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![Page 1: On the Difficulty of Some Shortest Paths Problems Amit Bhosle Department of Computer Science University of California, Santa Barbara.](https://reader035.fdocuments.us/reader035/viewer/2022062715/56649d7e5503460f94a60707/html5/thumbnails/1.jpg)
On the Difficulty of Some Shortest Paths Problems
Amit Bhosle
Department of Computer Science
University of California, Santa Barbara
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Overview : What We Will Discuss Today
• Introduction – Replacement Paths : the central problem
• Motivation
• Upper bounds :– Undirected : O(1) single source shortest paths computation [HS01]
– Directed : O(1) all pair shortest paths computation
• Lower bounds :– All pair shortest paths (APSP) [KKP]
– N-pairs shortest paths (NPSP)
– Replacement paths problem
– Related problems
• Conclusion
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The Replacement Paths Problem
• Given a weighted graph G(V,E), two nodes s and t, and a shortest • path P = {s=v0 , v1 , v2 ,…,vk=t} from s to t.
• Find the shortest path from s to t in the graphs :– Edge version : G - {ei} for all 0 i k where ei = {vi , vi+1}– Node version : G - {vi} for all 0 < i < k .
• G assumed to be 2-connected.• G may be directed or undirected.• We focus on the edge version (lower bounds hold for node version).
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Why Study the Replacement Paths Problem ?
Applications :
– Computational mechanism design (VCG auctions).
payment(e) = d (s,t ; G \ e) – d (s,t ; G |w(e)=0 )
Weight of thereplacement path for e Needs constant time per edge after an initial
single source shortest path computation
Payments need to be computed for all edges of the ( s, t ) shortest path.
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Why Study the Replacement Paths Problem ? (Contd.)
Applications :
– Network routing for single link failure recovery.
– Directly yields the Most Vital Edge in the shortest path domain.
– Sub-problem in other problems e.g. K - Shortest paths.
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Naïve Algorithm
• Recompute : Compute the s - t shortest paths separately in each of the k graphs.
G - {ei } for all 1 i < k.
• Time Complexity : Since k = O(n) in worst case, the naïve algorithm may require O( n (m+n log n) ) time.
• Simple, but inefficient (at least) for undirected graphs.
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Upper Bounds
• Undirected edge version solved recently by Hershberger and Suri [HS01] in O(m + n log n) time.
• Nardelli, et al. [NPW01] solved the node version in the same time bounds.
• Directed version can be solved in O(APSP) time. – In some models, the All Pairs Shortest Paths problem can be
solved in sub-cubic time whereas the naïve algorithm may still be cubic in the worst case.
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Undirected Version Upper Bound [Hershberger-Suri]
s te
p
Lemma : If p Vt|e , path(p,t) cannot contain e
Vt|eVs|e
q
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Undirected Version Upper Bound (Contd.)
s te
Vt|eVs|e u
v
Weight of the path using edge (u,v) is :d (s,u) + w (u,v) + d (v,t)
Ee
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Undirected Version Upper Bound (Contd.)
• Assign cost to each edge (u,v) E(Vs|e ,Vt|e) as :
cost (u,v) = d(s,u) + w(u,v) + d(v,t)
• Immediately suggests an algorithm : (weight of) the replacement path for the edge e is
MIN(u,v) : u Vs|e, v Vt|e{ cost(u,v) }
• But, arbitrarily computing these values could yield no improvement – there can be (m) edges in each cut !
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Undirected Version Upper Bound (Contd.)
• From Ei , delete edges ending at vi+1 and add edges starting at vi+1 to get Ei+1 : we need O ( deg(v i+1)) heap operations.
• Standard heap implementation yields an O(m log n) algorithm.
A slightly modified algorithm using F-Heaps runs in O(m+n log n) time.
s tei ei+1
vi vi+1
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Difficulty in Directed Version
s te
vVt|e
Vs|e
The basic lemma does not hold for digraphs : Even though v Vt|e , path(v,t) contains edge e .
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Directed Replacement Paths : O(APSP)
• The naïve algorithm requires O( k (m+n log n )), where k is the number of edges on the shortest path.
• We present an algorithm which runs in O(k.n) time after an initial APSP computation.
• APSP has the trivial lower bound of (n 2) and k is at most n-1. Thus, the running time remains dominated by the APSP computation.
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Directed Replacement Paths : O(APSP) (Contd.)
We start by computing the All Pairs Shortest Paths in G \ P(s,t)where P = {e0 , e2 ,…, ek-1 } is the shortest path from s to t .
s=v0t=vk
v
ei
Vs|i
vj vj+1
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Directed Replacement Paths : O(APSP) (Contd.)
• For any edge ei , Vs|i can be constructed in O(n) time.
• The best path through a node v Vi is :
path v (s,t | i ) = MIN kj = i+1 { d(s,v ) + d(v,vj ) + d(vj ,t ) }.
We shall see later that this can be done in constant time per node.
• The replacement path for the edge ei is :
MIN v V s|i { path v (s,t | i ) }.
This requires O(n) time per edge on P, since | Vi | is O (n).
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Directed Replacement Paths : O(APSP) (Contd.)
s=v0t=vk
v
ei
ei+1
We compute the replacement paths for ei before ej for i > j.
path v (s,t | i ) = MIN { path v (s,t | i+1 ) ,
d(s,v ) + d(v ,vi+1 ) + d(vi+1 ,t ) }
vi+1
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Directed Replacement Paths : O(APSP) (Contd.)
• Time required remains dominated by the initial APSP computation.
(All other steps together require O (n 2) )
• Implies an O(k.APSP) time algorithm for the k-shortest paths.
• O ( APSP ) :
– Fast matrix multiplication :
O(n3 (log log n / log n)) ; O(C0.681 n2.575 ).
– Comparison based : O ( n ( m + n log n) )
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All Pairs Shortest Paths Lower Bound [Karger-Koller-Phillips]
n nm / n
Any algorithm must compare all the mn paths.
Otherwise, some modification of edges enables us to fool the algorithm.
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N-Pairs Shortest Paths (NPSP)
n nm / n
Any algorithm must investigate all the (m n ) tripleswhere m = O (n n)
We shall reduce an NPSP instance to a Replacement Paths problem instance, thus applying the lower bound of the former to the latter.
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Replacement Paths Lower Bound
• The lower bound is based on the n-pairs shortest paths lower bound.
• Some restrictions need to be imposed since the NPSP lower bound is not sufficiently robust. But still includes all known algorithms.
• Reduction basically says that an O (T (m,n)) algorithm for the replacement paths implies an O (T (m,n)) algorithm for NPSP.
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Replacement Paths Lower Bound (Contd.)
vi vi+1
H
s t
(i+1).W(n-i).W
W = nemax
emax H
sjtj
0 0000 0 0
(n-k).W p.W
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Replacement Paths Lower Bound (Contd.)
• Crucial property : Replacement path for edge ei = {vi ,vi+1} leaves the spine P at vi , enters H at sj , leaves H at tj and joins back at vi+1.
• That is, each replacement path in G computes an (sj ,tj ) distance in H.
• The reduction is linear time and does not increase the input complexity.
• NPSP has a (weak) lower bound of (m n). With some restrictions, we apply the lower bound to the replacement paths problem.
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Lower Bounds on Related Problems
Similar bounds on some problems closely related to the replacement paths problem :
– K-shortest paths : (m n) per output path.
– <Length> <hop count> metric : (m n).
– Subpaths deletions : (m k) for k subpaths deleted.
– Replacement SP trees : (mn).
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K-Shortest Paths
• Given a digraph G, and two nodes s and t , and an integer k, one needs to find k shortest paths from s to t in order of increasing weights.
• All known algorithms for k-SP use replacement paths as a subroutine : note that the shortest replacement path is the second shortest path from s to t.
– Basic idea : ith shortest path differs from all the i -1 shorter paths in atleast one edge. Thus, new candidates are generated by finding the replacement paths for these shorter paths.
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K-Shortest Paths (Contd.)
• Typically, all the algorithms for the k-SPs invoke one replacement paths subroutine per output path.
• Thus, the lower bound of (m n) applies to all algorithms for the k-SPs which use replacement paths as a subroutine.
• Is it possible to determine the shortest replacement path without computing all the replacement paths for a given shortest path ?
• The non-simple k-SPs can be computed in O(m + n log n + k) time using a technique not based on replacement paths.
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<length> <hop count>
• Ties among shortest paths are sometimes broken in favor of the one with fewer edges.
• Archer & Tardos suggested the <length> <hop count> metric for replacement paths. This metric also favors paths with fewer edges.
• A slight modification in the basic construction allows us to establish a lower bound of (m n) for computing replacement paths with this metric.
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<length> <hop count> (Contd.)
vi vi+1
H
s
(i+1)W(n-i)W
W = nemaxsj
tj
0 0000 0
vn
0
v3n=t
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<length> <hop count> (Contd.)
Weight of a path with d detours with detour r skipping kr edges of P is exactly equal to :
(3n + 4d - kr ) {(dn + kr )W + lr }
No. of edges Weight of the path
Thus, the metric is minimized when d = 1.If this detour skips k edges, the weight of the path becomes (3n + 4 - k ) {(n+k)W + l }
Thus, k = 1 for optimal replacement path and again, each replacement path computes an (si ,tj ) distance in H.
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Replacement Shortest Paths Trees
• Given a digraph G, a node s, and the shortest paths tree Ts = {e1 ,e2 ,…, en-1} of s in G, compute the shortest path tree of s in each of the n-1 graphs G \ ei .
• Naïve algorithm :
Recompute - compute the shortest path tree of s in each of the.
Graphs G \ ei for all 1 i n-1.
• Time complexity : O (mn + n2 log n).
• We prove a lower bound of (mn) for this problem.
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Replacement Shortest Paths Trees (Contd.)
vi t = vn
H
0
(n-i).W
W = n.emax
Si
0 0000 0 0
Sn
s = v0
Sk
vk
(n-k).W
0
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Replacement Shortest Paths Trees (Contd.)
• Crucial properties :
– The shortest path tree in G of s = v0 contains the shortest path tree of sn in H as a subtree.
– The shortest path tree in G \ {ei = (vi , vi+1)} contains the shortest path tree of si+1 in H as a subtree.
• Thus, all the replacement shortest paths trees in G essentially yield the all-pair-shortest-paths in H.
• Using the lower bound of (mn) for the APSP, we arrive at the same lower bound for the problem at hand.
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Subpaths Deletions
• Given a digraph G, two nodes s and t, the shortest path P from s to t and a set Q = {q1 , q2 ,…, qk} of sub-paths of P. One needs to compute the shortest path from s to t in the graphs G \ qi 1 i k.
• Naïve algorithm.
– Recompute : compute the s - t shortest path in the graphs G \ qi for all 1 i k.
– Time complexity : O (k (m+n log n)).
– We prove a lower bound of (m k) and an upper bound of O(APSP + k.n ) for the problem at hand.
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Subpaths Deletions (Contd.)
si dj
H
(n-i)W
W = nemaxSi
0 0000 0 0
Dj
s = s0sn dn t=d0
(n-j)W
0 0
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Subpaths Deletions (Contd.)
• Crucial property : The replacement path from s to t in the graph.
G \ { pi = (si , dj )} follows P till si , enters H at Si , leaves H at Dj , joins P back at dj and follows P till t.
• Thus, the replacement path for each deleted sub-path computes an (si , dj ) distance in H.
• Using the lower bound of the k-pair shortest paths we arrive at a lower bound of (m k) for this problem where the number of sub-paths to be deleted is O(k).
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Subpaths Deletions : Upper Bound
• The k replacement paths can be computed in O(APSP + k.n).
• Exactly similar to the O(APSP) algorithm for the replacement paths problem : We first sort the sub-paths by their right end points and then continue in the same fashion as earlier.
• Worst case optimal : (m k) and both m and k could be as large as O(n 2), but considerable gap between the upper and lower bounds for smaller values of k and/or m.
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Concluding Remarks
• These are the first super-linear lower bounds for the problem (s).
• Still a significant gap between the lower bound of (m n) and the trivial upper bound of O(mn + n 2 log n).
• K-shortest paths : Can one solve the problem without using the replacement paths as a subroutine ?
• Faster APSP algorithms can improve the trivial upper bound (these would have to be non path-comparison based).
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Thank You