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9
АЛГОРИТМИЗАЦИЯ И ПРОГРАММИРОВАНИЕ ALGORITHMIC AND PROGRAMMING
УДК 004.272.2
А. А. Prihozhy1, O. N. Karasik2
^elarusian National Technical University 2ISsoft Solutions (part of Coherent Solutions)
ADVANCED HETEROGENEOUS BLOCK-PARALLEL ALL-PAIRS SHORTEST PATH ALGORITHM
The problem of finding shortest paths between all pairs of vertices in a large-size graph has many application domains in industry, technology, science, economics, and society. The algorithms solving the problem and proposed in the literature target either lowering a computational complexity or efficient exploitation of computational resources. This paper proposes the advanced heterogeneous block-parallel shortest paths algorithm that is a result of further development and improvement of known blocked algorithms. Starting from the homogeneous blocked algorithm, it distinguishes four types of blocks: diagonal, vertical of cross, horizontal of cross, and peripheral. To speed up the computations, separate algorithms for all block types have been developed, which reduce the number of iterations in nested loops and account for the sequential reference locality of data in CPU caches. The algorithms improve the spatial and temporal reference locality in big data processing. Experiments carried out on a server equipped with two Intel Xeon E5-2620 v4 processors have shown the speedup of up to 60-70% the proposed single- and multi-threaded advanced heterogeneous blocked algorithms yield over the single-and multiple-threaded homogeneous blocked Floyd - Warshall algorithms.
Keywords: shortest path, blocked algorithm, heterogeneous algorithm, multi-core system, throughput.
For citation: Prihozhy А. А., Karasik O. N. Advanced heterogeneous block-parallel all-pairs shortest path algorithm. Proceedings of BSTU, issue 3, Physics and Mathematics. Informatics, 2023, no. 1 (266), pp. 77-83. DOI: 10.52065/2520-6141-2023-266-1-13.
А. А. Прихожий1, О. Н. Карасик2
'Белорусский национальный технический университет
2 Иностранное производственное унитарное предприятие «Иссофт Солюшенз»
УСОВЕРШЕНСТВОВАННЫЙ РАЗНОРОДНЫЙ БЛОЧНО-ПАРАЛЛЕЛЬНЫЙ АЛГОРИТМ ПОИСКА КРАТЧАЙШИХ ПУТЕЙ НА ГРАФЕ
Задача нахождения кратчайших путей между всеми парами вершин графа имеет множество областей применения в промышленности, технике, науке, экономике и обществе. Предложенные в литературе алгоритмы, решающие эту задачу на графах большого размера, направлены либо на снижение вычислительной сложности, либо на эффективное использование ресурсов вычислительной системы. В статье предлагается усовершенствованный разнородный блочно-параллель-ный алгоритм поиска кратчайших путей, который является результатом дальнейшего развития и улучшения известных блочных алгоритмов. Отталкиваясь от однородного блочного алгоритма, он различает четыре типа блока: диагональный, вертикальный перекрестный, горизонтальный перекрестный и периферийный. Для ускорения вычислений в статье разработаны отдельные алгоритмы для каждого типа блока, уменьшающие количество итераций во вложенных циклах и учитывающие строковое размещение данных в кэшах процессора. Алгоритмы улучшают пространственную и временную локальность при обработке больших данных. Эксперименты, проведенные на сервере, оснащенном двумя процессорами Intel Xeon E5-2620 v4, показали повышение производительности предложенных одно- и многопоточных усовершенствованных разнородных блочных алгоритмов до 60-70% по сравнению с одно- и многопоточными блочными алгоритмами Флойда - Уоршелла.
Ключевые слова: кратчайший путь, блочный алгоритм, разнородный алгоритм, многоядерная система, производительность.
Для цитирования: Прихожий А. А., Карасик О. Н. Усовершенствованный разнородный блочно-параллельный алгоритм поиска кратчайших путей на графе // Труды БГТУ. Сер. 3, Физико-математические науки и информатика. 2023. № 1 (266). С. 77-83. DOI: 10.52065/2520-6141-2023-266-1-13 (На англ.).
Introduction. The problem of searching for shortest paths in weighted graphs is widely exploited to find optimal solutions in industry, technology, science, economics, and society. The algorithms [1-15] of solving the problem either try to decrease the computational complexity, or to efficiently account for the features of computer architectures to speedup computations. In the paper, we consider all pairs shortest path problem on large graphs. The blocked algorithms [10-15] aim for efficient parallelization on multi-processor (multi-core) systems and utilization of CPU hierarchical memory. The contribution of the paper is the development of an advanced fast blocked heterogeneous algorithm for shortest paths search in two versions: single- and multiple-threaded parallel.
Main part. Let a graph G = (V, E) be constructed of a vertex set V of cardinality N and a set E of edges represented by matrix Wof weights. If wi,j Ф it is a weight of edge (i,. j), otherwise (i,. j) g E. In the blocked Floyd - Warshall (BFW) Algorithm 1, a matrix of path distances is decomposed into matrix B[M x M] of blocks with block-size S and block-count M=N / S. Figure 1 illustrates the process and order of calculating blocks of D0, C1, C2 and P3 types. The universal BCA Algorithm 2 is a classic Floyd - Warshall algorithm, in which bLj, b2i:k and b3k,j are elements ofblocks B1, B2 and B3 respectively. It recalculates each block element S times; therefore, has the computational complexity of 0(S3). The universalism of BCA is a source of slowing down the computations in BFW.
In the paper, instead of the single BCA, we introduce four other D0, C1, C2 and P3 algorithms, which have the same names as the block types have, and have the numbers of 1, 2, 2 and 3 arguments respectively. The objective of our work is to make the new algorithms D0, C1, C2 and P3 faster by accounting for features of the block types and the data dependences between blocks. Each block has the row-major layout in CPU hierarchical memory and support spatial locality during BFW operation. At M = 2, BFW calls the four algorithms in the order as follows:
D0(Bo,o), C1(B1,o, Bo,o), C2(Bo,1, Bo,o);
P3(BU, B1,o, Bo,1), D0(Bu), C1 (Bo,1, Bu);
C2(B1 ,o, Bu), P3(Bo,o, Bo,1, B1,o).
The following procedure lies in the basis of D0, C1, C2 and P3 construction:
Algorithm 1: Blocked Floyd-Warshall (BFW)_
Input: A number N of graph vertices Input: A matrix W[NxN] of graph edge weights Input: A size S of block Input: A number M ^ N/ S of blocks Output: A blocked matrix B[MxM] of path distances B ^ W
for m ^ 0 to M - 1 do
Bm,mm+1 ^ BCA(Bm,m, Bmm, Bmm) // D0 for v ^ 0 to M - 1 do if v ± m then Bv,mm+1 ^ BCA(Bv,m, Bv,m, Bm,m) // C1 Bm,vm+1 ^ BCA(Bm,v, Bm,m, Bm,v) // C2 for v ^ 0 to M - 1 do if v ^ m then for u ^ 0 to M - 1 do if u ^ m then Bv,u ^ BCA (Bv,u, Bv,m, Bm,u) // P3 return B
Algorithm 2: Block calculation BCA
Input: S is size of block Input: B1, B2, B3 are input blocks Output: B1 is recalculated block for k ^ 0 to S - 1 do for i ^ 0 to S - 1 do for j ^ 0 to S - 1 do sum ^ b2i,k + b3k,j if b1y > sum then b1i,J ^ sum;
return B1
12 3 4
РЗ Cl РЗ РЗ
С2 DO C2 С2
РЗ Cl РЗ РЗ
РЗ Cl РЗ РЗ
Fig. 1. Illustration of BFW operation: cross moves from top-left to bottom-right corner; C1 and C2 are executed after D0, and P3 after C1 and C2
The graph is extended by stepwise adding of vertices.
A sequence B1(1)...B1(k)...B1(S) of states of the Bl block is generated where k is a current number of rows and/or columns the block has.
The elements of row k and/or column k that are path distances from and to vertex k respectively are calculated over elements of matrix Bl(k - 1).
In their turn, the matrix Bl(k - 1) is recalculated into Bl(k) over the elements of row k and/or column k.
Algorithm DO. Since DO calculates block B1 over itself without involving other blocks, both row k and column k are added to Bl(k - 1). DO starts operation from B:(1) of size 1 x 1 and step-by-step increases the block size (Fig. 2). The above-described procedure calculates row k, column k, and B1(k) over B1(k - 1).
We have applied a set of formal transformations to the procedure and obtained Algorithm 3 (DO) of calculating the diagonal block. The transformations include 1) reordering of operations; 2) resynchro-nizing operation executions from neighbour iterations of loop to calculate b1y(k - 1) over b 1y(k - 2), b1i,k-\(k - 1), and b1k-1,j(k - 1), and to calculate b11}k(k) and b1kj(k) over b1y(k - 1), b1i,k(k - 1) and b1jk - 1) within the same iteration; 3) merging nested loops. DO calculates blocks b1y(k - 1), i, j = 0...k - 1 as:
b1i, j (k -1) = min {b\t J (k - 2), z}; z = 61,k _,(k -1) + b1k-1 j (k -1).
The element b1i:k(k) of column k is calculated using the equation as follows:
b1,k(k) = min (b1,j(k-1) + b1 jk(k-1)).
j=0...k-1v '
The element b 1k,j(k) of row k is calculated by the following equation:
bl
k,; (k) = г min 1 (, (k -1) + b1,; (k -1)).
In algorithm DO, two loops along i and j have the iteration scheme that produces k iterations, which is less than S iterations in BCA. The loops consequently process matrices of growing size [1 x 1], [2 x 2] ... [S x S], which indicate the DO has a temporal locality property. Comparison of DO and BCA shows that DO has smaller number of iterations in the nested loops and reduces the cache pressure. Reduction p in the number of executions of the most nested loop body is evaluated with
p(DO / BCA) = 6/(2+3/ S+1/ S2).
When S ^ ~ the reduction p(DO /BCA) ^ 3. The number of accesses to memory is also reduced.
Algorithm C1. Since it calculates a vertical block B1 of cross over itself and block B3, column k is added to Bl(k - 1) without adding row k (Fig. 3). C1 starts operation with B:(1) of size Sx1 and increases the block-size in column-wise manner.
Algorithm 3: Calculating diagonal block DO
Input: A diagonal block B1 Input: A size S of block Output: A recalculated diagonal block B1 for k ^ 2 to S do k1 ^ k - 1 for i ^ 1 to k1 do for j ^ 1 to k1 do 52 ^ b1i,k1 + b1k1j
50 ^ biy + b1j,k
51 ^ b1k,i + blj
k1 ^ S - 1
for i ^ 1 to k1 - 1 do for j ^ 1 to k1 - 1 do
52 ^ b1i,k1 + bik1j return B1
if b 1 ij > S2 then b 1 ij ^ S2 if b 1 i,k > so then b 1 i,k ^ so if b 1kj > S1 then b 1kj ^ S1
if bl; / > 52 then blij ^ 52
k - 1
k - 1
k
b1i
b1k
■H ГШ
b1ki - b1/ O
k
Fig. 2. Recurrent calculation of diagonal block DO
O
k
O
k
0 j k - 1k 0 j k
S
B1 (k - 2) B1 (k - 1) j k - 1
b b1i,k - 1 b1k
S
b3
k -
b3
B3
IHI
B1(k)
S
Fig. 3. Recurrent calculation of block C1 of cross
0
k
S
The above-described procedure calculates column k and Bl(k) over Bl(k - 1). We have applied to the procedure the transformations like those from the D0 case and obtained Algorithm 4 (C1) of calculating the vertical block of cross. Algorithm C1 calculates b 1ij(k - 1), i = 0...S - 1, j = 0...k - 2 using equation
b1i k (k) = min<
fbi,j (k - 2), lb!- ,k-,(k -1) + b3
j ,k-J
The equation as follows aims for calculating element b1i,k(k) of column k.
b1
, k (k) = . njiin1(b1, j (k -1) + b3 j, k).
j =0...k-1 N J J '
In algorithm C1, the loop along j has the iteration scheme that produces k iterations, which is less than S iterations in BCA. The loop consequently processes matrices of growing size [S * 1], [S * 2] ... [S * S], which indicate that C1 has a property of temporal reference locality. Comparison of C1 against BCA shows that C1 decreases the cache pressure. In C1, the overall number of iterations of the most nested loop is smaller than in BCA, and the reduction is evaluated by
p(C1/BCA) = 2/(1 + 5/S + 4/S2).
Algorithm 4: Calculating vertical block C1 of cross
Input: A block B1 Input: A block B3 Input: A size S of block Output: A recalculated block B1 for k ^ 1 to S - 1 do k1 ^ k - 1; for i ^ 0 to S - 1 do for j ^ 0 to k1 do S2 ^ b1i,k1 + b3k1j S0 ^ bhj + b3jk k1 ^ S - 1 for i ^ 0 to S - 1 do for j ^ 0 to k1 - 1 do
if b 1 ij > S2 then b 1 ij ^ S2 if b 1 i,k > S0 then b 1 i,k ^ S0
S2 ^ b1i,k1 + b3k1j return B1
if b 1ij > S2 then b1ij ^ S2
When S ^ the reduction p(C1 / BCA) ^ 2. The number of accesses to memory is also reduced.
Algorithm C2. Since it calculates a horizontal block B1 of cross over itself and block B2, only row k is added to Bl(k - 1) without adding column k (Fig. 4). C2 starts operation with B:(1) of size 1xS and increases the block size in row-wise manner.
The above-described procedure calculates row k and block Bl(k) over B1(k - 1). Transformations like those from the DO case have been applied to the procedure, yielding Algorithm 5 (C2) of calculating the horizontal block of cross. C2 calculates b1i,j(k - 1), i = 0...k - 2, j = 0...S - 1 using equation
fb1i j (k - 2), ] b1ll (k -1) = min \ J
J |b2i k-1 (k -1) + b1k-u (k - 1).J
Algorithm 5: Calculating horizontal block C2 of cross Input: A block B1 Input: A block B2 Input: A size S of block Output: A recalculated block B1 for k ^ 1 to S - 1 do k1 ^ k - 1; for i ^ 0 to k1 do for j ^ 0 to S - 1 do s2 ^ b2ik1 + b 1k1j if b > s2 then b ^ s2 so ^ b2k,i + b 1ij if b 1k,j > so then b 1k,j ^ so k1 ^ S - 1
for i ^ 0 to k1 - 1 do for j ^ 0 to S - 1 do
S2 ^ b2i,k1 + b 1k1j if b 1i,j > S2 then b 1i,j ^ S2 return B1
The following equation aims for calculating element b1ik(k) of column k.
b1k,j (k) = i mm (b2 k,. + M,j (k -1)).
In algorithm C2, the loop along i has the iteration scheme that produces k iterations, which is less than S iterations in BCA. The loop sequentially processes matrices of growing size [1 x S], [2 x S] ... [S x S], which indicate the C2 obtains a property of temporal reference locality that reduces the cache pressure. In C2, the reduction of the overall number of iterations of the most nested loop and of the overall number of accesses to memory is the same as in C1.
The nest of three loops across k, i and j of algorithms DO, C1 and C2, carries out computation of block B1. It calculates variables sO, s1 and s2, and matrix elements b 1ij, b1i,k and b1k,j upon elements of blocks B1, B2 and B3. It traverses the rows and columns k - 1, k and i multiple times. In contrast to the rows which support sequential reference locality in hierarchical memory, the columns are deployed in different lines and increase the data traffic in hierarchical cache memory. To improve the data locality, we have transformed the algorithms to collect the column elements in one-dimensional arrays and then access them intensively.
Algorithm P3. It calculates a peripheral block B1 over itself and the B2 and B3 blocks. Since the graph extension-based procedure cannot be used effectively
in this case, we derive P3 from transforming the classical Floyd-Warshall Algorithm 2. To provide the sequential reference locality for all blocks B1, B2 and B3, we have reordered the three k - i - j nested loops to the nest i - k - j of loops.
Results. We implemented the homogeneous blocked Floyd-Warshall algorithm (BFW) and the advanced heterogeneous blocked algorithm (HBA) in C++ language using GNU GCC compiler v10.2.0. Each algorithm has two implementation versions: single-thread and multiple-threaded. The algorithms were parallelized at task level and compiled to multiple-threaded applications by means of OpenMP 4.5. We carried out the vectorisation of operations in all program codes to speed up the computations. The experiments were done on a rack server equipped with two Intel Xeon E5-2620 v4 processors. Each processor contains 8 cores and 16 hardware threads. Every core is equipped with private L1 (32 KB) and L2 (256 KB) caches, and the cores within each processor share inclusive L3 (20 MB) cache.
To measure algorithm parameters on the multi-core system architecture, we used the Intel VTune Profiler 2021.8. The experiments were conducted on randomly generated complete weighted graphs. The paper describes results for graphs of 4800 vertices. Each experiment was carried out multiple times and verified.
Table 1 reports the execution time of the single-thread sequential BFW and HBA. BFW consumed from 35.4 to 54.0 sec of CPU, meanwhile HBA consumed from 29.6 to 35.2 sec. The speedup of HBA over BFW is from 6.1% to 59.6%.
Table 1
Execution time and comparison of single-thread HBA and BFW on graph 4800
Block- Block BFW, HBA, Speedup,
matrix count sec sec %
2x2 4 53.99 33.82 59.6
3x3 9 35.40 29.61 19.6
4x4 16 35.83 31.90 12.3
5x5 25 36.15 32.41 11.5
6x6 36 36.03 33.13 8.8
8x8 64 37.34 35.19 6.1
Table 2 reports the execution time of multiple-threaded BFW and HBA parallelized by OpenMP. BFW consumed from 11.2 to 57.5 sec of CPU, meanwhile HBA consumed from 10.9 to 43.6 sec. The speedup of HBA over BFW is from 0.4% to 71.2%.
We explain the reduction of speedup at increasing M by the rapid growth of the number of peripheral blocks and by the insignificant speedup of P3 over BCA, and by the fact that P3 loses DO, C1 and C2 significantly.
Table 2
Execution time and comparison of multiple-threaded HBA and BFW on graph 4800
Block- Block BFW, HBA, Speedup,
matrix count sec sec %
2x2 4 56.01 32.73 71.2
3x3 9 57.50 38.27 50.2
4x4 16 50.37 35.64 41.3
5x5 25 56.19 43.64 28.8
6x6 36 31.89 31.76 0.4
8x8 64 11.21 10.94 2.4
Indeed, the fraction of P3 blocks in the total block count is evaluated as (M - 1)2 / M2 and is 0.25, 0.44, 0.56, 0.64, 0.69, 0.77 and 0.94 for
M = 2, 3, 4, 5, 6, 8 and 32 respectively. Therefore, HBA is more effective for non-large values of M.
Conclusion. The proposed advanced heterogeneous block-parallel all pairs shortest path algorithm no matter single- or multiple-threaded considers the features of four types of blocks and overcomes the homogeneous blocked Floyd-Warshall algorithm by up to 60-70% regarding the reduction of execution time on multi-core systems. The D0 algorithm of calculating the diagonal block is a promising alternative to the classic Floyd-Warshall algorithm with respect to the throughput since it has the property of spatial and temporal references locality and can handle efficiently the processor's hierarchical memory.
References
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Information about the authors
Prihozhy Anatoly Alekseevich - DSc (Engineering), Professor, Professor, Department of Computer and System Software. Belarusian National Technical University (65, Nezavisimosti ave., 220013, Minsk, Republic of Belarus). E-mail: prihozhy@yahoo.com
Karasik Oleg Nikolayevich - PhD (Engineering), Lead Engineer, ISsoft Solutions (5, Chapaeva str., 220034, Minsk, Republic of Belarus). E-mail: karasik.oleg.nikolaevich@gmail.com
Информация об авторах
Прихожий Анатолий Алексеевич - доктор технических наук, профессор, профессор кафедры программного обеспечения информационных систем и технологий. Белорусский национальный технический университет (220013, г. Минск, пр. Независимости 65, Республика Беларусь). E-mail: prihozhy@yahoo.com
Карасик Олег Николаевич - кандидат технических наук, ведущий инженер. Иностранное производственное унитарное предприятие «Иссофт Солюшенз». (220034, г. Минск, ул. Чапаева, 5, Республика Беларусь). E-mail: karasik.oleg.nikolaevich@gmail.com
Поступила после доработки 15.11.2022 Трулы БГТУ Серия 3 № 1 2023
