Parallel Graph Algorithms

Parallel Graph Algorithms
Kamesh Madduri Lawrence Berkeley National Laboratory CS267/EngC233 Spring 2010 April 15, 2010

Lecture Outline Applications Review of key results
Case studies: Graph traversal-based problems, parallel algorithms Breadth-First Search Single-source Shortest paths Betweenness Centrality Community Identification

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.

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

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 Matroids, 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.

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)

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.

Graph –theoretic problems in social networks
Community identification: clustering Targeted advertising: centrality Information spreading: modeling Image Source: Nexus (Facebook application)

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

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

Characterizing Graph-theoretic computations (2/2)
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

Graph traversal-based parallel algorithms, case studies Breadth-First Search Single-source Shortest paths Betweenness Centrality Community Identification

History 1735: “Seven Bridges of Königsberg” problem, resolved by Euler, first result in graph theory. … 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”

The PRAM model Idealized parallel shared memory system model
Unbounded number of synchronous processors; no synchronization, communication cost; no parallel overhead EREW, CREW Measuring performance: space and time complexity; total number of operations (work)

The Helman-JaJa model Extension to the PRAM model for shared memory algorithm design and analysis. T(n, p) is measured by the triplet –TM(n, p), TC(n, p), B(n, p) TM(n, p): maximum number of non-contiguous main memory accesses required by any processor TC(n, p): upper bound on the maximum local computational complexity of any of the processors B(n, p): number of barrier synchronizations.

PRAM Pros and Cons Pros Cons Simple and clean semantics.
The majority of theoretical parallel algorithms are specified with 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.

Building blocks of classical PRAM graph algorithms
Prefix sums List ranking Euler tours, Pointer jumping, Symmetry breaking Vertex collapse Tree contraction

Prefix Sums Input: A, an array of n elements; associative binary operation Output: O(n) work, O(log n) time, n processors B(3,1) C(3,2) B(2,1) C(2,1) B(2,2) C(2,2) B(1,1) C(1,1) B(1,2) C(1,2) B(1,3) C(1,3) B(1,4) C(1,4) B(0,1) C(0,1) B(0,2) C(0,2) B(0,3) C(0,3) B(0,4) C(0,4) B(0,5) C(0,5) B(0,6) C(0,6) B(0,7) C(0,7) B(0,8) C(0,8)

Parallel Prefix X: array of n elements stored in arbitrary order. For each element i, let X(i).value be its value and X(i).next be the index of its successor. For binary associative operator Θ, compute X(i).prefix such that X(head).prefix = X (head).value, and X(i).prefix = X(i).value Θ X(predecessor).prefix where head is the first element i is not equal to head, and predecessor is the node preceding i. List ranking: special case of parallel prefix, values initially set to 1, and addition is the associative operator.

List ranking Illustration
Ordered list (X.next values) Random list (X.next values) 2 3 4 5 6 7 8 9 4 6 5 7 8 3 2 9

List Ranking key idea 1. Chop X randomly into s pieces
2. Traverse each piece using a serial algorithm. 3. Compute the global rank of each element using the result computed from the second step. In the Helman-JaJa model, TM(n,p) = O(n/p).

Connected Components Building block for many graph algorithms
Minimum spanning tree, biconnected components, planarity testing, etc. Representative of the “graft-and-shortcut” approach CRCW PRAM algorithms [Shiloach & Vishkin ’82]: O(log n) time, O((m+n) logn) work [Gazit ’91]: randomized, optimal, O(log n) time. CREW algorithms – [Han & Wagner ’90]: O(log2n) time, O((m+nlog n) logn) work.

Shiloach-Vishkin algorithm
Input: n isolated vertices and m PRAM processors. Each processor Pi grafts a tree rooted at vertex vi to the tree that contains one of its neighbors u under the constraints u< vi Grafting creates k ≥ 1 connected subgraphs, and each subgraph is then shortcut so that the depth of the trees reduce at least by half. Repeat graft and shortcut until no more grafting is possible. Runs on arbitrary CRCW PRAM in O(log n) time with O(m) processors. Helman-JaJa model: TM = (3m/p + 2)log n, TB = 4log n. graft shortcut 4 2 4 2 1 3 1 3 1,4 2,3

An example higher-level algorithm
Typically composed of several low-level efficient algorithms. Tarjan-Vishkin’s biconnected components algorithm: O(log n) time, O(m+n) time. Compute spanning tree T for the input graph G. Compute Eulerian circuit for T. Root the tree at an arbitrary vertex. Preorder numbering of all the vertices. Label edges using vertex numbering. Connected components using the Shiloach-Vishkin algorithm.

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.

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.

Graph Algorithms on today’s systems
Concurrency Simulating PRAM algorithms: hardware limits of memory bandwidth, number of outstanding memory references; synchronization Locality Work-efficiency Try to improve cache locality, but avoid “too much” superfluous computation

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

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 (x86), DMA support (Cell), wide SIMD, floating point-centric cores (GPUs). 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.

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

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

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

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

Performance Observations
Youtube social network Flickr social network Graph expansion Edge filtering

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

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

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)

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

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

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.

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)

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

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)

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.

∆ - 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

∆ - 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

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

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

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

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

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

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

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

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

Classify edges as “heavy” and “light”

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

Relax heavy edges. No reinsertions in this step.

No. of phases (machine-independent performance count)
high diameter low diameter

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

Last non-empty bucket (machine-independent performance count)
Fewer buckets, more parallelism

Number of bucket insertions (machine-independent performance count)

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

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.

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

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

Parallel BC Algorithm Illustration
5 8 1 7 3 4 6 9 2

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

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

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

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

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!

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

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

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

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

Centrality Analysis applied to Protein Interaction Networks
43 interactions Protein Ensembl ID ENSG Kelch-like protein 8

Community Identification
Implicit communities in large-scale networks are of interest in many cases. WWW Social networks Biological networks Formulated as a graph clustering problem. Informally, identify/extract “dense” sub-graphs. Several different objective functions exist. Metrics based on intra-cluster vs. inter-cluster edges, community sizes, number of communities, overlap … Highly studied research problem 100s of papers yearly in CS, Social Sciences, Physics, Comp. Biology, Applied Math journals and conferences.

Agglomerative Clustering, Parallelization
Bottom-up approach: Start with n singleton communities, iteratively merge pairs to form larger communities. What measure to minimize/maximize? A measure known as modularity How do we order merges? Use a priority queue Parallelization: perform multiple “independent” merges simultaneously.

Graph Analysis with cache-based multicore systems
SNAP: Small-world Network Analysis and Partitioning. 10-100x faster than competing graph analysis software. Parallelism, heuristics exploiting graph topology. Can process graphs with billions of vertices and edges. Open-source: Optimizations for multicore systems: improving cache locality, minimize synchronization overhead with atomics. D.A. Bader and K. Madduri, IPDPS 2008, Parallel Computing 2008.

Designing fast parallel graph algorithms
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 constant 106 108 1012 Peta+ 104 log n # of passes over data ~ n Data size (n: # of vertices/edges)

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

Future Research Challenges
New methods/analytics: Modeling network dynamics; persistent monitoring of dynamically changing properties. Software: Portable, high-performance, extensible routines. Emerging systems: how do we best utilize GPUs, flash disks? Analytics, Summarization, Visualization BIG DATA Modeling & Simulation

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