Space Efficient Linear Time Algorithms for BFS, DFS and Applications
Research on space efficient graph algorithms, particularly for st -connectivity, has a long history including the celebrated polynomial time, O (lg n ) bits 1 algorithm in undirected graphs by Reingold J. JACM. 55 (4) ( 2008 ), and polynomial time, n / 2 Θ ( lg n ) bits algorithm in directed graphs...
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| Veröffentlicht in: | Theory of computing systems Jg. 62; H. 8; S. 1736 - 1762 |
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| Abstract | Research on space efficient graph algorithms, particularly for
st
-connectivity, has a long history including the celebrated polynomial time,
O
(lg
n
) bits
1
algorithm in undirected graphs by Reingold J. JACM.
55
(4) (
2008
), and polynomial time,
n
/
2
Θ
(
lg
n
)
bits algorithm in directed graphs by Barnes et al. SICOMP.
27
(5), 1273–1282 (
1998
). Recent works by Asano et al. ISAAC (
2014
) and Elmasry et al. STACS (
2015
), reconsidered classical fundamental graph algorithms focusing on improving the space complexity. Elmasry et al. gave, among others, an implementation of breadth first search (BFS) in a graph
G
with
n
vertices and
m
edges, taking the optimal
O
(
m
+
n
) time using
O
(
n
) bits improving the naïve
O
(
n
lg
n
) bits implementation. Similarly, Asano et al. provided space efficient implementations for performing depth first search (DFS) in a graph
G
. We continue this line of work focusing on improving the space requirement for implementing a few classical graph algorithms. Our first result is a simple data structure that can maintain any subset
S
of a universe of
u
elements using just
u
+
o
(
u
) bits and supports in constant time, apart from the standard insert, delete and membership queries, the operation
findany
that finds and returns any element of the set (or outputs that the set is empty). It can also enumerate all elements present currently in the set in no particular order in
O
(
k
+ 1) time where
k
is the number of elements currently belonging to the set. While this data structure supports a weaker set of operations than that of Elmasry et al. STACS (
2015
), it is simple, more space efficient and is sufficient to support a BFS implementation optimally in
O
(
m
+
n
) time using at most 2
n
+
o
(
n
) bits. Later, we further improve the space requirement of BFS to at most
n
lg 3 +
o
(
n
) bits albeit with a slight increase in running time to
O
(
m
lg
n
f
(
n
)) time where
f
(
n
) is any extremely slow growing function of
n
, and the
o
term in the space is a function of
f
(
n
). We also discuss a similar time-space tradeoff result for finding a minimum weight spanning tree of a weighted (bounded by polynomial in
n
) undirected graph using
n
+
O
(
n
/
f
(
n
)) bits and
O
(
m
lg
n
f
(
n
)) time, for any function
f
(
n
) such that 1 ≤
f
(
n
) ≤
n
. For DFS in a graph
G
, we provide an implementation taking
O
(
m
+
n
) time and
O
(
n
lg
m
/
n
) bits. This partially answers at least for sparse graphs, a question asked by Asano et al. ISAAC (
2014
) whether DFS can be performed in
O
(
m
+
n
) time and using
O
(
n
) bits, and also simultaneously improves the DFS result of Elmasry et al. STACS (
2015
). Using our DFS algorithm and other careful implementations, we show how one can also test for biconnectivity, 2-edge connectivity, and find cut vertices and bridges of a given undirected graph within the same time and space bounds; earlier classical linear time algorithms for these problems used Ω(
n
lg
n
) bits of space. |
|---|---|
| AbstractList | Research on space efficient graph algorithms, particularly for
st
-connectivity, has a long history including the celebrated polynomial time,
O
(lg
n
) bits
1
algorithm in undirected graphs by Reingold J. JACM.
55
(4) (
2008
), and polynomial time,
n
/
2
Θ
(
lg
n
)
bits algorithm in directed graphs by Barnes et al. SICOMP.
27
(5), 1273–1282 (
1998
). Recent works by Asano et al. ISAAC (
2014
) and Elmasry et al. STACS (
2015
), reconsidered classical fundamental graph algorithms focusing on improving the space complexity. Elmasry et al. gave, among others, an implementation of breadth first search (BFS) in a graph
G
with
n
vertices and
m
edges, taking the optimal
O
(
m
+
n
) time using
O
(
n
) bits improving the naïve
O
(
n
lg
n
) bits implementation. Similarly, Asano et al. provided space efficient implementations for performing depth first search (DFS) in a graph
G
. We continue this line of work focusing on improving the space requirement for implementing a few classical graph algorithms. Our first result is a simple data structure that can maintain any subset
S
of a universe of
u
elements using just
u
+
o
(
u
) bits and supports in constant time, apart from the standard insert, delete and membership queries, the operation
findany
that finds and returns any element of the set (or outputs that the set is empty). It can also enumerate all elements present currently in the set in no particular order in
O
(
k
+ 1) time where
k
is the number of elements currently belonging to the set. While this data structure supports a weaker set of operations than that of Elmasry et al. STACS (
2015
), it is simple, more space efficient and is sufficient to support a BFS implementation optimally in
O
(
m
+
n
) time using at most 2
n
+
o
(
n
) bits. Later, we further improve the space requirement of BFS to at most
n
lg 3 +
o
(
n
) bits albeit with a slight increase in running time to
O
(
m
lg
n
f
(
n
)) time where
f
(
n
) is any extremely slow growing function of
n
, and the
o
term in the space is a function of
f
(
n
). We also discuss a similar time-space tradeoff result for finding a minimum weight spanning tree of a weighted (bounded by polynomial in
n
) undirected graph using
n
+
O
(
n
/
f
(
n
)) bits and
O
(
m
lg
n
f
(
n
)) time, for any function
f
(
n
) such that 1 ≤
f
(
n
) ≤
n
. For DFS in a graph
G
, we provide an implementation taking
O
(
m
+
n
) time and
O
(
n
lg
m
/
n
) bits. This partially answers at least for sparse graphs, a question asked by Asano et al. ISAAC (
2014
) whether DFS can be performed in
O
(
m
+
n
) time and using
O
(
n
) bits, and also simultaneously improves the DFS result of Elmasry et al. STACS (
2015
). Using our DFS algorithm and other careful implementations, we show how one can also test for biconnectivity, 2-edge connectivity, and find cut vertices and bridges of a given undirected graph within the same time and space bounds; earlier classical linear time algorithms for these problems used Ω(
n
lg
n
) bits of space. Research on space efficient graph algorithms, particularly for st-connectivity, has a long history including the celebrated polynomial time, O(lg n) bits1 algorithm in undirected graphs by Reingold J. JACM. 55(4) (2008), and polynomial time, n/2Θ(lgn)\(n/2^{\Theta (\sqrt {\lg n})}\) bits algorithm in directed graphs by Barnes et al. SICOMP. 27(5), 1273–1282 (1998). Recent works by Asano et al. ISAAC (2014) and Elmasry et al. STACS (2015), reconsidered classical fundamental graph algorithms focusing on improving the space complexity. Elmasry et al. gave, among others, an implementation of breadth first search (BFS) in a graph G with n vertices and m edges, taking the optimal O(m + n) time using O(n) bits improving the naïve O(n lg n) bits implementation. Similarly, Asano et al. provided space efficient implementations for performing depth first search (DFS) in a graph G. We continue this line of work focusing on improving the space requirement for implementing a few classical graph algorithms. Our first result is a simple data structure that can maintain any subset S of a universe of u elements using just u + o(u) bits and supports in constant time, apart from the standard insert, delete and membership queries, the operation findany that finds and returns any element of the set (or outputs that the set is empty). It can also enumerate all elements present currently in the set in no particular order in O(k + 1) time where k is the number of elements currently belonging to the set. While this data structure supports a weaker set of operations than that of Elmasry et al. STACS (2015), it is simple, more space efficient and is sufficient to support a BFS implementation optimally in O(m + n) time using at most 2n + o(n) bits. Later, we further improve the space requirement of BFS to at most n lg 3 + o(n) bits albeit with a slight increase in running time to O(m lg nf(n)) time where f(n) is any extremely slow growing function of n, and the o term in the space is a function of f(n). We also discuss a similar time-space tradeoff result for finding a minimum weight spanning tree of a weighted (bounded by polynomial in n) undirected graph using n + O(n/f(n)) bits and O(m lg nf(n)) time, for any function f(n) such that 1 ≤ f(n) ≤ n. For DFS in a graph G, we provide an implementation taking O(m + n) time and O(n lg m/n) bits. This partially answers at least for sparse graphs, a question asked by Asano et al. ISAAC (2014) whether DFS can be performed in O (m + n) time and using O(n) bits, and also simultaneously improves the DFS result of Elmasry et al. STACS (2015). Using our DFS algorithm and other careful implementations, we show how one can also test for biconnectivity, 2-edge connectivity, and find cut vertices and bridges of a given undirected graph within the same time and space bounds; earlier classical linear time algorithms for these problems used Ω(n lg n) bits of space. |
| Author | Satti, Srinivasa Rao Chakraborty, Sankardeep Banerjee, Niranka Raman, Venkatesh |
| Author_xml | – sequence: 1 givenname: Niranka surname: Banerjee fullname: Banerjee, Niranka organization: The Institute of Mathematical Sciences, HBNI, CIT Campus – sequence: 2 givenname: Sankardeep orcidid: 0000-0002-0719-160X surname: Chakraborty fullname: Chakraborty, Sankardeep email: sankardeep@imsc.res.in organization: The Institute of Mathematical Sciences, HBNI, CIT Campus – sequence: 3 givenname: Venkatesh surname: Raman fullname: Raman, Venkatesh organization: The Institute of Mathematical Sciences, HBNI, CIT Campus – sequence: 4 givenname: Srinivasa Rao surname: Satti fullname: Satti, Srinivasa Rao organization: Seoul National University |
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| Keywords | DFS Biconnectivity Topological sort Space efficient algorithms Succinct dictionary Chain decomposition BFS |
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| References | Banerjee, N., Chakraborty, S., Raman, V.: Improved space efficient algorithms for BFS, DFS and applications. In: 22nd COCOON, vol. 9797, pp 119–130. Springer, LNCS (2016) Chakraborty, S., Raman, V., Satti, S.R.: Biconnectivity, chain decomposition and st-numbering using O(n) bits. In: 27th ISAAC, pp 22:1–22:13 (2016) Munro, J.I.: Tables. In: FSTTCS LNCS, vol. 1180, pp 37–42 (1996) Poyias, A., Puglisi, S.J., Raman, R.: Compact dynamic rewritable (cdrw) arrays To appear in 19th ALENEX (2017) AhoAVHopcroftJEUllmanJDThe design and analysis of computer algorithms1974BostonAddison-Wesley0326.68005 ClarkDCompact pat trees1996CanadaPhD thesis. University of Waterloo ReingoldOUndirected connectivity in log-spaceJ. ACM200855417:117:24244501410.1145/1391289.13912911315.68156 Elmasry, A., Hagerup, T., Kammer, F.: Space-efficient basic graph algorithms. In: 32nd STACS, pp 288–301 (2015) BrȯnnimannHChanTMSpace-efficient algorithms for computing the convex hull of a simple polygonal line in linear timeComput. Geom.20063427582222288310.1016/j.comgeo.2005.11.005 Darwish, O., Elmasry, A.: Optimal time-space tradeoff for the 2d convex-hull problem. In: 22nd ESA, pp 284–295 (2014) FredericksonGNUpper bounds for time-space trade-offs in sorting and selectionJ. Comput. Syst. Sci.1987341192688129210.1016/0022-0000(87)90002-X0642.68122 WillardDELog-logarithmic worst-case range queries are possible in space theta(n)Inf. Process. Lett.1983172818410.1016/0020-0190(83)90075-30509.68106 BarbaLKormanMLangermanSSadakaneKSilveiraRISpace-time trade-offs for stack-based algorithmsAlgorithmica201572410971129335768610.1007/s00453-014-9893-51328.68290 Batagelj, V., Zaversnik, M.: An O(m) algorithm for cores decomposition of networks. CoRR, arXiv:0310049 (2003) Gupta, A., Hon, W., Shah, R., Vitter, J.S.: A framework for dynamizing succinct data structures. In: 34th ICALP, pp 521–532. Proceedings, Wroclaw (2007) Raman, R., Raman, V., Rao, S.S.: Succinct dynamic data structures. In: 7th WADS, LNCS 2125, pp 426–437 (2001) HonWKSadakaneKSungWKSuccinct data structures for searchable partial sums with optimal worst-case performanceTheor Comput. Sci.20114123951765186285766910.1016/j.tcs.2011.05.0231226.68032 Brodnik, A., Carlsson, S., Demaine, E.D., Munro, J.I., Sedgewick, R.: Resizable arrays in optimal time and space. In: 6th WADS, LNCS, vol. 1663, pp 37–48 (1999) CormenTHLeisersonCERivestRLSteinCIntroduction to algorithms20093rd edn.CambridgeMIT Press1187.68679 Chakraborty, D., Pavan, A., Tewari, R., Vinodchandran, N.V., Yang, L.: New time-space upperbounds for directed reachability in high-genus and h-minor-free graphs. In: FSTTCS, pp 585–595 (2014) TarjanREA note on finding the bridges of a graphInf. Process. Lett.19742616016134948310.1016/0020-0190(74)90003-90282.68018 AsanoTBuchinKBuchinMKormanMMulzerWRoteGSchulzAReprint of: Memory-constrained algorithms for simple polygonsComput. Geom.2014473469479313357410.1016/j.comgeo.2013.11.00406249163 Ellen, F.: Constant time operations for words of length w. Unpublished Manuscript Elmasry, A., Juhl, D.D., Katajainen, J., Satti, S.R.: Selection from read-only memory with limited workspace, vol. 554, pp 64–73 (2014) Dodis, Y., Patrascu, M., Thorup, M.: Changing base without losing space. In: 42nd STOC, pp 593–602 (2010) SchmidtJMStructure and constructions of 3-connected graphs2011BerlinPhD thesis, Free University of Berlin Brandes, U.: Eager st-ordering. In: ESA, pp 247–256 (2002) Chakraborty, S., Jo, S., Satti, S.R.: Improved space-efficient linear time algorithms for some classical graph problems. In: 15th CTW (2017) Kammer, F., Kratsch, D., Laudahn, M.: Space-efficient biconnected components and recognition of outerplanar graphs. In: 41st MFCS, pp 56:1–56:14 (2016) Asano, T., Izumi, T., Kiyomi, M., Konagaya, M., Ono, H., Otachi, Y., Schweitzer, P., Tarui, J., Uehara, R.: Depth-first search using O(n) bits. In: 25th ISAAC, LNCS, vol. 8889, pp 553–564 (2014) Chan, T.M., Munro, J.I., Raman, V.: Selection and sorting in the ”restore” model. In: 25th SODA, pp 995–1004 (2014) SchmidtJMA simple test on 2-vertex- and 2-edge-connectivityInf. Process. Lett.20131137241244301798610.1016/j.ipl.2013.01.0161259.05173 TarjanREDepth-first search and linear graph algorithmsSIAM J. Comput.19721214616030417810.1137/02010100251.05107 AsanoTMulzerWRoteGWangYConstant-work-space algorithms for geometric problemsJoCG2011214668282539406693710 AroraSBarakBComputational complexity - a modern approach2009CambridgeCambridge University Press10.1017/CBO97805118040901193.68112 Asano, T., Kirkpatrick, D.G., Nakagawa, K., Watanabe, O.: Ȯ(√ n)-space and polynomial-time algorithm for planar directed graph reachability. In: 39th MFCS LNCS, vol. 8634, pp 45–56 (2014) MunroJIPatersonMSelection and sorting with limited storageTheor. Comput. Sci.19801231532358931210.1016/0304-3975(80)90061-40441.68067 BrodalGSChaudhuriSRadhakrishnanJThe randomized complexity of maintaining the minimumNord. J. Comput.19963433735114360180870.68086 Chakraborty, S., Satti, S.R.: Space-efficient algorithms for maximum cardinality search, stack BFS, queue BFS and applications. In: 23rd COCOON, pp 87–98 (2017) Muthukrishnan, S.: Data streams: Algorithms and applications. Found. Trends Theor. Comput. Sci. 1(2) (2005) MehlhornKData structures and algorithms 1: sorting and searching, volume 1 of eatcs monographs on theoretical computer science1984BerlinSpringer10.1007/978-3-642-69672-5 Banerjee, N., Chakraborty, S., Raman, V., Roy, S., Saurabh, S.: Time-space tradeoffs for dynamic programming in trees and bounded treewidth graphs. In: 21st COCOON, vol. 9198, pp 349–360. Springer, LNCS (2015) BarnesGBussJRuzzoWSchieberBA sublinear space, polynomial time algorithm for directed s-t connectivitySICOMP199827512731282162306010.1137/S00975397932831510908.05080 GabowHNPath-based depth-first search for strong and biconnected componentsInf. Process. Lett.2000743-4107114176155110.1016/S0020-0190(00)00051-X1339.68301 FredrikssonKKilpelȧinenPPractically efficient array initializationSoftw. Pract. Exper.201646443546710.1002/spe.2314 Elmasry, A., Kammer, F.: Space-efficient plane-sweep algorithms. In: 27th ISAAC, pp 30:1–30:13 (2016) Eppstein, D., Loffler, M., Strash, D: Listing all maximal cliques in large sparse real-world graphs. ACM Journal of Experimental Algorithmics, 18 (2013) MunroJIRamanVSelection from read-only memory and sorting with minimum data movementTheor. Comput. Sci.19961652311323141189010.1016/0304-3975(95)00225-10872.68045 Mortensen, C.W., Pagh, R., Patrascu, M.: On dynamic range reporting in one dimension. In: 37th STOC, pp 104–111 (2005) Hagerup, T., Kammer, F.: Succinct choice dictionaries. CoRR, arXiv:1604.06058 (2016) ChakrabortySRamanVSattiSRBiconnectivity, st-numbering and other applications of DFS using O(n) bitsJ. Comput. Syst. Sci.2017906379369810610.1016/j.jcss.2017.06.0061375.68089 T Asano (9841_CR3) 2014; 47 TH Cormen (9841_CR23) 2009 9841_CR44 D Clark (9841_CR22) 1996 9841_CR45 9841_CR43 9841_CR40 AV Aho (9841_CR1) 1974 K Mehlhorn (9841_CR38) 1984 DE Willard (9841_CR51) 1983; 17 L Barba (9841_CR9) 2015; 72 9841_CR39 JM Schmidt (9841_CR48) 2013; 113 9841_CR34 RE Tarjan (9841_CR49) 1972; 1 H Brȯnnimann (9841_CR15) 2006; 34 WK Hon (9841_CR36) 2011; 412 9841_CR37 9841_CR35 JI Munro (9841_CR42) 1996; 165 9841_CR30 T Asano (9841_CR6) 2011; 2 JI Munro (9841_CR41) 1980; 12 9841_CR28 9841_CR29 K Fredriksson (9841_CR32) 2016; 46 GS Brodal (9841_CR13) 1996; 3 9841_CR20 9841_CR21 9841_CR26 9841_CR27 GN Frederickson (9841_CR31) 1987; 34 9841_CR24 9841_CR25 RE Tarjan (9841_CR50) 1974; 2 G Barnes (9841_CR10) 1998; 27 9841_CR4 9841_CR5 HN Gabow (9841_CR33) 2000; 74 9841_CR8 9841_CR7 S Arora (9841_CR2) 2009 9841_CR17 9841_CR18 O Reingold (9841_CR46) 2008; 55 9841_CR11 9841_CR12 9841_CR16 9841_CR14 S Chakraborty (9841_CR19) 2017; 90 JM Schmidt (9841_CR47) 2011 |
| References_xml | – reference: Dodis, Y., Patrascu, M., Thorup, M.: Changing base without losing space. In: 42nd STOC, pp 593–602 (2010) – reference: GabowHNPath-based depth-first search for strong and biconnected componentsInf. Process. Lett.2000743-4107114176155110.1016/S0020-0190(00)00051-X1339.68301 – reference: AroraSBarakBComputational complexity - a modern approach2009CambridgeCambridge University Press10.1017/CBO97805118040901193.68112 – reference: BrodalGSChaudhuriSRadhakrishnanJThe randomized complexity of maintaining the minimumNord. J. Comput.19963433735114360180870.68086 – reference: Mortensen, C.W., Pagh, R., Patrascu, M.: On dynamic range reporting in one dimension. In: 37th STOC, pp 104–111 (2005) – reference: Raman, R., Raman, V., Rao, S.S.: Succinct dynamic data structures. In: 7th WADS, LNCS 2125, pp 426–437 (2001) – reference: BarbaLKormanMLangermanSSadakaneKSilveiraRISpace-time trade-offs for stack-based algorithmsAlgorithmica201572410971129335768610.1007/s00453-014-9893-51328.68290 – reference: MunroJIRamanVSelection from read-only memory and sorting with minimum data movementTheor. Comput. Sci.19961652311323141189010.1016/0304-3975(95)00225-10872.68045 – reference: Chakraborty, S., Raman, V., Satti, S.R.: Biconnectivity, chain decomposition and st-numbering using O(n) bits. In: 27th ISAAC, pp 22:1–22:13 (2016) – reference: TarjanREA note on finding the bridges of a graphInf. Process. Lett.19742616016134948310.1016/0020-0190(74)90003-90282.68018 – reference: SchmidtJMA simple test on 2-vertex- and 2-edge-connectivityInf. Process. Lett.20131137241244301798610.1016/j.ipl.2013.01.0161259.05173 – reference: Chakraborty, S., Jo, S., Satti, S.R.: Improved space-efficient linear time algorithms for some classical graph problems. In: 15th CTW (2017) – reference: MehlhornKData structures and algorithms 1: sorting and searching, volume 1 of eatcs monographs on theoretical computer science1984BerlinSpringer10.1007/978-3-642-69672-5 – reference: ClarkDCompact pat trees1996CanadaPhD thesis. University of Waterloo – reference: AhoAVHopcroftJEUllmanJDThe design and analysis of computer algorithms1974BostonAddison-Wesley0326.68005 – reference: Batagelj, V., Zaversnik, M.: An O(m) algorithm for cores decomposition of networks. CoRR, arXiv:0310049 (2003) – reference: Brandes, U.: Eager st-ordering. In: ESA, pp 247–256 (2002) – reference: Chan, T.M., Munro, J.I., Raman, V.: Selection and sorting in the ”restore” model. In: 25th SODA, pp 995–1004 (2014) – reference: Elmasry, A., Juhl, D.D., Katajainen, J., Satti, S.R.: Selection from read-only memory with limited workspace, vol. 554, pp 64–73 (2014) – reference: HonWKSadakaneKSungWKSuccinct data structures for searchable partial sums with optimal worst-case performanceTheor Comput. Sci.20114123951765186285766910.1016/j.tcs.2011.05.0231226.68032 – reference: Asano, T., Izumi, T., Kiyomi, M., Konagaya, M., Ono, H., Otachi, Y., Schweitzer, P., Tarui, J., Uehara, R.: Depth-first search using O(n) bits. In: 25th ISAAC, LNCS, vol. 8889, pp 553–564 (2014) – reference: Chakraborty, S., Satti, S.R.: Space-efficient algorithms for maximum cardinality search, stack BFS, queue BFS and applications. In: 23rd COCOON, pp 87–98 (2017) – reference: Elmasry, A., Kammer, F.: Space-efficient plane-sweep algorithms. In: 27th ISAAC, pp 30:1–30:13 (2016) – reference: ChakrabortySRamanVSattiSRBiconnectivity, st-numbering and other applications of DFS using O(n) bitsJ. Comput. Syst. Sci.2017906379369810610.1016/j.jcss.2017.06.0061375.68089 – reference: Kammer, F., Kratsch, D., Laudahn, M.: Space-efficient biconnected components and recognition of outerplanar graphs. In: 41st MFCS, pp 56:1–56:14 (2016) – reference: WillardDELog-logarithmic worst-case range queries are possible in space theta(n)Inf. Process. Lett.1983172818410.1016/0020-0190(83)90075-30509.68106 – reference: TarjanREDepth-first search and linear graph algorithmsSIAM J. Comput.19721214616030417810.1137/02010100251.05107 – reference: FredrikssonKKilpelȧinenPPractically efficient array initializationSoftw. Pract. Exper.201646443546710.1002/spe.2314 – reference: Gupta, A., Hon, W., Shah, R., Vitter, J.S.: A framework for dynamizing succinct data structures. In: 34th ICALP, pp 521–532. Proceedings, Wroclaw (2007) – reference: Eppstein, D., Loffler, M., Strash, D: Listing all maximal cliques in large sparse real-world graphs. ACM Journal of Experimental Algorithmics, 18 (2013) – reference: Asano, T., Kirkpatrick, D.G., Nakagawa, K., Watanabe, O.: Ȯ(√ n)-space and polynomial-time algorithm for planar directed graph reachability. In: 39th MFCS LNCS, vol. 8634, pp 45–56 (2014) – reference: Banerjee, N., Chakraborty, S., Raman, V., Roy, S., Saurabh, S.: Time-space tradeoffs for dynamic programming in trees and bounded treewidth graphs. In: 21st COCOON, vol. 9198, pp 349–360. Springer, LNCS (2015) – reference: Poyias, A., Puglisi, S.J., Raman, R.: Compact dynamic rewritable (cdrw) arrays To appear in 19th ALENEX (2017) – reference: ReingoldOUndirected connectivity in log-spaceJ. ACM200855417:117:24244501410.1145/1391289.13912911315.68156 – reference: Ellen, F.: Constant time operations for words of length w. Unpublished Manuscript – reference: CormenTHLeisersonCERivestRLSteinCIntroduction to algorithms20093rd edn.CambridgeMIT Press1187.68679 – reference: Banerjee, N., Chakraborty, S., Raman, V.: Improved space efficient algorithms for BFS, DFS and applications. In: 22nd COCOON, vol. 9797, pp 119–130. Springer, LNCS (2016) – reference: Munro, J.I.: Tables. In: FSTTCS LNCS, vol. 1180, pp 37–42 (1996) – reference: AsanoTBuchinKBuchinMKormanMMulzerWRoteGSchulzAReprint of: Memory-constrained algorithms for simple polygonsComput. Geom.2014473469479313357410.1016/j.comgeo.2013.11.00406249163 – reference: FredericksonGNUpper bounds for time-space trade-offs in sorting and selectionJ. Comput. Syst. Sci.1987341192688129210.1016/0022-0000(87)90002-X0642.68122 – reference: Darwish, O., Elmasry, A.: Optimal time-space tradeoff for the 2d convex-hull problem. In: 22nd ESA, pp 284–295 (2014) – reference: Elmasry, A., Hagerup, T., Kammer, F.: Space-efficient basic graph algorithms. In: 32nd STACS, pp 288–301 (2015) – reference: SchmidtJMStructure and constructions of 3-connected graphs2011BerlinPhD thesis, Free University of Berlin – reference: Brodnik, A., Carlsson, S., Demaine, E.D., Munro, J.I., Sedgewick, R.: Resizable arrays in optimal time and space. In: 6th WADS, LNCS, vol. 1663, pp 37–48 (1999) – reference: Muthukrishnan, S.: Data streams: Algorithms and applications. Found. Trends Theor. Comput. Sci. 1(2) (2005) – reference: BrȯnnimannHChanTMSpace-efficient algorithms for computing the convex hull of a simple polygonal line in linear timeComput. Geom.20063427582222288310.1016/j.comgeo.2005.11.005 – reference: BarnesGBussJRuzzoWSchieberBA sublinear space, polynomial time algorithm for directed s-t connectivitySICOMP199827512731282162306010.1137/S00975397932831510908.05080 – reference: AsanoTMulzerWRoteGWangYConstant-work-space algorithms for geometric problemsJoCG2011214668282539406693710 – reference: MunroJIPatersonMSelection and sorting with limited storageTheor. Comput. Sci.19801231532358931210.1016/0304-3975(80)90061-40441.68067 – reference: Hagerup, T., Kammer, F.: Succinct choice dictionaries. CoRR, arXiv:1604.06058 (2016) – reference: Chakraborty, D., Pavan, A., Tewari, R., Vinodchandran, N.V., Yang, L.: New time-space upperbounds for directed reachability in high-genus and h-minor-free graphs. 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st
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O
(lg
n
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1... Research on space efficient graph algorithms, particularly for st-connectivity, has a long history including the celebrated polynomial time, O(lg n) bits1... |
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