shortest path algorithms using dynamic breadth-first search

Shortest Path Algorithms Using Dynamic Breadth-First Search* Donald Goldfarb Department of Industrial Engineering and Operations Research, Columbia University, New York, New York, 10027

Jianxiu Hao and Sheng-Roan Kai GTE Laboratories Incorporated, 40 Sylvan Road, Waltham, Massachusetts 02254

A new O(nm) label-correcting algorithm is presented for finding shortest paths from a given node to all other nodes in a network of n nodes and m arcs or finding a directed cycle of negative length. In this algorithm, a node is scanned on the k-th scanning step only if its “label depth”-i.e., the length of the path corresponding to the distance label - equals k . Variants of this algorithm are discussed. and computational results show that several of these are very efficient. A new criterion for detecting a negative cycle that is also based on the label depth of a node is given. and Computational tests show that it is extremely effective.

1. INTRODUCTION

We consider the problem of finding directed shortest paths from a given node r to all other nodes in a network, or finding a directed cycle of negative length. The shortest path (SP) problem described above is a fundamental and important optimization problem as many combinatorial optimization and network flow problems can be either formulated directly as SP problems or require the solution of SP subproblems (e.g., see [6,10,12]).

During the last 3 decades, numerous algorithms have been proposed for solving the SP problem. For general surveys, see [2,4, 131. Many of these methods are variants of the well-known Bellman-Ford-Moore (BFM) label- correcting algorithm [1,5,11].

In this paper, we present yet another variant of the BFM algorithm. Our algorithm has a worst-case complexity of O(nrn) and, hence, from a complexity viewpoint, it is neither better nor worse than the best existing methods. How- ever, it is designed to scan as few arcs as possible while retaining an @inn)

“This research was supported in part by NSF grants DMS-8s- 12277 and CDR-84-21402 and ONR contract N0014-87-K0214.

NETWORKS, Vol. 21 (1991) 29-50 @ 1991 John Wiley & Sons, Inc. CCC 0028-3045/91/01029-022$04.00

30 GOLDFARB, HAO, AND KAI

time bound. Consequently, it is efficient in practice as well as in theory, as is borne out by our computational tests.

The key idea underlying our algorithnm is the use of the “label depth” of a node, which is defined as the number of arcs in the path used to determine the current “distance” label of that node, to decide if a node should be scanned. Like breadth-first search, our algorithm performs a sequence of steps in which, on step k , only those nodes with depth (label depth in our algorithm) equal to k are scanned. Because the label depth of a node can change in our shortest path algorithm, we call it the Dynamic Breadth-First Search (DBFS) algorithm. We also use the label depths of nodes to provide a very effective criterion for detecting a negative cycle if one exists in the network.

In the next section, we introduce some preliminaries and review the BFM algorithm. The DBFS shortest path algorithm is presented and analyzed in Section 3. In Section 4, we discuss several variants of the DBFS algorithm that are designed to further reduce the number of nodes scanned, including one that is based on the simplex method. In Section 5 , we present the results of computational tests comparing the DBFS algorithm, variants of it, and other variants of the BFM algorithm on SP problems without negative cycles. These results estabalish the efficacy of our approach. Finally, in Sections 6 and 7, we present a new criterion for detecting negative cycles and some computational results that demonstrate that it is extremely effective.

2. PRELIMINARIES AND LABEL-CORRECTING ALGORITHMS

Let G = (N,A) denote a finite directed graph having node-set N of cardinality n and arc-set A of cardinality m. For each arc ( u , v ) E A , let cu,, be its length, which may be positive, zero, or negative. Given a fixed node r E N , called the root, the shortest path (SP) problem is to find directed paths from r to all other nodes in G of shortest length or to find a directed cycle of negative length (negative cycle). For the moment we shall assume that there is no negative cycle in G. The case where there are negative cycles is deferred until Section 6.

For each node u E N , we shall define its forward star or adjacency Zist as F(u) = { v E NI ( u , v ) E A } and use y*(u) to denote the length of a shortest path from r to u. From the principle of optimality of dynamic programming, these shortest path lengths satisfy the system of equations:

y ( v ) = minb(u) + cu,,l u # v , ( u , v ) E A } for all v E N\{r},

where y ( r ) = 0. The BFM algorithm solves this system by initially setting

SP ALGORITHMS USING BREADTH-FIRST SEARCH 31

and then successively computing better and better approximations yl(v) 2

yz(v) 2 y3(v) 2 . . . to y*(v) from

It is easily shown that this process terminates with yk(v) = y*(v), for all v E N, for k i n - 1 if and only if there are no negative cycles.

In the BFM algorithm, y k ( v ) is called the label of node v. Observe that during the k-th label-correcting step old labels yk(u), u # v are used in (2) to compute an improved label yk+,(v) for node v even if improved labels Yk+,(u) have already been computed for some nodes u.

As in successive approximation methods for solving systems of linear equations, it is preferable to use improved labels if they are available during a label- correcting pass (the Gauss-Seidel method) instead of always using old labels (the Jacobi method). This is the idea behind several variants of the BFM method, including one due to Yen [15] that processes an ordered list of nodes alternatively forwards and backwards. During a forward (backward) scan, only the inward arcs of the node being scanned whose tails come before (after) it in the node list, and, hence, which have improved labels, are scanned. Conse- quently, each scan processes, on the average, half of the arcs in the network. Another variant, the Partitioning Shortest Path (PSP) algorithm due to Glover et al. [7] scans only the outward arcs of nodes whose distance labels were improved during the last step and uses improved labels as soon as they become available.

3. THE DYNAMIC BREADTH-FIRST SEARCH ALGORITHM

For each node v E N , let the smallest number of arcs in a shortest path from r to v be denoted by l(v) and let L ( k ) = {v E Nll(v) = k}. We shall say that a label-correcting algorithm satisfies Property D (D stands for “depth”) i f , for all k, on the k-th label-correcting step (or scan), all nodes in L(k) - i.e.. the forward stars of such nodes - are scanned. Clearly, any label-correcting algorithm that starts with the labels specified in (1) and satisfies Property D will terminate after at most n - 2 steps. Also, let the label depth of a node v , which we denote by d(v), be defined as the number of arcs in the path used to determine a label y(v).

The BFM algorithm satisfies Property D, but many variants of it do not. For example, the PSP algorithm in its most basic form that uses current labels and maintains a queue of scan-eligible nodes - i.e., those nodes whose labels have improved - does not. By thinking of the queue of scan-eligible nodes as two queues in tandem in the PSP algorithm, the end of a step is determined by the emptying of the first queue, at which point the two queues are interchanged. This algorithm does not satisfy Property D because it allows d(v) to become greater than k + 1 prior to the k-th step and, hence, allows nodes to prematurely become permanently labeled. In particular, a node v with shortest path depth


l(v) = k + 1 can become permanently labeled prior to step k , and, hence, such a node will not be scanned during step k .

Since such premature permanent labelings are fortuitous, it makes sense to scan on step k only those nodes with label depth k . This is the basic idea behind the DBFS algorithm given below. In it, p ( v ) denotes the predecessor of node v on the path used to determine the current node label y ( v ) . The set C1 contains nodes that are candidates for scanning on the current step, whereas C2 contains nodes that are candidates for scanning on the next step.

For the moment, we assume that the network has no negative cycles. In Section 6, we show how to modify the algorithm to detect negative cycles.

Algorithm DBFS

(0) Initialization : Set y ( v ) = c,,,, p ( v ) = r, and d ( v ) = 1, f o r all v E F ( r ) , and y ( v ) = a, p ( v ) = 0, and d ( v ) = 0, for all v 4 F(r) . Set y(r) = 0, C1 t F ( r ) , C2 t0, and k = 1.

If C1 = 0 , go to ( 3 ) ; otherwise select any node u E C1 and set C1 tC l \{u} . If d (u ) = k , go to (2) ; otherwise repeat this step.

For all v E F(u) , i f y , + cu,, < y,, set y , = y , + c,,, and p ( v ) = u, and if d ( v ) # k + 1, set C2 t C 2 U {v} and d ( v ) = k + 1. Return to (1).

If C2 = 0, stop; the SP problem is solved [ y ( u ) is the length of a shortest path f rom r to u and the tree of shortest paths can be recovered from p ( . ) ] . Otherwise, set C1 c C 2 , C2 t 0 and k t k + 1. Go to (1).

(1) Select a Node to Scan:

(2) Scan Forward Star of Selected Node:

( 3 ) Set Up Candidate Set f o r Next Stage:

Before proving that the DBFS algorithm satisfies Property D and, hence, that it has a worst-case complexity of O(mn), we note that d ( v ) in the algorithm is, in fact, the label depth of node v, if y ( v ) # a, since it is clear that d ( v ) is set equal to d ( u ) + 1 whenever y ( v ) is set equal to y ( u ) + cu,,. Moreover, at the start of step k , C1 consists entirely of those nodes u with d ( u ) = k , since on the previous step these nodes and only these nodes are put into C2. Since during step k a node u in C1 is scanned only if d ( u ) remains equal to k , the algorithm does what it was designed to do.

Lemma 1. At the start of step k , L (k ) C C1 and y ( v ) = y*(v) for all v E L(k) . Moreover, during step k , all nodes v in L(k) are scanned-i.e., Algorithm DBFS satisfies Property D.

Proof. It is clear from the initialization step that the lemma holds for k = 1. Now assume that it is true for k = k, and suppose that l ( v ) = I? + 1. Then, there is a shortest path from r to v that has (u,v) as its last arc and l(u) = k. By the induction hypothesis, at the start of iteration k u E C1, and u is scanned during


iteration k . Since there may be more than one shortest path to v of length k + 1. let u be the first node on such a path that is scanned by the algorithm. Before ii is scanned y(v) > y*(v); hence, y(v) will be set equal to y*(v) when u is scanned and v added to C2. Since C1 is set equal to C2 at the end of the iteration. the first part of the induction is proved. Since y(v) = y*(v), d(v) cannot increase during iteration k + 1; hence, d(v) = k + 1 during iteration k + 1 and node v will be scanned during that iteration. This completes the second part of the induction.

It now immediately follows from our remarks at the beginning of Section 3 and from the fact that each iteration of Algorithm DBFS requires at most O(m) operations that:

Theorem 1. Algorithm DBFS solves the SP problem in at most O(nm) time.

4. VARIANTS OF THE DBFS ALGORITHM

At any time during the execution of Algorithm DBFS, there is a unique tree T that is determined by the labeling process up to that point. Specifically, T is defined by the list of predecessor pointers p ( u ) . If we let yT(v) denote the length of the path in T from r to v, then the label y(v) is said to sharp if y(v) = v T ( v ) . In Algorithm DBFS, the labels y(v) are not necessarily sharp and the label depth d(v) is not necessarily equal to the depth of node v in T. If y(v) is not sharp, y T ( v ) < y(v); hence, the label of any node, say y ( w ) , that is reduced by scanning v, cannot become equal to y*(w), and clearly the effort expended in scanning v is wasted.

The number of nodes, and, hence, arcs, scanned by Algorithm DBFS can be reduced if only nodes with sharp labels are scanned. An efficient way to implement this strategy is through the simplex method. In the simplex method for the SP problem, T is the basis tree, the label y(v) = yT(v) of node v is the dual variable corresponding to that node, and d(v) is the depth in T of node

In step (2) of a DBFS simplex variant, if y(u) + cU,” < y(v) [i.e., the reduced cost C,,,, = c,,,,, + y(u) - y(v), of arc (u,v) is negative], then the labels of all nodes in the subtree T, that “hangs” from node v are decreased by lCu,v/, not just y(v). Consequently, the number of pivots (i.e., multinode label updates) required by the simplex variant is, in general, less than the number of single- node label updates required by the standard label-correcting version of Algo- rithm DBFS. For implementational purposes, usually a preorder thread and a reverse preorder thread are maintained in addition to the predecessor pointer p(v) and the depth d(v). This facilitates updating the basis tree T and the labels y(w) and depths d ( w ) of all nodes w in the subtree T,. Aside from these updating aspects, our DBFS simplex variant is identical to Algorithm DBFS, except that (i) we use a marker m(v) rather than d(v) to indicate whether or not v has been put into C2, since in stage k , d(v) = k + 1 does not guarantee that in v is in C2; (ii) we set m(v) = d(v) for all v E N , in step (0); and (iii) we replace step ( 2 ) by

V.


( 2 ’ ) For all v E F ( u ) , i fy (u) + cu,, I y ( v ) and m ( v ) # k + 1 , set C2 t C 2 U {v} and m ( v ) = k + 1 ; and i f y ( u ) + cU,” < y ( v ) , do a simplex pivot with (u ,v ) as the entering arc and the current basic arc whose head is v as the leaving arc, and update T and y ( w ) and d ( w ) for all w E T,. Return to ( 1 ) .

The major drawback of the simplex variant of the DBFS algorithm is that each pivot is expensive. Not only are the labels and depths of all nodes in the subtree T, updated, but the list structures, which are used to represent the basis tree T , need to be updated. In an attempt to avoid scanning the forward stars of nodes whose labels are not sharp, we also implemented a variant of Algorithm DBFS, in which the depth criterion “d(u) = k” for scanning a node after removal from the candidate set C1 was replaced by “ d ( p ( u ) ) = k - 1.” In the next section, this variant is referred to as Algorithm DBFS1. We also tried using the criteria “ d ( p ( p ( u ) ) ) = k - 2” (Algorithm DBFS2), “ d ( p ( p ( p ( u ) ) ) ) = k - 3” (Algorithm DBFS3), and “ d ( p ( p ( p ( p ( u ) ) ) ) ) = k - 4” (Algorithm DBFS4). Note that in order to efficiently implement Algorithms DBFS2, DBFS3, and DBFS4 we defined p ( r ) = r l and d(r1) = - 1 , p ( r 1 ) = r 2 and d(r2) = -2, and p ( r 2 ) = r3 and d(r3) = -3.

The likelihood that a node is sharp increases as the number predecessors used in the depth test is increased, but so does the work involved. As our computational results in the next section show, the greater the depth of the optimal shortest path tree, the more worthwhile it is to check further along the predecessor path of a node. We note that Desrochers [3] suggested following the predecessor path for a node all the way back to r in order to determine whether or not the label of a node is sharp. Clearly, the DBFS-simplex variant achieves the same effect with more efficiency unless the optimal shortest path tree is shallow. We also note that the DBFS simplex variant is essentially a specialization to the shortest path problem of the network simplex algorithm using the “restricted most negative” rule proposed by Roohy-Laleh [ 141 to prevent exponentially long sequences of degenerate pivots [S].

5. COMPUTATIONAL COMPARISONS

We computationally compared the performance of the DBFS algorithm and its variants described in the previous section with the BFM and PSP algorithms. In this section and those that follow, when we refer to the BFM algorithm we shall mean that version of the standard method that starts with the labels specified in ( 1 ) and that scans the forward stars of all nodes sequentially, always using current and, hence, possibly improved labels. All of the tests reported in this section were on shortest path problems without negative cycles. These problems were randomly generated as follows.

The number of nodes, n , the approximate maximum depth, d , of a node in the optimal tree T * of shortest paths, and the approximate average in-degree, 6, of a node, and average cost, C, of an arc in G were specified. Let t! = [ (n - l ) / d ] and ran[l ,q] represent a random integer uniformly distributed between 1 and q. Specifying node 1 as the root, we first constructed an optimal


tree T * with approximate maximum depth d by assigning node 1 to layer zero and then successively choosing ran[ 1,2t] nodes randomly from those nodes in N\{l} that had not yet been selected, putting them in the next layer, and connecting each of them by a backward arc to a random node in the current layer. Then, for each node, an additional ran[1,26] arcs into it were generated with unique but randonm tails. Finally, each arc in T* was assigned a cost of zero and each arc not in T* was assigned a positive cost ran[l,2c].

By appropriately choosing node potentials ~ ( u ) for all nodes u E N , and replacing each arc cost cU.&. by ct4.,, + ~ ( u ) - ~ ( v ) , we can make some of the arc costs negative. Since all the algorithms that we tested are totally unaffected by the choice of ~ ( u ) , our test results also apply to problems with negative arc costs, even though the arc costs actually generated were all nonnegative.

For our computational tests we generated 200 random problems, comprising 10 replicates of 20 different problem types. The input specifications for each problem type and the average number of arcs, m , in the network, and the average maximum depth of T *, 2, are presented in Table I. The input parameter value of 25 for d has an asterisk next to it for the first four problem types, because in those problems, instead of assigning a cost of zero to each arc in T * , a cost of ran[l,2E] was assigned. Consequently, in these cases. T " was not an optimal tree of shortest paths, and the problems generated were similar to ones that would have resulted had we generated a random network with nonnegative costs, i.e., one that did not try to specify the maximum depth of an optimal tree. In such random networks, the longest length of a shortest path is very likely to be quite short, as the values of d for problem types 1-4 in Table I show.

In Table 11, we present the average time it took the DBFS algorithm, the variants of it described in Section 4, the BFM algorithm, and the PSP algorithm to solve the 10 problems of each problem type. The times given are CPU seconds for the SUN-3 and do not include input and output times. All algorithms were coded in FORTRAN and were compiled using the F 77-SUN FORTRAN Compiler (without optimization).

The ratios of the average times reported in Table I1 with respect to the average times required by the PSP algorithm are presented in Table 111. The results reported in Tables I1 and 111 show that there is very little difference between the performance of the DBFS and PSP algorithms on the problems tested. However, as the density of the network, or equivalently, the average degree of a node, increases, the performance of the DBFS algorithm relative to that of the PSP algorithm improves. The results for all the DBFS variants also indicate that as the density of the network increases it becomes more worthwhile to do extra work to try to avoid scanning nodes whose labels are not sharp and, hence, whose true depth at iteration k is not equal to k . As these tables illustrate, the relative improvement increases as more distant predecessors are checked and is greatest for the simplex variant, DBFSS, in which all labels are sharp.

As the total times in these tables show, Algorithm, DBFS4 was, on the average, about 20% faster than was the PSP algorithm. It is interesting that


TABLE I. Problem attributes.

Problem n d d E A d

1 1000 2 1000 3 1000 4 1000

5 lo00 6 1000 7 1000 8 1000

9 1000 10 1000 11 1000 12 1000 13 1000 14 1000 15 1000 16 1000

17 2500 18 2500 19 5000 20 5000

9 14 19 24

9 14 19 24

9 14 19 24

9 14 19 24

5 11 6

11

25 * 20,000 25 * 20,000 25 * 20,000 25 * 20,000 25 20,000 25 20,000 25 20,000 25 20,000

50 20,000 50 20,000 50 20,000 50 20,000

100 20,000 100 20,000 100 20,000 100 20,000

25 20,000 100 20,000 25 20,000

100 20,000

9795 11.1 14,384 10.4 20,130 11.0 25,335 11.5

9719 24.3 14,408 23.2 20,025 24.0 25,408 23.9

9764 47.2 14,451 49 .O 20,309 48.2 25,296 44.2

9754 95.2 14,445 95.2 19,946 91.7 25,408 94.1

14,244 24.8 29,942 98.2 28,423 24.4 59,912 96.5

our simplex variant, DBFSS, was not much slower than were any of the other algorithms tested; in fact, it was only about 10% slower than was the BFM algorithm on the average. As we shall see in the next section, Algorithm DBFSS is particularly effective when there are negative cycles.

For each set of 10 runs whose average CPU time is reported in Table 11, we present in Tables IV-VI the average number of nodes whose forward stars are scanned, the average number of nodes that become candidates for scanning [i.e., which are put on the candidate queue (Cl/C2)], and the average number of labels that are updated, respectively.

We gathered these statistics by repeating all of the timing runs reported in Table 11, after adding the appropriate code for each algorithm for this purpose. It is easily verified from Tables I1 and IV that the CPU times for Algorithm PSP and all of the DBFS algorithms other than the simplex variant are roughly proportional to the numbers of arcs scanned - i.e., to the numbers of nodes scanned multiplied by the average out-degree of a node.

The ratio of the total average number of nodes scanned to the total average number of nodes that are candidates (see Tables IV and V) is an interesting measure of how effective an algorithm is at only scanning nodes that “need” to be scanned - the smaller the ratio, the more effective the algorithm. This ratio is 1.0 for the PSP algorithm, since every node that is added to Cl/C2 is scanned. For the DBFS algorithm and variants of it, this ratio decreases as more effort is expended to avoid scanning nodes whose labels are not sharp.

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17

1.23

9 1 .O

OO

1.04

7 0.

942

0.91

2 0.

900

0.90

6 1.

685

18

1.26

3 1.

000

1.01

5 0.

894

0.85

8 0.

837

0.83

0 1.

212

19

1.24

8 1 .O

OO

1.03

7 0.

931

0.90

8 0.

898

0.90

3 1.

682

20

1.25

5 1.O

OO

1.00

8 0.

904

0.85

6 0.

836

0.82

2 1.

203

Tot

al

1.16

6 1.O

OO

1.00

3 0.

881

0.84

2 0.

823

0.81

4 1.

266

1.31

9 E

TA

BL

E I

V.

Prob

lem

B

FM

PSP

DB

FS

DB

FSl

DB

FS2

DB

FS3

DB

FS4

DB

FSS

1 23

60

2003

19

68

1727

16

85

1679

16

79

1686

2

2582

21

93

2119

18

29

1785

17

78

1778

18

06

3 27

00

2249

21

88

1883

18

53

1848

18

47

1900

4

2699

22

91

2189

18

92

1846

18

40

1840

19

12

Ave

rage

num

ber

of n

odes

sca

nned

.

5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

Tot

al

4407

41

55

4412

43

85

5804

64

09

5896

58

46

7724

77

97

7417

76

83

9960

19

,394

19

,760

37

,537

168,

927

4239

37

78

4111

40

46

5420

59

52

5829

57

29

7547

76

79

7074

76

05

9120

18

,026

18

,177

35

,346

158,

414

4157

37

25

3950

39

24

5547

59

21

5593

55

69

7383

75

10

6883

74

32

9202

17

,907

18

,183

34

,972

156,

322

3440

31

69

3365

33

32

4468

49

15

4676

46

02

6333

63

99

5959

62

50

7814

15

,150

15

,407

30

,245

132,

855

3129

29

89

3 105

31

28

4092

45

11

4373

42

20

5894

59

09

5579

58

09

7185

14

,191

14

,271

27

,857

123,

411

2966

28

92

2999

29

74

3919

42

41

4233

41

01

5653

56

35

5341

54

47

6872

13

,582

13

,806

26

,600

118,

406

2914

28

33

2960

29

20

3825

39

60

4065

39

25

5437

54

19

5177

53

20

6746

13

,217

13

,465

25

,770

11 5,

097

2771

26

96

2821

28

29

3412

35

60

3624

35

82

4397

44

92

4462

44

62

5678

10

,450

9,

660

18,2

95

94,4

95

TA

BL

E V

. A

vera

ge n

umbe

r of

can

dida

te n

odes

. P

P 0

Prob

lem

B

FM

PSP

DB

FS

DB

FSl

DB

FS2

DB

FS3

DB

FS4

DB

FSS

1

7992

20

03

2449

23

86

2376

23

75

2374

23

45

m

2 77

92

2193

27

71

2697

26

91

2688

26

88

2781

3

8091

22

49

2960

28

85

2878

28

77

2877

30

75

4 82

9 1

2291

30

53

2986

29

73

2973

29

73

3261

1

D

5 15

,584

42

39

4718

43

00

409 1

39

61

3927

37

15

-0

D

6 14

,985

37

78

4377

41

26

4027

39

65

3924

38

06

15,6

84

4111

47

50

4497

43

14

4240

42

16

4170

z

7 8 15

,184

40

46

4795

45

32

4419

43

02

4256

43

71

0

9 26

,273

54

20

6197

54

59

5195

50

68

4996

44

33

10

28,0

71

5952

67

08

6107

58

30

5624

53

74

4814

11

26

,573

58

29

6514

59

88

5788

56

91

5556

52

49

12

24,7

75

5729

65

98

604 1

57

70

5591

55

47

5344

13

49,9

50

7547

81

57

7563

72

59

7088

68

92

5633

14

48

,251

76

79

8426

77

80

7424

72

13

7038

59

94

15

48,6

51

7074

79

37

7467

72

16

704 1

69

10

6191

16

49

,150

76

05

8548

78

54

7561

72

64

7169

64

20

17

41,4

83

9120

10

,244

94

87

9048

88

22

8732

80

15

18

130,

197

18,0

26

20,0

28

18,4

44

17,8

24

17,3

82

17,1

08

14,4

12

19

85,4

82

18,1

77

20,3

15

18,7

93

18,0

48

17,7

30

17,4

71

16,2

50

20

260,

947

35,3

46

39,2

55

36,7

61

35,0

51

34,1

20

33,4

49

28,3

65

Tot

al

913,

406

158,

414

178,

800

166,

153

159,

783

156,

115

153.

477

138.

644

E F

TA

BL

E V

I.

Ave

rage

num

ber

of l

abel

upd

ates

.

Prob

lem

B

FM

PSP

1 39

27

3175

2

4840

37

85

3 54

86

4240

4

5717

45

72

5 63

32

5739

6

6586

56

51

7 72

98

6366

8

7749

65

98

9 79

15

7093

10

91

03

8113

11

92

31

8384

12

95

40

8586

13

10

,141

95

03

14

10,8

77

10,1

25

15

11,1

85

9863

16

11

,523

10

.714

17

12

,961

1 1

,544

18

26

,432

23

,622

19

25

,737

23

,077

20

51

,781

46

,634

T

otal

24

4,36

1 21

7,38

4

DB

FS

3162

36

95

4193

44

69

5665

55

80

6191

64

29

7223

80

76

8105

83

85

9367

99

19

9647

10

,470

11

,642

23

,529

23

,125

46

,310

21

5,16

4

DB

FSl

DB

FS2

DB

FS3

DB

FS4

DB

FSS

3054

35

51

4036

43

26

5153

52

31

5827

60

58

6365

73

34

7428

76

76

8613

91

21

8992

95

98

10,6

91

21,5

65

21,2

09

42,9

96

198,

824

3038

35

41

4026

43

07

4927

51

16

5626

59

29

6075

70

27

7206

73

75

8260

87

22

8705

92

65

10,2

00

20,8

49

20,3

64

41,1

39

191,

697

3036

35

37

4025

43

06

4787

50

51

5547

58

05

5939

68

12

7 102

72

87

8076

85

03

8520

89

57

9961

20

,363

20

,022

40

,148

18

7,78

4

3036

35

37

4025

43

06

4752

50

08

5520

57

56

5866

65

56

6964

71

40

7875

83

24

8383

88

57

9868

20

,076

19

,751

39

,455

18

5 ,05

5

3244

38

44

4377

46

83

5333

56

25

6161

65

00

6490

74

90

7713

80

75

8590

92

39

9356

97

68

10,9

00

22,4

94

21,7

51

43,0

33

204,

756

-


Specifically, this ratio is 0.891 for DBFS, 0.800 for DBFS1, 0.772 for DBFS2, 0.758 for DBFS3, 0.750 for DBFS4, and 0.682 for DBFSS. In the BFM algorithm, all nodes are considered in sequence; every node is a candidate when it is considered. The BFM algorithm has a low ratio of scanned to candidate nodes, not because it scans few nodes (it scans more nodes than any of the other algorithms), but rather because it considers so many nodes as candidates.

The average numbers of label updates required by the algorithms are reported in Table VI. These results are not very surprising except for those for Algorithm DBFSS. For this algorithm the number of updates computed for a run was the sum of the numbers of nodes in the subtrees T,, scanned over all pivots, i.e., every time a label (dual variable) changed it was counted. When compared with the most efficient algorithm, Algorithm DBFS4, Algorithm DBFSS is seen not to update many labels needlessly. In fact, in our tests, Algorithm DBFSS always updates fewer labels than does Algorithm BFM, and except on the first four sets of problems (which have low maximal depth), Algorithm DBFSS updates fewer labels than do both the generic PSP and DBFS algorithms.

6. NEGATIVE CYCLES

It is well known that if during the k-th stage of the BFM algorithm the node labels of at least n - k distinct nodes decrease, then the network contains a negative cycle (e.g., see [lo]; Theorem 11.2). This criterion also applies to the DBFS algorithm and its variants and to other polynomial-time variants of the BFM algorithm such as the PSP algorithm.

By using the label depth of a node, a much weaker condition can be given to test for a negative cycle in the DBFS, PSP, and BFM algorithms, enabling these algorithms to terminate much earlier when such cycles are present. If we let n( j ) equal the number of nodes with label depth d(v) = j , this condition is embodied in the following theorem.

Theorem 2. If during stage k of Algorithms DBFS (or any of its variants), PSP, or BFM, n( j ) = 0 for some j = 1, . . . ,k - 1, then the network contains a negative cycle.

Proof. This theorem is a consequence of the fact that for all of these algorithms y(v) = y*(v) for all v E L(k) at the start of stage k and, thereafter, if

rn there are no negative cycles in the network.

Clearly, the condition of this theorem will be satisfied if the labels of at least II - k distinct nodes decrease during stage k . Because the label depth of a node can only increase in the DBFS algorithm and if it does so during stage k it becomes equal to k + 1, implementing this condition within Algorithm DBFS is particularly simple. Specifically, we need to initialize n(1) = IF(r)l , n(0) = n - n(l), and n( j ) = 0, j = 1, . . . ,n in step (0) of Algorithm DBFS and replace the “if d(v) # k + 1” clause of step (2) by


Z f d ( v ) f k + 1, set n ( d ( v ) ) c n ( d ( v ) ) - 1.

If n ( d ( v ) ) = 0, stop; the network contains a negative cycle. Otherwise, set C2 t C 2 U { v } , d ( v ) = k + 1, and n ( k + 1) = n(k + 1) + 1.

It is obvious that Algorithm DBFS, modified in the above manner to detect negative cycles, satisfies Theorem 1 whether or not such cycles are present in the network. If the algorithm terminates because of the presence of a negative cycle, one such cycle can be identified by tracing a path from node v [corresponding to n(d(v)) = 01 using the predecessor pointer, marking each node visited, until a node is reached that was previously visited and consequently marked.

In the simplex variant of the DBFS algorithm, as in all shortest path simplex algorithms, a negative cycle is easily detected since y (u ) will decrease if and only if the fundamental cycle determined by the entering arc (u,v) and the basis tree T is a negative cycle. Consequently, shortest path simplex algorithms are particularly effective in detecting such cycles.

To implement the condition of Theorem 2 within the PSP algorithm and the BFM algorithm using current (i.e., improved) labels, we need to maintain and update the label depths of all nodes as is done in Algorithm DBFS. When the label y ( v ) and label depth d(v) of a node is changed during stage k , it is necessary to check whether d(v) (the old label depth) is less than k , as well as checking if n ( d ( v ) ) = 0. It is also necessary to check if n ( k ) = 0 at the start of stage k .

During stage k of the BFM and PSP algorithms, it is possible for n(j) to equal zero for j > k while there are nodes v with d(v) > j , even if the network has no negative cycles. This can occur because d(v) is not the “actual” depth of node v in the tree determined by the predecessor array p ( .). Therefore, it can appear form the label depth that there is a break, and, a cycle, in this tree when there is none.

To illustrate this, consider the network corresponding to the directed graph G = (N,A), where N = {1,2,3,4,5,6,7}, A = A l U A4, A l = {(1,2),(1,5),(2,3), (3,4),(4,7),(5,4),(7,6)}, and A4 = {(1,3),(1,4),(3,6)}, and cu,,. = 1 for all arcs ( u , v ) E A l and c,,,, = 4 for all arcs (u ,v ) E A4. If the BFM and PSP algorithms are applied to this network, where the initial queue of scan eligible nodes in the latter algorithm is {2,3,4,5}, then during stage 1 of the BFM algorithm and during stage 2 of the PSP algorithm immediately after arc (7,6) is scanned and node 6 is relabeled, d(1) = 0, 4 2 ) = 4 5 ) = 1, 4 3 ) = 4 4 ) = 2, 4 7 ) = 4, and 4 6 ) = 5. This situation cannot occur in the DBFS algorithm because d(v ) 5 k + 1 for all nodes v during stage k .

7. COMPUTATIONAL COMPARISONS ON NETWORKS WITH NEGATIVE CYCLES

We now present results of computational tests comparing versions of the BFM, PSP, and DBFS algorithms that use the depth condition of Theorem 2 to detect a negative cycle with versions that use the standard condition for detecting such a cycle based on the number of node labels that change at each stage. The former versions are distinguished by a “D” at the end of their names.


We also included the simplex variant DBFSS of the DBFS algorithm in our tests.

For our tests, we generated sets of 10 random problems for each of 24 different problem types, using the generation procedure described in Section 5. Specifically, the 24 problem types were obtained by generating the four types of problems listed as Problems 5-8 in Table I and then adding one arc of negative cost between a randomly chosen pair of nodes in the “optimal” tree T*, a specified number of arcs apart in T*. For each of the four original problem types (without negative cycles), we obtained six different types of problems by setting the negative cost of the additional arc to -1, -3000, or - 10,000, and specifying the difference in depths of the head and tail of this arc as either 5 or 15 and, hence, the length of the resulting negative cycle as 6 or 16. The attributes of the resulting 24 problem types are listed in Table VII.

In the first eight problems, the created cycle had a cost of -1; hence, the networks in these problem had exactly one cycle. Because of the large negative cost of the created cycle in the remaining problems, the networks in these problems had many negative cycles. In general, those problems with a negative

TABLE VII. Problem attributes (with negative cycles). Cycle Cycle

Problem cost length n m E

1 2 3 4

5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

-1 -1 -1 -1

-1 -1 -1 -1

- 3000 - 3000 - 3000 -3000 -3000 - 3000 -3000 - 3000

- 10,000 -10,Ooo - 10,000 - 10,000 - 10,000 -10,Ooo -10,000 - 10,000

6 6 6 6

16 16 16 16 6 6 6 6

16 16 16 16 6 6 6 6

16 16 16 16

1000 1000 1000 1000

1000 1000 1000 1000

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

9719 14,408 20,025 25,408

9719 14,408 20,025 25,408

9719 14,408 20,025 25,408

9719 14,408 20,025 25,408

9719 14,408 20,025 25,408

9,719 14,408

25,408 20,025

20,000 20,000 20,000 20,000

20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000


cycle cost of -10,000 had more negative cycles than did the problems with a negative cycle cost of -3000.

In Table VIII, we present the average time it took each of the seven algorithms to detect a negative cycle in the 10 problems in each problem type. The times given and CPU seconds for the SUN-3, and all codes were compiled using the F77-SUN FORTRAN compiler (with optimization). The times given do not include the time required to identify a negative cycle when one was detected.

The simplex variant DBFSS was clearly the best-performing algorithm, with Algorithm DBFSD a close second. On the average, the versions of the DBFS and PSP algorithms that used the depth condition to detect a negative cycle- henceforth, referred to as “D-versions” - were approximately 30 times faster than were the versions that detected a negative cycle based on the number of label changes. The latter versions took extremely long amounts of time to solve the last 16 types of problems-i.e., those with many negative cycles. As can be seen from Table IX, which gives the average number of stages for each

TABLE VIII. Average CPU time (before detecting a negative cycle).

Problem BFM BFMD PSP

1 4.578 2 7.494 3 5.834 4 9.260 5 4.944 6 6.184 7 7.860 8 8.938 9 5.962

10 6.128 11 4.457 12 5.368 13 6.058 14 7.776 15 2.126 16 6.572 17 2.214 18 1.688 19 0.572 20 0.708 21 1.842 22 1.560 23 0.445 24 0.542

0.438 0.514 0.680 0.694 0.418 0.478 0.608 0.657

0.352 0.390 0.529 0.570

0.388 0.394 0.530 0.567

0.278 0.312 0.384 0.380 0.272 0.320 0.373 0.398

1.020 2.996 2.048 4.137 1.480 2.172 3.502 3.838

12.940 16.612 23.140 25.646

11.108 15.154 21.728 22.718 12.852 15.326 19.308 18.156 12.384 15.278 17.552 21.252

PSPD DBFS DBFSD DBFSS

0.316 0.406 0.529 0.570 0.320 0.416 0.528 0.572

0.338 0.414 0.534 0.579

0.336 0.410 0.528 0.592 0.282 0.334 0.398 0.412 0.266 0.332 0.396 0.435

1.006 2.928 1.962 4.054

1.466 2.120 3.394 3.731

12.250 14.948 20.234 21.279

9.966 14.438 19.170 19.786 11.010 12.236 11.390 12.150

10.508 12.340 14.678 15.074

Total 109.11 10.92 302.35 10.24 252.12

0.262 0.342 0.446 0.496 0.258 0.344 0.450 0. 505

0.278 0.344 0.456 0.520

0.278 0.356 0.448 0.523 0.228 0.292 0.350 0.376 0.236 0.292 0.332 0.384

8.80

0.242 0.310 0.374 0.462 0.286 0.366 0.458 0.507 0.246 0.294 0.352 0.406

0.246 0.320 0.396 0.443 0.216 0.256 0.332 0.394 0.214 0.282 0.350 0.400

8.15


be seen from Table IX, which gives the average number of stages for each algorithm and problem type, these versions took a long time to solve all problem types because so many stages were necessary before a negative cycle could be detected. These algorithms required fewer stages for the last 16 types of problems-i.e., those with many cycles- than for the first eight types of problems- i.e., those with exactly one cycle. But as Tables X and XI show, the number of nodes that were scanned and nodes that were candidates increased substan- tially for these algorithms when the number of negative cycles in the network increased.

These tables also show that the D-versions of the DBFS and PSP algorithms, on the other hand, performed slightly better, in general, the more negative cycles there were in the network, as did Algorithms BFMD and DBFSS. More- over, a comparison with the results reported in Section 5 shows, taking into account the fact that those results were obtained from code that was compiled without optimization, the performance of all of the “D” algorithms and Algo- rithm DBFSS does not degrade when there are negative cycles.

TABLE IX. Average number of stages (before detecting a negative cycle). Problem BFM BFMD PSP PSPD DBFS DBFSD DBFSS

1 935 2 793 3 893 4 825 5 907 6 896 7 887 8 864

9 198 10 131 11 70 12 71

13 228 14 196 15 33 16 90 17 62 18 31 19 8 20 9

21 50 22 30 23 7 24 7

Total 8221

26 970 28 970 29 24 925 25 924 27 27 970 29 970 30 24 933 26 932 28 22 950 34 949 34 21 937 32 937 34 20 922 29 924 31 18 920 29 92 1 30

10 594 16 576 18 8 498 12 459 14 8 496 12 461 13 7 413 11 370 12

11 524 17 496 18 8 533 14 55 1 15 8 486 12 456 13 7 384 11 336 13 6 448 8 402 10 5 390 7 349 9 5 331 6 222 8 5 244 5 206 8 6 429 8 392 10 6 351 7 334 9 5 276 6 284 8 4 290 6 247 8

29 1 14,214 390 13,668 429

11 12 11 14

21 21 20 20

13 10 8

10

14 14 11 12 9 7 7 9 9 9 9 9

290

TA

BL

E X

.

Prob

lem

B

FM

BF

MD

PS

P P

SPD

D

BF

S D

BF

SD

DR

FS

S

Ave

rage

num

ber

of n

odes

sca

nned

(be

fore

det

ectin

g a

nega

tive

cycl

e).

1

2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

Tot

al

40,1

86

85,4

33

38.4

81

73,2

32

49,9

32

57,4

75

64,5

10

67,9

11

125,

980

96,3

07

53,2

34

54,5

71

126,

918

121,

351

25,1

28

69,1

66

47,4

16

26,5

06

6800

72

66

39,9

85

25,0

91

5281

54

98

1.31

3.6S

8

4758

46

79

4693

45

96

463 1

45

20

4699

44

47

4675

40

63

4162

41

21

5022

40

35

4242

38

80

3437

30

54

2802

25

46

3477

31

49

2603

24

27

94,7

18

17.4

01

38,4

22

19.6

23

35,4

54

25,4

78

27,6

73

34,0

49

31,9

51

224,

918

213,

793

226,

s 28

215,

012

193,

249

196,

047

212,

653

192,

232

219,

933

194,

077

187,

375

150,

574

2 14,

886

195,

966

170,

885

176,

297

3,41

4,47

6

4161

41

65

4244

40

53

4172

42

2 1

4309

40

70

4734

45

12

4689

45

98

4622

44

55

4605

45

53

3810

37

42

358 1

34

44

3822

37

19

3559

35

29

99.4

40

17,2

85

38,0

86

19,3

83

35,1

76

25,3

43

27,4

44

33,7

83

31,7

56

214,

448

194,

731

200,

179

180,

410

174,

350

189,

388

190,

796

170,

064

187,

336

156,

091

110,

475

99,9

26

182,

499

158,

550

143,

314

126,O

O 1

2,90

6.8 1

4

4059

40

43

4087

39

24

4086

41

05

4095

39

34

4432

40

25

4132

40

76

4360

42

25

4097

41

43

3488

33

01

305 1

28

93

3447

33

94

2987

29

47

91,3

31

2072

22

17

2126

23

88

2649

26

98

2701

26

96

2060

20

22

1888

20

58

2125

22

43

2159

22

03

1755

17

27

1783

19

41

1810

18

71

1883

19

35

51,0

10

% F B 5 G) =i

cn c

v, z G)

W

W

U

-I

I!

W

VJ

-I

VJ

W F ? F B P

U

TA

BL

E X

I.

Ave

rage

num

ber

of c

andi

date

nod

es (

befo

re d

etec

ting

a ne

gativ

e cy

cle)

. ~~

Prob

lem

B

FM

BFM

D

PSP

PSPD

D

BFS

D

BFS

D

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

Tot

al

933,

565

792,

007

891,

907

823,

176

905,

193

894,

304

885,

413

862,

636

197,

202

129,

969

69,1

30

70,4

29

227,

672

195,

604

31,9

68

89,0

10

61,5

38

30,2

69

7592

79

92

49,2

50

29,3

70

5994

61

93

8,19

7,38

3

25,1

74

23,0

76

26,8

73

23,5

76

21,2

78

20,6

79

19,3

80

17,8

82

9890

74

92

7492

67

93

10,8

89

7892

72

92

6593

5894

48

95

4395

39

96

5894

51

94

4195

38

96

280,

610

17,4

01

38,4

22

19,6

23

35,4

54

25,4

78

27,6

73

34,0

49

31,9

51

224,

9 18

213,

793

226,

528

215,

012

193,

249

196,

047

212,

653

192,

232

219,

933

194,

077

187,

375

150,

574

214,

886

195,

966

170,

885

176,

297

3,41

4,47

6

4161

41

65

4244

40

53

4172

42

2 1

4309

40

70

4734

45

12

4689

45

98

4622

44

55

4605

45

53

3881

37

42

3581

34

44

3822

37

19

3559

35

29

99,4

40

17,9

22

38,8

55

20,2

10

36,0

95

26,0

04

28,2

04

34,6

62

32,6

89

227,

679

21 1,

923

222,

636

207,

632

179,

954

200,

178

203,

902

187,

357

21 7,

349

188,

070

140,

239

135,

591

202,

282

182,

198

176,

949

161,

910

3,28

0,49

0

4682

47

56

4892

47

82

4703

48

20

4922

48

08

5401

51

55

5449

55

72

5253

52

95

5368

56

02

4694

47

28

4734

47

66

4644

48

23

4668

47

87

119,

301

DB

FSS

c)

0

3101

c :

3535

3744

3685

38

32

3389

D

rn m

3951

,O

4028

b

2

3088

0

3160

2

x

31

80

3452

3183

34

23

3507

35

93

2830

29

46

3190

33

50

2886

30

97

3271

33

42

80,5

81

TA

BL

E X

U.

Ave

rage

num

ber

of l

abel

upd

ates

(be

fore

det

ecti

ng a

neg

ativ

e cy

cle)

. Pr

oble

m

BFM

B

FMD

P

SP

PS

PD

DB

FS

DB

FSD

D

BFS

S

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

Tot

al

42,1

17

87,9

33

41,3

65

76,4

29

51,8

77

59,9

14

67,4

05

71,0

90

145,

854

120,

788

76,3

91

137,

286

138,

446

31,8

44

87,6

15

65,1

32

44,0

29

13,4

17

15,4

30

54,1

87

39,0

65

10,6

29

11,7

69

1.56

0.21

8

70,2

06

6673

70

97

7539

74

60

6552

69

54

7588

76

1 1

7361

74

10

8263

83

48

7566

72

06

8172

81

80

6744

69

44

7465

73

18

6600

71

04

7334

75

97

177,

372

18,9

41

40,3

52

21,9

43

37,9

80

27,0

14

29,6

04

36,3

68

34,4

76

254,

622

252,

133

282,

569

276,

871

204,

857

217,

494

241,

654

229,

610

297,

824

284.

985

295,

852

250,

873

263,

122

256,

577

247,

098

286,

540

4,38

9,35

9

5702

60

96

6566

65

81

5709

61

52

6629

65

96

6383

66

04

7232

74

19

6224

65

08

7129

73

76

5763

62

0 1

6474

67

29

5671

61

40

6539

67

67

155,

190

18,8

69

40.0

89

21.6

83

37,7

52

26,9

52

29,4

39

36,1

35

34,3

46

245,

210

233,

418

254,

815

241,

425

186,

864

21 1,

704

219,

822

206,

611

261,

259

236,

933

183,

436

182,

778

228,

095

215,

625

2 18,

475

2 13,

840

3,78

5,60

4

563 1

59

91

6366

64

4 1

5652

60

55

6397

64

65

6406

64

69

705 1

74

02

6234

66

06

6962

74

18

5838

63

09

6605

69

12

5758

63

97

6594

69

37

154,

896

4935

54

32

5619

62

6 1

5724

61

70

6635

68

50

4988

55

43

5830

63

71

5560

61

98

6582

68

39

5054

56

96

6458

65

49

522 1

59

66

6569

66

72

143,

722


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Received November 1989 Accepted March 1990

shortest path algorithms using dynamic breadth-first search

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