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Graph Algorithms Carl Tropper Department of Computer Science McGill University

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Definitions An undirected graph G is a pair (V,E), where V is a finite set of points called vertices and E is a finite set of edges. An edge e ∈ E is an unordered pair (u,v), where u,v ∈ V. In a directed graph, the edge e is an ordered pair (u,v). An edge (u,v) is incident from vertex u and is incident to vertex v. A path from a vertex v to a vertex u is a sequence of vertices where v 0 = v, v k = u, and (v i, v i+1 ) ∈ E for I = 0, 1,…, k-1. The length of a path is defined as the number of edges in the path.

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Directed and undirected

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More definitions An undirected graph is connected if every pair of vertices is connected by a path. A forest is an acyclic graph, and a tree is a connected acyclic graph. A graph that has weights associated with each edge is called a weighted graph.

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How to represent a graph 1 Adjacency matrix represents undirected graph (n 2 )space to store matrix

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How to represent a graph 2 An ordered list of nodes Weights are inside nodes (E) space to store the list

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Which is better? If graph is sparse, use pointers/list If dense, then go with the adjacency matrix

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MST: Prim’s Algorithm A spanning tree of an undirected graph G is a subgraph of G which is (1) a tree and (2) contains all the vertices of G. In a weighted graph, the weight of a subgraph is the sum of the weights of the edges in the subgraph. A minimum spanning tree (MST) for a weighted undirected graph is a spanning tree with minimum weight.

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MST on the right

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Prim is a greedy algorithm Is recursive Start by selecting an arbitrary vertex, include it into the current MST (the root) Grow the current MST by inserting into it the vertex closest to one of the vertices already in current MST (the vertex which (1) is outside the MST and (2) adds the smallest weight to the MST)

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Prim

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Formal Prim Lines 10-13 executed n-1 times 12 and 13 executed O(n) times Prim is (n 2 )

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Parallel Prim Let p be the number of processes, and let n be the number of vertices. The adjacency matrix is partitioned in 1-D block fashion, with distance vector d partitioned accordingly. In each step, a processor selects the locally closest node, followed by a global (all to one) reduction to select globally closest node. One processor (say P 0 ) contains the MST. This node is inserted into MST, and the choice broadcast (one to all) to all processors. Each processor updates its part of the d vector locally.

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Parallel Prim picture

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Analysis The cost to select the minimum entry is (n/p) The cost of a broadcast is (log p). The cost of local update of the d vector is (n/p). The parallel time per iteration is (n/p + log p). The total parallel time is given by (n 2 /p + n log p). The corresponding isoefficiency is (p 2 log 2 p).

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Single source shortest path Dijkstra For a weighted graph G = (V,E,w), the single- source shortest paths problem is to find the shortest paths from a vertex v ∈ V to all other vertices in V. Dijkstra's algorithm is similar to Prim's algorithm. It maintains a set of nodes for which the shortest paths are known. It grows this set based on the node closest to source using one of the nodes in the current shortest path set. The difference: Prim stores the the cost of the minimal cost edge connecting a vertex in VT to u, Dijkstra stores minimal cost to reach u Greedy algorithm

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Single source analysis The weighted adjacency matrix is partitioned using the 1-D block mapping. Each process selects, locally, the node closest to the source, followed by a global reduction to select next node. The node is broadcast to all processors and the l-vector updated. The difference: Prim stores the the cost of the minimal cost edge connecting a vertex in VT to u, Dijkstra stores minimal cost to reach u The parallel performance of Dijkstra's algorithm is identical to that of Prim's algorithm.

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All Pairs Shortest Path Given a weighted graph G(V,E,w), the all-pairs shortest paths problem is to find the shortest paths between all pairs of vertices v i, v j ∈ V. Look at two versions of Dijkstra And Floyd’s algorithm

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First of all Execute n instances of the single- source shortest path problem, one for each of the n source vertices. Complexity is (n 3 ) because complexity of shortest path algorithm is (n 2 )

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Two strategies Source partitioned-execute n shortest path problems on n processors. Each of the n nodes gets to be a source node. Source parallel-partition adjacency matrix a la Prim

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Dijkstra: Source Partitioned Use n processors, each processor P i finds the shortest paths from vertex v i to all other vertices by executing Dijkstra's sequential single-source shortest paths algorithm. It requires no interprocess communication (provided that the adjacency matrix is replicated at all processes). The parallel run time of this formulation is: Θ(n 2 ). While the algorithm is cost optimal, it can only use n processors. Therefore, the isoefficiency due to concurrency is p 3.

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Dijkstra: Source Parallel Want to keep more then n processors busy Given p processors (p > n), each single source shortest path problem is executed by one on one of n partitions Use p/n processors for each of the problems From before, the parallel time is: T P = Θ(n 3 /p) {computation}+ Θ(n log p) {communication} For cost optimality, we have p = O(n 2 /log n) and the isoefficiency is Θ((p log p) 1.5 ).

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Floyd’s For any pair of vertices v i, v j ∈ V, consider all paths from v i to v j whose intermediate vertices belong to the set {v 1,v 2,…,v k }. Let p i (, k j ) (of weight d i (, k j ) be the minimum-weight path among them. If vertex v k is not in the shortest path from v i to v j, then p i (, k j ) is the same as p i (, k j -1). If f v k is in p i (, k j ), then we can break p i (, k j ) into two paths - one from v i to v k and one from v k to v j. Each of these paths uses vertices from {v 1,v 2,…,v k-1 }.

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Recurrence This equation must be computed for each pair of nodes and for k = 1, n. The serial complexity is O(n 3 ). From our observations, the following recurrence relation follows:

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Floyd This program computes the all-pairs shortest paths of the graph G = (V,E) with adjacency matrix A.

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Parallel Floyd Matrix D (k) is divided into p blocks of size (n / √p) x (n / √p) {n 2 /p elements per block} Each processor updates its part of the matrix during each iteration. To compute d l (, k k -1) processor P i,j must get d l (, k k -1) and d k (, k r -1). In general, during the k th iteration, each of the √p processes containing part of the k th row send it to the √p - 1 processes in the same column. Similarly, each of the √p processes containing part of the k th column sends it to the √p - 1 processes in the same row.

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Matrix D k (a) Matrix D (k) distributed by 2-D block mapping into √p x √p subblocks, and (b) the subblock of D (k) assigned to process P i,j.

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Communication In general, during the k th iteration, each of the √p processes containing part of the k th row send it to the √p - 1 processes in the same column. Similarly, each of the √p processes containing part of the k th column sends it to the √p - 1 processes in the same row.

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Communication Patterns a) Communication patterns used in the 2-D block mapping. When computing d i (, k j ), information must be sent to the highlighted process from two other processes along the same row and column. (b) The row and column of √p processes that contain the k th row and column send them along process columns and rows.

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Floyd-Parallel Formulation Floyd's parallel formulation using the 2-D block mapping. P *,j denotes all the processes in the j th column, and Pi, * denotes all the processes in the i th row. The matrix D (0) is the adjacency matrix.

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Analysis During each iteration of the algorithm, the k th row and k th column of processors perform a one-to-all broadcast along their rows/columns. The size of this broadcast is n/√p elements, taking time Θ((n log p)/ √p). The synchronization step takes time Θ(log p). The computation time is Θ(n 2 /p) There are n iterations of the algorithm {k=1,..,n} The parallel run time of the 2-D block mapping formulation of Floyd's algorithm is

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Analysis II The above formulation can use O(n 2 / log 2 n) processors cost-optimally. The isoefficiency of this formulation is Θ(p 1.5 log 3 p). This algorithm can be further improved by relaxing the strict synchronization after each iteration Go to next slide

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Pipelining Floyd The synchronization step in parallel Floyd's algorithm can be removed without affecting the correctness of the algorithm. A process starts working on the k th iteration as soon as it has computed the (k-1) th iteration and has the relevant parts of the D (k- 1) matrix.

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Pipelining Floyd Communication protocol followed in the pipelined 2-D block mapping formulation of Floyd's algorithm. Assume that process 4 at time t has just computed a segment of the k th column of the D (k-1) matrix. It sends the segment to processes 3 and 5. These processes receive the segment at time t + 1 (where the time unit is the time it takes for a matrix segment to travel over the communication link between adjacent processes). Similarly, processes farther away from process 4 receive the segment later. Process 1 (at the boundary) does not forward the segment after receiving it.

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Pipelining Analysis In each step, n/√p elements of the first row are sent from process P i,j to P i+1,j. Similarly, elements of the first column are sent from process P i,j to process P i,j+1. Each such step takes time Θ(n/√p). After Θ(√p) steps, process P √p,√p gets the relevant elements of the first row and first column in time Θ(n). The values of successive rows and columns follow after time Θ(n 2 /p) in a pipelined mode. Process P √p,√p finishes its share of the shortest path computation in time Θ(n 3 /p) + Θ(n). When process P √p,√p has finished the (n-1) th iteration, it sends the relevant values of the n th row and column to the other processes.

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Pipelining Analysis The pipelined formulation of Floyd's algorithm uses up to O(n 2 ) processes efficiently. The corresponding isoefficiency is Θ(p 1.5 ). The overall parallel run time of this formulation is

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Comparison of shortest path algorithms Assumption: Parallel architecture has O(p) bisection bandwith {minimum communication between 2 halves of network} Conclusion: Pipelined Floyd is the most scalable-lowest iso and run time and can use (n 2) processors

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Transitive Closure of a Graph If G = (V,E) is a graph, then the transitive closure of G is defined as the graph G* = (V,E*), where E * = {(v i,v j ) | there is a path from v i to v j in G} The connectivity matrix of G is a matrix A * = (a i *,j ) such that a i *,j = 1 if there is a path from v i to v j or i = j, and a i *,j = ∞ otherwise. To compute A * we assign a weight of 1 to each edge of E and use an all-pairs shortest paths algorithm on the graph.

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Connected Components of a Graph G=(V,E) V=C 1 C 2 ……… C k u,v belong to C i iff there is a path from u to v and vice versa Picture

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Depth First Search Perform DFS on the graph to get a forest - each tree in the forest corresponds to a connected component. b has the components Each component is a tree

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Parallel Component algorithm Partition adjacency matrix into p sub- graphs G i and assign each G i to a process P i Each P i computes spanning forest of G i Merge spanning forests pair wise until there is one spanning forest

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Parallel Components

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Ops for merging Algorithm uses disjoint sets of edges. Ops for the disjoint sets: find(x) –returns a pointer to the representative element of the set containing x. Each set has its own unique representative. union(x, y) –unites the sets containing the elements x and y.

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Merging Ops For merging forest A into forest B, for each edge (u,v) of A, a find operation is performed to determine if the vertices are in the same tree of B. If not, then the two trees (sets) of B containing u and v are united by a union operation. Otherwise, no union operation is necessary. Merging A and B requires at most 2(n-1) find operations and (n-1) union operations.

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Parallel Block Mapping The n x n adjacency matrix is partitioned into p blocks. Each processor can compute its local spanning forest in time Θ(n 2 /p). Merging is done by embedding a logical tree into the topology. There are log p merging stages, and each takes time Θ(n). Thus, the cost due to merging is Θ(n log p). During each merging stage, spanning forests are sent between nearest neighbors. Θ(n) edges of the spanning forest are transmitted

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Performance of Block Mapping The performance is similar to Prim and Dijkstra’s single shortest path algorithm For a cost-optimal formulation p = O(n / log n). The corresponding isoefficiency is Θ(p 2 log 2 p).

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Sparse Graphs A graph G = (V,E) is sparse if |E| is much smaller than |V| 2

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Algorithms for sparse graphs Can reduce the complexity of dense graph algorithms by making use of adjacency list instead of adjacency graph –E.g. Prim’s algorithm is Θ(n 2 ) for a dense matrix and Θ(|E| log n) for a sparse matrix Key to good performance of dense matrix algorithms was the ability to assign roughly equal workloads to all of the processors and to keep the communication local Floyd-assigned equal size blocks from the adjacency matrix of consecutive rows and columns=>local communication

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Sparse difficulties Partitioning adjacency matrix is harder then it looks Assign equal # vertices to processors + their adjacency lists. Some processors may have more links then others. Assign equal # links=> need to split adjacency list of a vertex among processors=>lots of communication Works for “certain” structures-next slide

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Grid graph-a certain structure If vertices have more or less the same degree it works.

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Maximal Independent Set A set of vertices I ⊂ V is called independent if no pair of vertices in I is connected via an edge in G. An independent set is called maximal if by including any other vertex not in I, the independence property is violated.

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Who cares? Maximal independent sets can be used to find how many parallel tasks from a task graph can be executed Used in graph coloring algorithms

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Simple MIS algorithm start with empty MIS I, and assign all vertices to a candidate set C. Vertex v from C is moved into I and all vertices adjacent to v are removed from C. This process is repeated until C is empty. Problem: serial algorithm

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Luby’s algorithm In the beginning C is set equal to V, I is empty Assign random numbers to all of the nodes of the graph If vertex has a smaller random # the all of its adjacent vertices, then put it in I Delete all of its adjacent neighbors from C Repeat above steps until C is empty Takes O(log |V|) steps on the average Luby invented this algorithm for graph coloring

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Parallel Luby for Shared Address Space 3 arrays of size |V| I: idependence array I(i)=1 if i is in MIS. Initially all I(i)=0 R: random number array C: candidate array C(i)=1 if i is a candidate Partition C among p processes Each process generates random number for each of its vertices Each process checks to see if random numbers of its vertices are smaller then those of adjacent vertices Process zeroes entries corresponding to adjacent entries

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Single Source Shortest Paths 2 steps happen in each iteration Extract u (V-V T ) such that l[u]=min[l[v]], v V-V T For each vertex in (V-V T ), compute l[v]=min[l[v],l[u] + w(u,v)] Make sense to use adjacency list for last equation Use priority queue to store l values with smallest on top Priority queue implemented with binary heap

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Johnson Updating vertices in the heap is the big time sink-O(log n) per update and |E| updates=> (|E|log n) complexity

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Parallel Johnson-first attempts Use one processor P 0 to house the queue- other processes update l[v] and give them to P 0 Problems: Single queue is a bottleneck Small number of processes can be kept busy (|E|/|V|) So distribute queue to processes. Need low latency architecture to make this work

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First atttempts-continued Still have low speedup (O(log n)) if each update takes O(1)) Can extract multiple nodes from queue- all vertices u with same minimum distance Why? Can process nodes with same distance in any order Still not enough speedup

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Solution-speculative decomposition When process P i extracts the vertex u ∈ V i, it sends a message to processes that store vertices adjacent to u. Process P j, upon receiving this message, sets the value of l[v] stored in its priority queue to min{l[v],l[u] + w(u,v)}. If a shorter path has been discovered to node v, it is reinserted back into the local priority queue. The algorithm terminates only when all the queues become empty.

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Speculative decomposition Distributed Memory Partition queue among the processors Partition vertices and adjacency lists among the processors Each processor updates its local queue and sends the results to the processors with adjacent vertices Recipient processor updates its value for shortest path

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More precisely For example: u Pi and v Pj and (u.v) is an edge. P i extracts u and updates l[u] P i sends message to P j with new value for v, which is l[u]+w(u,v). P j compares l[u]+w(u,v) to sp[v] If smaller, then have new sp[v] If larger, then discard

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In the beginning Only the source queue has a non-empty priority queue Then the wave happens

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Mapping-2-D Block n/√p x n /√p mesh

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Analysis of 2-D At most O( p) processes are busy at any time because the wave moves diagonally (diagonal= p) Max speedup is S=W/W/√p= √p E=1/√p Lousy efficiency

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2-D Cyclic

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Analysis Improves things because vertices are further apart Bad news: more communication

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1-D Block Mapping

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1-D rules Better idea because as wave spreads more processors get involved concurrently Assume p/2 processes are busy, then S=W/W/p/2=p/2 E=1/2 Improvement over 2-D Bad side-uses O(n) processes

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