| SPT {a + , a − } |
Gallo 3232 GALLO G. 1980. Reoptimization procedures in shortest path problem. Rivista di matematica per le scienze economiche e sociali, 3(1): 3-13.
|
o ( m + n · d
max ) |
1980 |
| P2P {A + , A − } |
Ramalingam & Reps 5858 RAMALINGAM G & REPS T. 1996. An incremental algorithm for a generalization of the shortest-path problem. Journal of Algorithms, 21(2): 267-305.
|
O (|| δ || log || δ ||) 2
|
1996 |
|
|
|
δ is a measure of the change in the input and output. |
|
| APSP {A + , A − } |
King 4545 KING V. 1999. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In Foundations of Computer Science, 1999. 40th Annual Symposium on, pages 81-89. IEEE.
|
|
1999 |
|
|
|
b is an upper bound on the weight of the arcs. |
|
| SPT {a + , a − } |
Narváez et al. 5151 NARVÁEZ P, SIU K-Y & TZENG H-Y. 2000. New dynamic algorithms for shortest path tree computation. IEEE/ACM Transactions on Networking (TON), 8(6): 734-746.
|
|
2000 |
|
|
|
complexity concerns Bellman-Ford version, δ
d denote the minimum number of nodes that must change their distance or parentattributes (or both). |
|
| SPT {A + , A − } |
Narvaez et al. 5252 NARVAEZ P, SIU K-Y & TZENG H-Y. 2001. New dynamic spt algorithm based on a ball-and-string model. IEEE/ACM Transactions on Networking (TON), 9(6): 706-718.
|
O ( δ
e · ln( δ
n )) |
2001 |
|
|
|
with binary heap, where δ
e and δ
n are the maximum number of key decrements and the maximum size of the heap, respectively. |
|
| SPT {A + , A − } |
Pallottino & Scutellà 5555 PALLOTTINO S & SCUTELLÀ MG. 2003. A new algorithm for reoptimizing shortest paths when the arc costs change. Operations Research Letters, 31(2): 149-160, mar 2003.
|
O ( hm + min{ hn log n , C
p } + min{ n log n , C
d , kn } |
2003 |
|
|
|
Let K
+ and K
− the set of the edge incremented and decremented, then C
p and C
d measure the cost perturbation due to the arcs in K
+ and K
− , k ≤ min{ n − n
r , C
d }, where n
r is the dimension of the first fragment of the new solution, while h is the number of primal phases of the algorithm. |
|
| P2P {A + , A − } |
Thomas & White 6363 THOMAS BW & WHITE CC. 2007. The dynamic shortest path problem with anticipation. European Journal of Operational Research, 176(2): 836-854.
|
O (2 L ) |
2007 |
|
|
|
L = | A
I |, where A
I ∈ A is a set of observed arcs. |
|
| SPT {a + , a − } |
Buriol et al. 99 BURIOL LS, RESENDE MG & THORUP M. 2008. Speeding up dynamic shortest-path algorithms. INFORMS Journal on Computing, 20(2): 191-204.
|
see Ramalingam & Reps (1996) |
2008 |
| P2P {A + , A − } |
Nannicini et al. 4848 NANNICINI G, BAPTISTE P, KROB D & LIBERTI L. 2008. Fast paths in dynamic road networks. Proceedings of ROADEF, 8: 1-14.
|
O ( m + n · log n ) |
2008 |
|
|
|
with Fibonacci’s heap |
|
| SPT {A + , A − } |
Chan & Yang 1212 CHAN EP & YANG Y. 2009. Shortest path tree computation in dynamic graphs. IEEE Transactions on Computers, 58(4): 541-557.
|
complexity for each unit operation |
2009 |
| SPT {A + , A − } |
Tretyakov et al. 6565 TRETYAKOV K, ARMAS-CERVANTES A, GARCÍA-BAÑUELOS L, VILO J & DUMAS M. 2011. Fast fully dynamic landmark-based estimation of shortest path distances in very large graphs. In: Proceedings of the 20th ACM international conference on Information and knowledge management, pages 1785-1794. ACM.
|
O ( k
2 · D
2 ) |
2011 |
|
|
|
k is the number of landmarks, D is the diameter of the graph. |
|
| P2P {A + , A − } |
D’Andrea et al. 2323 D’ANDREA A, D’EMIDIO M, FRIGIONI D, LEUCCI S & PROIETTI G. 2013. Dynamically maintaining shortest path trees under batches of updates. In: International Colloquium on Structural Information and Communication Complexity, pages 286-297. Springer.
|
O ((| B | + | ( G , B )| · k ) · log n ) |
2013 |
|
|
|
let β be a batch of updates operations, |B| = Σ βi∈B | β
i |, | ( G , B )| is the sum over all batches of B of the number of vertices affected by each of these batches, and k = O ( ). |
|
| P2P {A + , A − } |
Hong et al. 3939 HONG J, PARK K, HAN Y, RASEL MK, VONVOU D & LEE Y-K. 2017. Disk-based shortest path discovery using distance index over large dynamic graphs. Information Sciences, 382-383: 201-215.
|
O ( m + m
out · m
in ) |
2017 |
|
|
|
min is the average value of the in degree, and m
out the average value of the out degree. |
|
