## Description

CS 314 Principles of Programming Languages

Project 3: Efficient Parallel Graph Matching

In this project, you will be asked

to implement two parallel graph matching algorithms. Your program should take a legal

matrix-market matrix file as input and output the matching results.

This document is not a complete specification of the project. You will encounter important design and implementation issues that need to be addressed in your project solution.

Identifying these issues is part of the project. As a result, you need to start early, allowing

time for possible revisions of your solution.

1 Background

1.1 The Graph Matching Problem

Given a graph G = (V, E), while V is the set of vertices (also called nodes) and E ⊂ |V |

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.

A matching M in G is a set of pairwise non-adjacent edges which means that no two edges

share a common vertex. A vertex is matched if it is an endpoint of one of the edges in

the matching. Otherwise the vertex is unmatched [3]. In Figure 1, we show three different

matchings for the same graph.

A maximum matching can be defined as a matching where the total weight of the

edges in the matching is maximized. In Figure 1, (c) is a maximum matching because the

total weight of the edges in the matching is 7, and there could be no other matching that

has total weight greater than 7 for this graph.

(a) (b) (c)

3

2 4

2 1 3

2 4

2 1 3

2 4

2 1

Figure 1: Graph Matching Examples

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1.2 Parallel Graph Matching Algorithms

The graph matching problem is not easy to parallelize. Most existing matching algorithms

such as blossom algorithm are embarrassingly sequential. Here we describe two handshaking-based algorithms [2] that are amenable to parallelization.

1.2.1 One-Way Handshaking Matching

Given a graph G, two vertices shake hands only when there is an edge between these two

and they are the strongest neighbor of each other. We will define “strongest neighbor”. The

edge that connects the two handshaking vertices is added to the matching M. We show an

example of one-way handshaking matching in Figure 2.

To identify handshaking vertices, each vertex v in G extends a hand to its strongest

neighbor. Here the strongest neighbor of a vertex v is the neighbor vertex that sits on the

maximum-weight edge incident to v. This is a greedy algorithm that aims to maximize the

total weight in the matching. If there are multiple incident edges of v that have maximum

weight, the neighbor vertex that has the smallest vertex index will be chosen as the

strongest neighbor. For example, in Figure 2(b), the strongest neighbor of vertex E is vertex

B because it has an edge weight of 4 and a smaller index (alphabetical order) than vertex F.

For each vertex, we check if its strongest neighbor extends a hand back. If so, these

two vertices are matched. Then the corresponding edge will be added to the matching. For

example, in Figure 2(c), vertex B extends a hand to vertex E and vertex E also extends a

hand to vertex B, so edge BE will be added to the matching.

At every pass of the process, we check all the vertices that are not matched from previous

passes (each vertex is checked once in each pass), and identify if there is any new edge that

can be added to the matching. We repeat this until no more edges can be added to the

matching. We show the passes in Figure 2(b), (c), (d),(e) and (f). The handshaking method

is highly data parallel, since each vertex can be processed independently and there is no data

race.

1.2.2 N-Way Handshaking Matching

In one-way handshaking matching, it is possible that one vertex will have extended hands

from multiple neighbors, but at most one of these neighbors can be matched. This may affect

the matching efficiency. For example, in Figure 2 (b), both vertex D and vertex H extend

hands to vertex E. However, since vertex E’s strongest neighbor is vertex B, so neither vertex

D nor H will be matched at this pass of handshaking in Figure 2 (b).

Instead of extending one hand, N-way handshaking matching allows each vertex to

extend N hands (N 1). We show an example of 2-way handshaking matching in Figure 3.

In Figure 3(b), each vertex extends (up to) 2 hands at once, which extends to its strongest

and second strongest neighbors. For example, vertex H extends one hand to vertex E which

is its strongest neighbor and another hand to vertex I which is the second strongest neighbor.

In the next step, we take the edges whose two end points do not shake hands out of

consideration (as if we “discard” edges). In Figure 3(b), there is no handshaking between

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D E F

A B C

G H I

1 2

3

4

1 2

2 4 4

1 3 1

(a) Original Graph

D E F

A B C

G H I

1 2

3

4

1 2

2 4 4

1 3 1

(b) Extend a Hand to

the Strongest Neighbor

D E F

A B C

G H I

1 2

3

4

1 2

2 4 4

1 3 1

(c) Check Handshaking

and Matching Vertices

D E F

A B C

G H I

1 2

3

4

1 2

2 4 4

1 3 1

(d) Repeat (b), (c) with

Unmatched Vertices

D E F

A B C

G H I

1 2

3

4

1 2

2 4 4

1 3 1

(e) Repeat (b), (c) with

Unmatched Vertices

D E F

A B C

G H I

1 2

3

4

1 2

2 4 4

1 3 1

(f) One-Way Handshaking

Matching Result

Figure 2: One-Way Handshaking Matching Example

vertex H and vertex E, so the edge HE is “as if” discarded (before next step in the same

pass). After this, we will obtain an updated graph with a max degree N (N=2 in Figure

3(c)), we will refer to this graph as N-way graph in the remaining of the project description.

We now do one-way handshaking matching on the updated N-way graph. The matching on the N-way graph may yield more matched vertices. For example, in Figure 3(e), both

vertex H and vertex D can be matched in the first pass, while compared with one-way matching in Figure 2, both vertex H and vertex D have to be matched in the second pass.

1.3 POSIX threads – pthreads

A thread is defined as an independent stream of instructions that can be scheduled to run in

its own context. Multiple threads run in multiple active contexts.

Historically, hardware vendors have implemented their own proprietary versions of threads.

For UNIX systems, a standardized C language threads programming interface has been specified by the IEEE POSIX 1003.1c standard. Implementations that adhere to this standard

are referred to as POSIX threads, or pthreads[1].

For more details on how to use pthreads API to write a parallel program and how to

compile a pthreads program, please refer to the pthread tutorial [1] and the recitations.

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D E F

A B C

G H I

1 2

3

4

1 2

2 4 4

1 3 1

(a) Original Graph

D E F

A B C

G H I

1 2

3

4

1 2

2 4 4

1 3 1

(b) Extend N=2 Hands

at One Time

D E F

A B C

G H I

2

4

2

2 4 4

(c) Discard Edges

without a Handshaking

D E F

A B C

G H I

2

4

2

2 4 4

(d) Do One-Way Handshaking on the new graph

(e) Check Handshaking

and Matching Vertices

D E F

A B C

G H I

2

4

2

2 4 4

D E F

A B C

G H I

1 2

3

4

1 2

2 4 4

1 3 1

(f) N-Way Handshaking

Matching Result

Figure 3: N-Way Handshaking Matching Example (N=2)

2 Implementation

In this project, you will be asked to do the following:

1) implement the parallel one-way handshaking matching algorithm

2) Extra Credit: implement the parallel N-way handshaking matching algorithm

2.1 Data Structure vertex v0 v1 v2 v3 v4 v5 v6 v7 v8

offset 0 2 5 7 10 14 17 19 22

index 3 1 4 2 0 5 1 …… 6 7 5

weight 2 1 4 2 1 4 2 …… 1 2 1

degree 2 3 2 3 4 3 2 3 2

Figure 4: Adjacency Array Data Structure

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Un-directed weighted graph In this project, we use adjacency lists to represent an undirected weighted graph. There are four arrays: index[], offset[], degree[], and weight[]. The

array index[] keeps the neighbor lists of all vertices, for instance, node v0’s neighbor list is

followed by neighbor v1’s neighbor list, and so on. The array offset[] stores where a node’s

neighbor list starts in the index[] array. The corresponding weight[] array stores the weight

of each edge. The array degree[] stores the degree of each vertex, which is the number of

neighbors for each vertex.

An example of adjacency array representation is shown in Figure 4. For vertex v2, its

degree is 2 (degree[2] = 2). The offset[2] value is 5 which means its neighbor list starts

at index[5], thus index[5] = 5 (corresponding to vertex v5), index[6] = 1 (corresponding to

vertex v1), and vertex v5 and vertex v1 are two neighbors of vertex v2.

Pleas be aware that within a neighbor list (of a specific vertex), the neighbors

are sorted in descending order such that the strongest neighbor is always placed

as the first item and the second strongest neighbor is placed as the second item

and so on. We already sorted the neighbors of each vertex for you. You do not

have to sort them to find the strongest neighbor, however, you do need to filter

out the nodes that are already matched from previous passes.

We have provided graph I/O functions and code for reading/parsing the graphs, the

pointers to the four arrays: index[], offset[], weight[], and degree[] are stored in GraphData

struct in the provided code package. Here is what GraphData looks like:

struct GraphData {

int nNodes;

int nEdges;

int *offset;

int *degree;

int *index;

double *weight;

}

In the main function of the provided code package, it calls readmm(inputF ile, &graph) to

read and parse an input graph file into the data object graph. Please check DataStructure.h

and utils.c for more details.

N-way graph In N-way handshaking matching algorithm, you need to generate the

N-way graph. We use two arrays to represent the N-way graph: nWayGraphDegree[] and

nWayGraph[]. nWayGraphDegree[] is an array that stores the degree of each vertex in the

N-way graph. nWayGraph[] represents the adjacency list of the nodes in the N-way graph.

Since every node in the N-way graph has at most N neighbors, we allocate N*node_number

elements for the nWayGraph[] array, such that the neighbors of the vertex vi are stored

starting from nW ayGraph[i * N] to at most nW ayGraph[i * N + nW ayGraphDegree[i]-1]

in descending order such that the strongest neighbor are placed first. In Figure 5, we show

an example of the 2-way graph.

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vertex 0 1 2 3 4 5 6 7 8

nWayGraph 2 1 4 2 5 1 4 2 1 …… 8

nWayGraphDegree 2 2 2 2 2 2 2 2 2

Figure 5: 2-Way Graph Data Structure

Matching results The matching algorithm runs for as many passes as it needs, until no

edge can be added into the matching. It is possible that some vertices cannot be matched to

any neighbor.

In this project, we use the array res[] to store the matching results. We provide several

defined constants to represent the status of a vertex in match.h. The array res[] is initialized

to UNMATCHED (-1) for each vertex. When the graph matching program terminates,

for vertex i, res[i] is either its matched vertex index or NO_MATCHED_NODES (-2)

which represents vertex i is not matched. In general:

res[i] =

j, if vertex i and vertex j are matched

−1, initial value

−2, vertex i is not matched

In the main function, it calls write_match_result(outputF ile, res, nNodes) to write

the matching results stored in res into output file outputF ile. nNodes represents the total

number of vertices.

2.2 Work Balancing by Vertices

i = current thread index

nodeToProcess = the list of nodes to process

workchunk = (numNodes + threadNum – 1) / threadNum

beg = i * workchunk

end = min(beg + workchunk, numNodes)

for each v in { nodeToProcess[beg], nodeToProcess[beg+1], …, nodeToProcess[end-1] }

do

… your code for matching

end for

Figure 6: Balancing Based on Vertex Number

To implement parallel handshaking-based matching algorithms, we need to map the tasks

into different threads in a load balanced way. In this project, each thread will be in charge of

a subset of vertices. A global barrier synchronization is performed after each pass of matching

(the barrier code is already provided for you). The algorithm distributes the vertices evenly

to the co-running threads. Assuming there are nNodes vertices to be processed, the total

number of threads is threadNum, each thread will process around nNodes/threadNum

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nodes. Since nNodes is not necessarily a multiply of threadNum, the last thread might

be assigned ≤ nNodes/threadNum vertices. The code snippet in Figure 6 shows how each

thread should find the set of the vertices it is in charge of. Please use this vertex balancing

method for all your parallel function implementation.

2.3 One-Way Handshaking

We have described the one-way handshaking matching algorithm in Section 1.2.1. In

one-way handshaking, each vertex extends a hand to its strongest neighbor. Next it checks

if there is a handshaking between itself and its strongest neighbor.

You only need to complete the following functions in oneway.c:

1. extend_one_hand(int threadId, int threadNum, GraphData graph, int nodeNum, int *nodeToProcess, int *res, int *strongNeighbor)

Function Description:

Each thread needs to be assigned a subset of vertices from nodeToProcess[] array. See

Section 2.2 for load balancing method. For each vertex with ID v in this subset, find

its strongest neighbor and store the vertex index in the array strongNeighbor[v]. The

pseudo code is shown below.

Vi = the set of vertices assigned to thread i

N(k) = the set of (sorted) neighbors for vertex k

for each v in Vi

for each u in N(v) do

if (res[u] == UNMATCHED) then

strongNeighbor[v] = u

break out loop u

end if

end for

end for

Input Parameters:

threadId: The thread index

threadNum: Total thread number

graph: The graph object

nodeNum: The number of vertices

nodeToProcess Each element is a vertex ID, and this array is usually used to pass the

unmatched vertices to the processing function. The size of the array is nodeNum.

res: The array that stores the matching status for each vertex. The size of the array is

the total number of vertices in the graph.

Output Parameters:

strongNeighbor: The array that stores the index of the strongest neighbor for each

node. The size of the array is the total number of vertices in the graph.

e.g. strongNeighbor[] = {3,4,5,4,1,2,3,4,7} for the example in Figure 2

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2. check_handshaking(int threadId, int threadNum, int nodeNum, int *nodeToProcess, int *strongNeighbor, int *res)

Function Description:

Each thread needs to be assigned a subset of vertices from nodeToProcess[] array.

See Section 2.2 for load balancing method. For each vertex v in this subset, given

its strongest neighbor strongNeighbor[v], update res[v] correspondingly. The pseudo

code is shown below.

Vi = the set of vertices assigned to thread i

for each v in Vi do

s = strongNeighbor[v]

if (v == strongNeighbor[s]) then

res[v] = s

end if

end for

Input Parameters:

threadId: Thread index

threadNum: Total thread number

nodeNum: The number of vertices

nodeToProcess Each element is a vertex ID, and this array is usually used to pass the

unmatched vertices to the processing function. The size of the array is nodeNum.

strongNeighbor: The array that stores the index of the strongest neighbor for each

vertex. The size of the array is the total number of vertices in the graph.

Output Parameters:

res The array that stores the matching status for each vertex. The size of the array is

the total number of vertices in the graph.

E.g. res[] = {-1,4,5,-1,1,2,-1,-1,-1} for the example in Figure 2

2.4 Extra Credit (20%): N-Way Handshaking Matching

For N-way handshaking matching, each vertex extends N hands, then the algorithm

neglects (as if “discards”) those edges whose two end nodes have no handshaking, resulting

in the N-way graph. Upon the N-way graph, one-way matching is performed.

You only need to implement the following functions in nways.c:

1. generate_n_way_graph(int threadId, int threadNum, GraphData graph,

int nWays, int nodeNum, int *nodeToProcess, int *res, int *nWayGraphDegree, int *nWayGraph)

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Function Description:

Each thread needs to be assigned a subset of vertices from nodeToProcess[] array. See

Section 2.2 for load balancing method. For each vertex v, find its N strongest neighbors. Then the algorithm obtains the N-way graph information which is represented by

nW ayGraphDegree[] and nW ayGraph[]. The pseudo code is shown below. Note that

the neighbors of each vertex in the N-way graph also need to be sorted such that the

strongest neighbor is placed first.

Vi = the set of vertices assigned to thread i

N(k) = the set of (sorted) neighbors for vertex k

for each v in Vi

degree = 0

for each u in N(v) in sorted order do

if (res[u] == UNMATCHED) then

nWayGraph[N * v + degree] = u

degree++

if (degree == nWays) then

break for loop u

end if

end if

end for

end for

Input:

threadId: Thread index

threadNum: Total thread number

graph: Graph object

nWays: the number of hands extended

nodeNum: The number of vertices

nodeToProcess: Each element is a vertex ID, and this array is usually used to pass the

unmatched vertices to the processing function. The size of the array is nodeNum.

res: The array that stores the matching status for each vertex. The size of the array is

the total number of vertices in the graph.

Output:

nWayGraphDegree: The array that stores the vertex degree of the N-way graph. The

size of the array is the total number of vertices in the graph,

nWayGraph: The array that stores the N strongest neighbors of each vertex. The size

of the array is the total number of vertices multiplied by N.

e.g. nW ayGraphDegree[] = {2,2,2,2,2,2,2,2,2}

nW ayGrap[] = {3,1,4,2,5,1,4,0,1,5,2,4,3,7,4,8,7,5} for the example in Figure 3

2. prune_n_way_graph(int threadId, int threadNum, int nWays, int nodeNum, int *nodeToProcess, int *nWayGraphDegree, int *nWayGraph, int

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*strongNeighbor)

Function Description:

Each thread needs to be assigned a subset of vertices from nodeToProcess[] array. See

Section 2.2 for load balancing method. For each vertex, it finds the strongest neighbor

in the N-way graph. The pseudo code is shown below. Note that the neighbors of each

vertex are sorted such that the strongest neighbor is placed first.

Vi = the set of vertices assigned to thread i

S(k) = the set of N strongest neighbors for

vertex k

for each v in Vi

for each u in S(v) in sorted order do

for each x in S(u) in sorted order do

If (x == v) then

strongNeighbor[x] = u

break for loop u

end if

end for

end for

end for

Input Parameters:

threadId: Thread index

threadNum: Total thread number

nWays: The number of hands extended

nodeNum: The number of vertices

nodeToProcess: Each element is a vertex ID, and this array is usually used to pass the

unmatched vertices to the processing function. The size of the array is nodeNum.

nWayGraphDegree: The array that stores the vertex degree of the N-way graph. The

size of the array is the total number of vertices in the graph,

nWayGraph: The array that stores the N strongest neighbors of each vertex. The size

of the array is the total number of vertices multiplied by N.

Output Parameters:

strongNeighbor: The array that stores the the strongest neighbor for each vertex. The

size of the array is the total number of vertices in the graph.

e.g. strongNeighbor[] = {3,4,5,0,1,2,-2,8,7} for the example in Figure 3

2.5 Filtering Matched Vertices

In handshaking algorithms, each pass except the last one will produce a set of matched

vertices. The vertices that are matched from previous passes need not to be considered in

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the later passes. You will need to filter out the nodes that are already matched before starting

a new pass. Then you can distribute the unmatched vertices evenly to different threads again.

At the first pass, all the vertices need to be processed and they are evenly distributed

to co-running threads. Starting from the second pass, you will need to count the number

of unmatched nodes. An example is shown in Figure 7. Each thread counts the number of

unmatched vertices for the set of vertices it is in charge of, and stores the counts in the array

nodeCount[]. In Figure 7, for instance, nodeCount[0] = 1 since only vertex 0 is not matched

among the set of vertices that are processed by thread 0.

Next, an exclusive prefix sum is performed on nodeCount array. An exclusive prefix sum

of an array x[] is another array y[] that:

y[i] =

0, if i = 0

Pi−1

j=0 x[j], otherwise

The purpose of using exclusive prefix sum is to find out where to store the unmatched

vertices in the array newNodeT oP rocess[]. For example, in Figure 7, nodeCount = {1, 1,

3}, after exclusive prefix sum, it is y ={0, 1, 2} and threads i know where to place the

unmatched vertices it is in charge of: starting from newNodeToProcess[y[i]]. Every thread

then stores the unmatched vertices it finds to the newNodeT oP rocess array that will be

used for next pass.

nodeToProcess 0 1 2 3 4 5 6 7 8

Thread 0 Thread 2

nodeCount 1 1

Thread 1

3

exclusive prefix sum 0 1 2

newNodeToProcess 0 3 6 7 8

0 unmatched vertex 1 matched vertex

Figure 7: Filter Matched Nodes from Vertex Array

You only need to complete the following functions in filter.c:

1. count_unmatched_vertices(int threadId, int threadNum, int nodeNum, int

*nodeToProcess, int *res, int *nodeCount)

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Function Description:

Each thread counts the unmatched vertices in the set of vertices it is in charge of. The

count by thread i is stored in nodeCount[i]. The pseudo code is shown below.

Vi = the set of vertices assigned to thread i

for each v in Vi do

if (res[v] == UNMATCHED) then

nodeCount[i]++

end if

end for

Input:

threadId: Thread index

threadNum: Total thread number

nodeNum: The number of vertices

nodeToProcess: Each element is a vertex ID, and this array is usually used to pass the

unmatched vertices to the processing function. The size of the array is nodeNum.

res: The array that stores the matching status. The size of the array is the number of

vertices.

Output:

nodeCount: Each element of the array stores the number of unmatched vertices collected by each thread. The size of the array is the total number of threads.

e.g. nodeCount[] = {1,1,3} for the example in Figure 2.

2. update_remain_nodes_index(int threadId, int threadNum, int *nodeToProcess, int *startLocations, int *res, int nodeNum, int newNodeToProcess)

Function Description:

Each thread stores the unmatched vertices it finds into the newNodeT oP rocess array

starting at the location provided in startLocations. The pseudo code is shown below.

Vi = the set of vertices assigned to thread i

for each v in Vi do

if (res[v] == UNMATCHED) then

offset = startLocations[i]++

newNodeToProcess[offset] = v

end if

end for

Input:

threadId: Thread index

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threadNum: Total thread number

nodeToProcess: Each element is a vertex ID, and this array is usually used to pass the

unmatched vertices to the processing function. The size of the array is nodeNum.

startLocations: The array that stores the start location of each thread in newNodeT oP rocess.

The size of the array is the number of threads plus one.

res: The array that stores the matching status. The size of the array is the number of

vertices.

nodeNum: The number of vertices to process

Output:

newNodeToProcess: Each element is a vertex ID. The size of the array is the number

of vertices.

newNodeT oP rocess[] = {0,3,6,7,8} for the example in Figure 3

.

3 Grading

We provide six real world matrices for you to test your code. In the code package, we have

provided the function for transforming the input matrix to GraphData graph in main.c. Six

matrices and four hidden matrices will be used to grade your program. Your programs will

be graded based on functionality. You will receive 0 credit if we cannot compile/run your

code on ilab machines. All grading will be done automatically on ilab machines.

4 How to Get Started

The code package for you to start with is provided on Sakai. Create your own directory

on the ilab cluster, and copy the entire provided project folder to your home directory or

any other one of your directories. Make sure that the read, write, and execute permissions

for groups and others are disabled (chmod go-rwx <directory_name). Please read the

README file before you start.

4.1 Compilation and Execution

Compilation: We have provided a Makefile in the sample code package.

make match should compile everything into a single executable called match.

make clean should remove all generated executable and intermediate files.

Execution: To see the usage of the program, please compile the program and simply run

./match and the usage information will be shown. Please DO NOT change the usage of

the program.

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Program Output: The program will output the matching results to the file you specify in

the arguments and print the time and the number of iterations to get the maximal matching

on the screen. We have handled this part for you. Please DO NOT change any other file

that is not oneway.c, nways.c, and filter.c.

4.2 Input Matrix List

The following six matrix files are provided for you to test your graph matching implementations. The matrix will be transformed to a weighted un-directed graph automatically by the

program. A micro test matrix “test.mtx” is provided in the code package. The test matrix is

the example we used in the project description.

af_shell10 https://sparse.tamu.edu/MM/Schenk_AFE/af_shell10.tar.gz

cant https://sparse.tamu.edu/MM/Williams/cant.tar.gz

consph https://sparse.tamu.edu/MM/Williams/consph.tar.gz

dawson5 https://sparse.tamu.edu/MM/GHS_indef/dawson5.tar.gz

hood https://sparse.tamu.edu/MM/GHS_psdef/hood.tar.gz

ldoor https://sparse.tamu.edu/MM/GHS_psdef/ldoor.tar.gz

If the disk space on ilab machine is not enough, please download one matrix at one time

or use /freespace to store the matrices.

5 What to Submit

You need to submit two files: oneway.c and filter.c. If you finish the extra credit part for

N-way handshaking, please also submit nways.c. Please DO NOT include any output

matching results file and input graph/matrix files in your submission.

6 Questions

All questions regarding this project should be posted on Sakai forum. Enjoy the project!

References

[1] Blaise Barney. Posix threads programming. https://computing.llnl.gov/tutorials/

pthreads/, 2017.

[2] Jonathan Cohen and Patrice Castonguay. Efficient graph matching and coloring on the

gpu.

[3] Wikipedia contributors. Matching (graph theory) — Wikipedia, the free encyclopedia. https://en.wikipedia.org/w/index.php?title=Matching_(graph_theory)

&oldid=869039028, 2018. [Online; accessed 20-November-2018].

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