Exact and Approximate Distances in Graphs

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Exact and Approximate Distances in Graphs A
Survey

• Uri Zwick
• Tel Aviv University

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Distances and Shortest Paths
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Variations
undirected directed
unweighted non-negative integer weights integer
weights non-negative real weights real weights
given pair(s) single source all pairs
deterministic randomized
exact results additive error multiplicative error
Spanners Distance oracles
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Models of Computation
Integer weights word RAM model Each weight is
contained in a w-bit word. Allowed to perform
additions, subtractions, comparisons, shifts,
ANDs, ORs, XORs, and other bit operations.
Real weights addition-comparison model The only
operations allowed on edge weights are additions
(subtractions) and comparisons. We again assume
random access capabilities.
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Single-Source Shortest PathsClassical results
BFS mn Unweighted graphs
Dijkstra 59 Fredman-Tarjan 87 mn log n Nonnegative real edge weights
Bellman 58Ford 62 m n General real edge weights
Goldberg 95 (Gabow-Tarjan 89) mn1/2log N Integer edge weights
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SSSP, Priority queues and Sorting
Time of Dijkstra m(decrease key)
n(extract min). Monotone priority queues are
enough. Dijkstras algorithm sorts the
distances. If n elements can be sorted in nf(n)
time, then SSSP can be solved in mf(n) time.
Thorup 96 For undirected graphs, the sorting
bottleneck can be avoided! Thorup 97
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Single-Source Shortest Pathsdirected graphs,
nonnegative integer edge weights, randomized
algorithms
Thorup 96 m loglog n
Thorup 96 m(n log n)/w1/2-?
Raman 97, Cherkassky-Goldberg-Silverstein 97 mnw1/4?
Raman 97 mn(log n)1/3?
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Single-Source Shortest Paths undirected graphs,
nonnegative integer edge weights, deterministic
algorithm
Thorup 97 mn
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Single-Source Shortest Paths deterministic
algorithms
nonnegative integer weights
Hagerup 00 m log w directed
positive real weights
Pettie- Ramachandran 01 m?(m,n)n loglog R undirected
Pettie- Ramachandran 01 mn log R directed
R ratio between largest and smallest edge
weights
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Open Problems SSSP

• Directed SSSP with real edge weights in o(mn)
  time?
• Directed SSSP with integer edge weights in
  o(mn1/2log N) time?
• Directed SSSP with non-negative integer edge
  weights in linear time?
• What is the complexity of the SSSP problem with
  non-negative weights in the addition-comparison
  model?

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All-Pairs Shortest Paths
Reference Complexity Weights Directed?
Johnson 77 mnn2 log n real Yes
Kar-Kol-Phi 93McGeoch 95 mnn2 log n real Yes
Hagerup 00 mnn2 loglog n integer Yes
Thorup 97 mn integer No
Pettie-Rama. 01 mn?(m,n) real No
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Min-Plus (Distance) Product
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Algebraic Product
The algebraic product of two n by n matrices over
a ring can be computed using n? algebraic
operations (additions, subtractions,
multiplications), where ?lt2.376.
Strassen 69, , Coppersmith-Winograd 90
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APSP and DP
If D is an n by n matrix containing the edge
weightsof a graph, then Dn is the distance
matrix.
APSP(n) ? DP(n) log n APSP(n) ? 6 ( DP(n/2) 2
DP(n/4) 4 DP(n/8) ) O(n2) DP(n) ? APSP(3n)
Furman 70, Munro 71, Fischer-Meyer 71
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All-Pairs Shortest Paths directed graphs, real
weights
Floyd 62 Warshall 62 n3
Fredman 76 n5/2
Fredman 76Takaoka 92 n3(loglog n/log n)1/2
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All-Pairs Shortest Paths undirected graphs,
weights from 1,2,,M
Galil-Margalit 92Alon-Naor 92 Seidel 92Shoshan-Zwick 99 Mn? lt Mn2.367
? exponent of fast matrix multiplication ? lt
2.376 Coppersmith-Winograd 90
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Seidels Algorithm unweighted undirected
graphsrunning time n? log n
Algorithm Seidel(A) if AJ then return J-I else C
? Seidel(A2) X ? CA , deg ? Ae-1 dij ? 2cij
xij lt cijdegj return D endif
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All-Pairs Shortest Paths directed graphs,
weights from -M,,0,,M
(Galil-Margalit 91) Zwick 98
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Rectangular Matrix Multiplication
n?
n
? - The largest constant such that these
algebraic products can be computed using n2o(1)
operations.
n
n?
n
?gt0.294
n?
n
Coppersmith 97
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Sampled Distance Products
n
n
F ? D for i ?1 to log3/2n do s ? (3/2)i B ?
rand(V,(10n ln n)/s) F ? min F , FV,BFB,V

n
B
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All-Pairs Shortest Paths directed graphs,
weights from 1,W(1?)-approximate distances
and paths
Zwick 98 (n?/?) log(W/?)
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Open Problems APSP

1. Are n5/2 additions-comparisons needed?
2. An n3-? time algorithm in the add-comp model,
   counting all operations?
3. An n3-?logM time algorithm?
4. An n5/2 time algorithm for unweighted directed
   graphs?

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An estimated distance ?(u,v) is of stretch t
iff ?(u,v) ? ?(u,v) ? t ?(u,v)
An estimated distance ?(u,v) is of surplus t
iff ?(u,v) ? ?(u,v) ? ?(u,v) t
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All-Pairs Almost Shortest Pathsunweighted,
undirected graphs
Reference Time Surplus
Aingworth-Chekuri-Indyk-Motwani 96 n5/2 2
Dor-Halperin-Zwick 96 n3/2m1/2 , n7/3 2
n5/3m1/3 , n11/5 4
kn2-1/km1/k kn21/(3k-4) 2(k-1)
Ignoring polylogarithmic factors
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All-Pairs Almost Shortest Pathsweighted
undirected graphs
Reference Time Stretch
Cohen-Zwick 97 n3/2m1/2 2
n7/3 7/3
n2 3
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Multiplicative/Additive Approximations
For every ?gt0, there is bb(?), such that an
estimated distance ?(u,v) satisfying ?(u,v) ?
?(u,v) ? (1?)?(u,v) b(?) , for every u?S
and v?V, can be computed in O(mn?Sn1?)
time.
(Elkin-Peleg 01) , Elkin 01
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Open Problems Approx. APSP

1. Improve the surplus/time tradeoff. Finite surplus
   in n2o(1) time?
2. Improve the stretch/time tradeoff. Stretch lt 3 in
   n2o(1) time?
3. Further explore multiplicate/additive
   approximations.

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Spanners
Let G be a weighted undirected graph. A subgraph
H of G is a t-spanner of G iff ?u,v?G, ?H(u,v)
? t ?G(u,v) .
Awerbuch 85 Peleg-Schäffer 89
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Example
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Theorem
For every kgt1, every weighted undirected graph on
n vertices has a (2k-1)-spanner with at most
n11/k edges.
Tight for k1,2,3,5. Conjectured to be tight for
any k equivalent to a girth conjecture of Erdös.
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Proof/Algorithm
Consider the edges in non-decreasing order of
weight. Add each edge to the spanner if it does
not close a cycle of size at most 2k. The
resulting graph is a (2k-1)-spanner and it does
not contain a cycle of size at most 2k. Hence the
number of edges is at most n11/k.
Althöfer, Das, Dobkin, Joseph, Soares 93
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If cycle?2k, then red edge can be removed.
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(a,b)-Spanners
Let G be an unweighted undirected graph. A
subgraph H of G is an (a,b)-spanner of G iff
?u,v?G, ?H(u,v) ? a ?G(u,v) b .
(Dor-Halperin-Zwick 96, a1) Peleg-Elkin 01
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(a,b)-Spanners
Reference Size (a,b)
Dor-Halperin-Zwick 96 n3/2 (1,2)
Elkin-Peleg 01 n1? (1?,b(?))
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Open Problems Spanners

1. Are there blt?, and ?gt0, such that every
   unweighted undirected graph on n vertices has a
   (1,b)-spanner with n3/2-? edges?

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All-Pairs Shortest Paths
n by ndistancematrix
Input graph
APSPalgorithm
The output matrix may be much larger than the
input graph !!!
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(Approximate) Distance Oracles
Preprocessingalgorithm
Input graph
Compactdata structure
Query answeringalgorithm
?(u,v)
(u,v)
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Approximate Distance Oracles
Reference Preproc. time Space Query time Stretch
Awerbuch-Berger-Cowen-Peleg 93 kmn1/k kn11/k kn1/k 64k
Cohen 93 kmn1/k kn11/k kn1/k 2k?
Thorup-Zwick 01 kmn1/k kn11/k k 2k-1
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Open Problems Distance Oracles

1. Deterministic construction of(2k-1,n11/k,k)-dist
   ance oracles in o(mn) time?
2. Constructing a (3,n3/2,1)-distance oracle in
   n2o(1) time?
3. Distance oracles withadditive errors?

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A Slightly updated version of the survey can be
found at
http//www.cs.tau.ac.il/zwick/papers/dist-survey-
esa.ps.gz
Please send suggestions/corrections/comments to
zwick_at_tau.ac.il