Centrality

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**Betweenness Centrality**

Centrality: Quantitative measure to capture the importance of a vertex/edge in a graph degree, closeness, eigenvalue, betweenness Betweenness Centrality ( : No. of shortest paths between s and t) Applied to several real-world networks Social interactions WWW Epidemiology Systems biology

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**Algorithms for Computing Betweenness**

All-pairs shortest path approach: compute the length and number of shortest paths between all s-t pairs (O(n3) time), sum up the fractional dependency values (O(n2) space). Brandes’ algorithm (2003): Augment a single-source shortest path computation to count paths; uses the Bellman criterion; O(mn) work and O(m+n) space.

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**Our New Parallel Algorithms**

Madduri, Bader (2006): parallel algorithms for computing exact and approximate betweenness centrality low-diameter sparse graphs (diameter D = O(log n), m = O(nlog n)) Exact algorithm: O(mn) work, O(m+n) space, O(nD+nm/p) time. Madduri et al. (2009): New parallel algorithm with lower synchronization overhead and fewer non-contiguous memory references In practice, 2-3X faster than previous algorithm Lock-free => better scalability on large parallel systems

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**Parallel BC Algorithm Consider an undirected, unweighted graph**

High-level idea: Level-synchronous parallel Breadth-First Search augmented to compute centrality scores Two steps traversal and path counting dependency accumulation

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**Parallel BC Algorithm Illustration**

5 8 1 7 3 4 6 9 2

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**Parallel BC Algorithm Illustration**

1. Traversal step: visit adjacent vertices, update distance and path counts. 5 8 1 7 3 4 6 9 source vertex 2

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**Parallel BC Algorithm Illustration**

1. Traversal step: visit adjacent vertices, update distance and path counts. S D P 5 8 1 1 7 3 4 6 9 1 source vertex 2 7 1 2 5

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**Parallel BC Algorithm Illustration**

1. Traversal step: visit adjacent vertices, update distance and path counts. S D P 5 8 1 1 2 2 7 7 3 4 6 9 8 3 1 source vertex 2 7 1 2 5 2 5 7 Level-synchronous approach: The adjacencies of all vertices in the current frontier can be visited in parallel

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**Parallel BC Algorithm Illustration**

Traversal step: at the end, we have all reachable vertices, their corresponding predecessor multisets, and D values. S D P 5 8 2 1 1 6 6 1 4 2 2 7 7 3 4 6 9 8 3 8 3 1 source vertex 2 8 7 1 2 5 2 5 7 6 Level-synchronous approach: The adjacencies of all vertices in the current frontier can be visited in parallel

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**Graph traversal step analysis**

Exploit concurrency in visiting adjacencies, as we assume that the graph diameter is small: O(log n) Upper bound on size of each predecessor multiset: In-degree Potential performance bottlenecks: atomic updates to predecessor multisets, atomic increments of path counts New algorithm: Based on observation that we don’t need to store “predecessor” vertices. Instead, we store successor edges along shortest paths. simplifies the accumulation step reduces an atomic operation in traversal step cache-friendly!

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**Graph Traversal Step locality analysis**

All the vertices are in a contiguous block (stack) for all vertices u at level d in parallel do for all adjacencies v of u in parallel do dv = D[v]; if (dv < 0) vis = fetch_and_add(&Visited[v], 1); if (vis == 0) D[v] = d+1; pS[count++] = v; fetch_and_add(&sigma[v], sigma[u]); fetch_and_add(&Scount[u], 1); if (dv == d + 1) All the adjacencies of a vertex are stored compactly (graph rep.) Non-contiguous memory access Non-contiguous memory access Non-contiguous memory access Store to S[u] Better cache utilization likely if D[v], Visited[v], sigma[v] are stored contiguously

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**Parallel BC Algorithm Illustration**

2. Accumulation step: Pop vertices from stack, update dependence scores. S Delta P 5 8 2 1 1 6 6 4 2 7 7 3 4 6 9 8 3 8 3 source vertex 2 8 7 2 5 5 7 6

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**Parallel BC Algorithm Illustration**

2. Accumulation step: Can also be done in a level-synchronous manner. S Delta P 5 8 2 1 1 6 6 4 2 7 7 3 4 6 9 8 3 8 3 source vertex 2 8 7 2 5 5 7 6

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**Accumulation step locality analysis**

All the vertices are in a contiguous block (stack) for level d = GraphDiameter-2 to 1 do for all vertices v at level d in parallel do for all w in S[v] in parallel do reduction(delta) delta_sum_v = delta[v] + (1 + delta[w]) * sigma[v]/sigma[w]; BC[v] = delta[v] = delta_sum_v; Each S[v] is a contiguous block Only floating point operation in code

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**Centrality Analysis applied to Protein Interaction Networks**

43 interactions Protein Ensembl ID ENSG Kelch-like protein 8

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**Designing parallel algorithms for large sparse graph analysis**

System requirements: High (on-chip memory, DRAM, network, IO) bandwidth. Solution: Efficiently utilize available memory bandwidth. Improve locality Algorithmic innovation to avoid corner cases. “RandomAccess”-like Locality Complexity Problem Data reduction/ Compression “Stream”-like Faster methods O(n) 104 106 108 1012 Peta+ O(nlog n) Work performed O(n2) Problem size (n: # of vertices/edges)

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Review of lecture Applications: Internet and WWW, Scientific computing, Data analysis, Surveillance Overview of parallel graph algorithms PRAM algorithms graph representation Parallel algorithm case studies BFS: locality, level-synchronous approach, multicore tuning Shortest paths: exploiting concurrency, parallel priority queue Betweenness centrality: importance of locality, atomics

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