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CS214 Assignment
创建日期：2024-06-27 15:02:42
标签： CS214
Assignment

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1 Maximal Independent Set (35pts)

In a graph G = (V; E), an independent set is a set of vertices V 0 2 V , such that 8u; v 2 V 0, v and u are not
neighbors. A maximal independent set (MIS) is an independent set S such that for any v 2 V nS, S [ fvg
is not an independent set. This means that adding any one more node into S will make it not independent.

Note that MIS of a graph is not unique. Assume the graph is maintained using CSR, which means you can
directly read the degree of each vertex and the neighbor list of each vertex.

Sequentially, there is a simple greedy algorithm to find one maximal independent set. First of all, let’s
assign a unique priority to each vertex, uniformly at random. Let p(v) be the priority of vertex v. We
process all vertices in the order of their priority, highest the first. We start with S = ;. All vertices are
initialized as active. When processing a pivot v, we check if it’s still active. If not, we skip to the next
vertex. If so, we add v into S, and mark v and all its neighbors as inactive. This is because we chose v in S,
so all its neighbors cannot be chosen in S then. We then continue to the next vertex, until all vertices have
been processed. See Figure 1 as an example. The numbers indicates the priority of the vertices. The orange
vertices are those selected in MIS. We start with O1 and add it to S. It will set O6 and O5 as inactive.

Then we process O2 , and add it to S. Then we process O3 , and we found it’s already inactive (set inactive
by O2 ), so we skip it. O4 is still active now and can be added to S. O5 , O6 , and O6 are inactive because
of O1 and O2 . O8 is active and can be added. We can follow the process and finally, S = f1; 2; 4; 8; 10; 13g.


This does not work in parallel, since it requires O(jV j) rounds to process all vertices. In this problem, we
will design a parallel algorithm to find an MIS. Although directly parallelizing the above algorithm seems
hard, some techniques in class that might be useful. In particular, we will use the idea we learnt in the
lecture about deterministic parallelism. The core idea is to find all vertices that do not need to wait for any
other vertices, and just process them all in parallel.

In the class about deterministic parallelism, we’ve seen how to parallelize an iterative algorithm, where
the algorithm is a list of tasks performed one after the other. The key idea is to make the parallel version
of the algorithm to have exactly the same effect (and execution) as the sequential version. If
there are iterations that do not dependent on other iterations (or, all those it depends on has been finished),
we say an iteration is ready. In each round, we execute all ready iterations (tasks). In particular, we
can consider the following (you can plug in the random permutation algorithm to help you understand the
general idea):

For a task i, is there any unfinished task j before task i that could block task i? If not, or if all such
tasks have finished, we say task i is ready. We can then start performing task i immediately, instead
of waiting until task i − 1 finish.

When multiple tasks are ready at the same time, we can execute them all in parallel.

Answer the following questions:

1. (5pts) How can we generate unique priorities to each vertex uniformly at random? How can we do
that in parallel, and what are the work and span?

2. (5pts) Consider a vertex v in the graph with priority p(v), what is the condition that makes v ready
such that v can be used as the pivot right away? Hint: consider v’s priority and all its neighbors’
priority.

3. (8pts) Based on your idea above, in Figure 1, which vertices can be processed and added in round 1,
2, 3, …? For general cases, which vertices will be executed in parallel in each round (in this question,
you don’t need to specify how to find such vertices)?

4. (10pts) Describe a parallel algorithm based on your idea in Problem 3. You need to specify more
details about how to find the vertices to be processed in each round. You do not need to prove the
cost of your algorithm.

5. (7pts) Prove that your algorithm has span O(D log n), where D is the longest chain in the graph with
decreasing priority. In other words, D is the length of the longest chain v1; v2; :::; vD in the graph such
that p(v1) > p(v2) > · · · > p(vD)

6. (bonus, 1 candy and 3pts) Assume you have a constant-cost atomic operation fetch-and-add. De
sign a parallel algorithm for the MIS problem with work O(m), and span bound the same as above
(O(D log n)).


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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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