20: Algorithms
Lecture All-Pairs Shortest Paths
The tree which fills the arms grew from the tiniest sprout; the tower of nine storeys rose from a (small) heap of earth; the journey of a thousand li commenced with a single step. -- Lao-Tzu, Tao Te Ching, chapter 64 (6th century BC), translated by J. Legge (1891) And I would walk five hundred miles, And I would walk five hundred more, Just to be the man who walks a thousand miles To fall down at your door. -- The Proclaimers, "Five Hundred Miles (I'm Gonna Be)", Sunshine on Leith (2001) Almost there. . . Almost there. . . -- Red Leader [Drewe Henley], Star Wars (1977)
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20.1
All-Pairs Shortest Paths
The Problem
In the previous lecture, we saw algorithms to find the shortest path from a source vertex s to a target vertex t in a directed graph. As it turns out, the best algorithms for this problem actually find the shortest path from s to every possible target (or from every possible source to t) by constructing a shortest path tree. The shortest path tree specifies two pieces of information for each node v in the graph: dist(v) is the length of the shortest path (if any) from s to v; pred(v) is the second-to-last vertex (if any) the shortest path (if any) from s to v. In this lecture, we want to generalize the shortest path problem even further. In the all pairs shortest path problem, we want to find the shortest path from every possible source to every possible destination. Specifically, for every pair of vertices u and v, we need to compute the following information: dist(u, v) is the length of the shortest path (if any) from u to v; pred(u, v) is the second-to-last vertex (if any) on the shortest path (if any) from u to v. For example, for any vertex v, we have dist(v, v) = 0 and pred(v, v) = NULL. If the shortest path from u to v is only one edge long, then dist(u, v) = w(u v) and pred(u, v) = u. If there is no shortest path from u to v--either because there's no path at all, or because there's a negative cycle--then dist(u, v) = and pred(v, v) = NULL. The output of our shortest path algorithms will be a pair of V V arrays encoding all V 2 distances and predecessors. Many maps include a distance matrix--to find the distance from (say) Champaign to (say) Columbus, you would look in the row labeled `Champaign' and the column labeled `Columbus'. In these notes, I'll focus almost exclusively on computing the distance array. The predecessor array, from which you would compute the actual shortest paths, can be computed with only minor additions to the algorithms I'll describe (hint, hint).
20.2
Lots of Single Sources
The obvious solution to the all-pairs shortest path problem is just to run a single-source shortest path algorithm V times, once for every possible source vertex! Specifically, to fill in the one-dimensional subarray dist[s, ], we invoke either Dijkstra's or Shimbel's algorithm starting at the source vertex s.
c Copyright 2010 Jeff Erickson. Released under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License (http://creativecommons.org/licenses/by-nc-sa/3.0/). Free distribution is strongly encouraged; commercial distribution is expressly forbidden. See http://www.cs.uiuc.edu/~jeffe/teaching/algorithms for the most recent revision.
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Lecture 20: All-Pairs Shortest Paths
OBVIOUSAPSP(V, E, w): for every vertex s dist[s, ] SSSP(V, E, w, s)
The running time of this algorithm depends on which single-source shortest path algorithm we use. If we use Shimbel's algorithm, the overall running time is (V 2 E) = O(V 4 ). If all the edge weights are non-negative, we can use Dijkstra's algorithm instead, which decreases the running time to (V E + V 2 log V) = O(V 3 ). For graphs with negative edge weights, Dijkstra's algorithm can take exponential time, so we can't get this improvement directly.
20.3
Reweighting
One idea that occurs to most people is increasing the weights of all the edges by the same amount so that all the weights become positive, and then applying Dijkstra's algorithm. Unfortunately, this simple idea doesn't work. Different paths change by different amounts, which means the shortest paths in the reweighted graph may not be the same as in the original graph.
3 s 2 4 4 2 t
Increasing all the edge weights by 2 changes the shortest path s to t .
However, there is a more complicated method for reweighting the edges in a graph. Suppose each vertex v has some associated cost c(v), which might be positive, negative, or zero. We can define a new weight function w as follows: w (u v) = c(u) + w(u v) - c(v) To give some intuition, imagine that when we leave vertex u, we have to pay an exit tax of c(u), and when we enter v, we get c(v) as an entrance gift. Now it's not too hard to show that the shortest paths with the new weight function w are exactly the same as the shortest paths with the original weight function w. In fact, for any path u v from one vertex u to another vertex v, we have w (u v) = c(u) + w(u v) - c(v).
We pay c(u) in exit fees, plus the original weight of of the path, minus the c(v) entrance gift. At every intermediate vertex x on the path, we get c(x) as an entrance gift, but then immediately pay it back as an exit tax!
20.4
Johnson's Algorithm
Johnson's all-pairs shortest path algorithm finds a cost c(v) for each vertex, so that when the graph is reweighted, every edge has non-negative weight. Suppose the graph has a vertex s that has a path to every other vertex. Johnson's algorithm computes the shortest paths from s to every other vertex, using Shimbel's algorithm (which doesn't care if the edge weights are negative), and then sets c(v) dist(s, v), so the new weight of every edge is w (u v) = dist(s, u) + w(u v) - dist(s, v). 2
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Lecture 20: All-Pairs Shortest Paths
Why are all these new weights non-negative? Because otherwise, Shimbel's algorithm wouldn't be finished! Recall that an edge u v is tense if dist(s, u) + w(u v) < dist(s, v), and that single-source shortest path algorithms eliminate all tense edges. The only exception is if the graph has a negative cycle, but then shortest paths aren't defined, and Johnson's algorithm simply aborts. But what if the graph doesn't have a vertex s that can reach everything? No matter where we start Shimbel's algorithm, some of those vertex costs will be infinite. Johnson's algorithm avoids this problem by adding a new vertex s to the graph, with zero-weight edges going from s to every other vertex, but no edges going back into s. This addition doesn't change the shortest paths between any other pair of vertices, because there are no paths into s. So here's Johnson's algorithm in all its glory.
JOHNSONAPSP(V, E, w) : create a new vertex s for every vertex v w(s v) 0 w(v s) dist[s, ] SHIMBEL(V, E, w, s) abort if SHIMBEL found a negative cycle for every edge (u, v) E w (u v) dist[s, u] + w(u v) - dist[s, v] for every vertex u dist[u, ] DIJKSTRA(V, E, w , u) for every vertex v dist[u, v] dist[u, v] - dist[s, u] + dist[s, v]
The algorithm spends (V ) time adding the artificial start vertex s, (V E) time running SHIMBEL, O(E) time reweighting the graph, and then (V E + V 2 log V ) running V passes of Dijkstra's algorithm. Thus, the overall running time is (V E + V 2 log V).
20.5
Dynamic Programming
There's a completely different solution to the all-pairs shortest path problem that uses dynamic programming instead of a single-source algorithm. For dense graphs where E = (V 2 ), the dynamic programming approach eventually leads to the same O(V 3 ) running time as Johnson's algorithm, but with a much simpler algorithm. In particular, the new algorithm avoids Dijkstra's algorithm, which gets its efficiency from Fibonacci heaps, which are rather easy to screw up in the implementation. In the rest of this lecture, I will assume that the input graph contains no negative cycles. As usual for dynamic programming algorithms, we first need to come up with a recursive formulation of the problem. Here is an "obvious" recursive definition for dist(u, v): dist(u, v) = 0 min dist(u, x) + w(x v)
x
if u = v otherwise
In other words, to find the shortest path from u to v, try all possible predecessors x, compute the shortest path from u to x, and then add the last edge u v. Unfortunately, this recurrence doesn't work! In order to compute dist(u, v), we first have to compute dist(u, x) for every other vertex x, but to compute any dist(u, x), we first need to compute dist(u, v). We're stuck in an infinite loop! To avoid this circular dependency, we need an additional parameter that decreases at each recursion, eventually reaching zero at the base case. One possibility is to include the number of edges in the 3
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Lecture 20: All-Pairs Shortest Paths
shortest path as this third magic parameter. So let's define dist(u, v, k) to be the length of the shortest path from u to v that uses at most k edges. Since we know that the shortest path between any two vertices has at most V - 1 vertices, what we're really trying to compute is dist(u, v, V - 1). After a little thought, we get the following recurrence. 0 if u = v if k = 0 and u = v dist(u, v, k) = min dist(u, x, k - 1) + w(x v) otherwise
x
Just like last time, the recurrence tries all possible predecessors of v in the shortest path, but now the recursion actually bottoms out when k = 0. Now it's not difficult to turn this recurrence into a dynamic programming algorithm. Even before we write down the algorithm, though, we can tell that its running time will be (V 4 ) simply because recurrence has four variables--u, v, k, and x--each of which can take on V different values. Except for the base cases, the algorithm itself is just four nested for loops. To make the algorithm a little shorter, let's assume that w(v v) = 0 for every vertex v.
DYNAMICPROGRAMMINGAPSP(V, E, w): for all vertices u for all vertices v if u = v dist[u, v, 0] 0 else dist[u, v, 0] for k 1 to V - 1 for all vertices u for all vertices v dist[u, v, k] for all vertices x if dist[u, v, k] > dist[u, x, k - 1] + w(x v) dist[u, v, k] dist[u, x, k - 1] + w(x v)
In fact, this algorithm is almost the same as running Shimbel's algorithm once from each source vertex. The main difference is the innermost loop, which in Shimbel's algorithm would read "for all edges x v". This simple change improves the running time to (V 2 E), assuming the graph is stored in an adjacency list. Other differences are just as in the dynamic programming development of Shimbel's algorithm--we don't need the inner loop over vertices v, and we only need a two-dimensional table. After the kth iteration of the main loop in the following algorithm, the value of dist[u, v] lies between the true shortest path distance from u to v and the value dist[u, v, k] computed in the previous algorithm.
SHIMBELAPSP(V, E, w): for all vertices u for all vertices v if u = v dist[u, v] 0 else dist[u, v] for k 1 to V - 1 for all vertices u for all edges x v if dist[u, v] > dist[u, x] + w(x v) dist[u, v] dist[u, x] + w(x v)
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Lecture 20: All-Pairs Shortest Paths
20.6
Divide and Conquer
But we can make a more significant improvement. The recurrence we just used broke the shortest path into a slightly shorter path and a single edge, by considering all predecessors. Instead, let's break it into two shorter paths at the middle vertex on the path. This idea gives us a different recurrence for dist(u, v, k). Once again, to simplify things, let's assume w(v v) = 0. dist(u, v, k) = w(u v) min dist(u, x, k/2) + dist(x, v, k/2)
x
if k = 1 otherwise
This recurrence only works when k is a power of two, since otherwise we might try to find the shortest path with a fractional number of edges! But that's not really a problem, since dist(u, v, 2 lg V ) gives us the overall shortest distance from u to v. Notice that we use the base case k = 1 instead of k = 0, since we can't use half an edge. Once again, a dynamic programming solution is straightforward. Even before we write down the algorithm, we can tell the running time is (V 3 log V)--we consider V possible values of u, v, and x, but only lg V possible values of k.
FASTDYNAMICPROGRAMMINGAPSP(V, E, w): for all vertices u for all vertices v dist[u, v, 0] w(u v) for i 1 to lg V k = 2i for all vertices u for all vertices v dist[u, v, i] for all vertices x if dist[u, v, i] > dist[u, x, i - 1] + dist[x, v, i - 1] dist[u, v, i] dist[u, x, i - 1] + dist[x, v, i - 1]
This algorithm is not the same as V invocations of any single-source algorithm; we can't replace the innermost loop over vertices with a loop over edges. However, we can remove the last dimension of the table, using dist[u, v] everywhere in place of dist[u, v, i], just as in Shimbel's algorithm, thereby reducing the space from O(V 3 ) to O(V 2 ).
FASTSHIMBELAPSP(V, E, w): for all vertices u for all vertices v dist[u, v] w(u v) for i 1 to lg V for all vertices u for all vertices v for all vertices x if dist[u, v] > dist[u, x] + dist[x, v] dist[u, v] dist[u, x] + dist[x, v]
20.7
Aside: `Funny' Matrix Multiplication
There is a very close connection observed (first by Shimbel, and later independently by Bellman) between computing shortest paths in a directed graph and computing powers of a square matrix. Compare the following algorithm for multiplying two n n matrices A and B with the inner loop of our first dynamic programming algorithm. (I've changed the variable names in the second algorithm slightly to make the similarity clearer.) 5
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Lecture 20: All-Pairs Shortest Paths
MATRIXMULTIPLY(A, B): for i 1 to n for j 1 to n C[i, j] 0 for k 1 to n C[i, j] C[i, j] + A[i, k] B[k, j] APSPINNERLOOP: for all vertices u for all vertices v D [u, v] for all vertices x D [u, v] min D [u, v], D[u, x] + w[x, v]
The only difference between these two algorithms is that we use addition instead of multiplication and minimization instead of addition. For this reason, the shortest path inner loop is often referred to as `funny' matrix multiplication. DYNAMICPROGRAMMINGAPSP is the standard iterative algorithm for computing the (V - 1)th `funny power' of the weight matrix w. The first set of for loops sets up the `funny identity matrix', with zeros on the main diagonal and infinity everywhere else. Then each iteration of the second main for loop computes the next `funny power'. FASTDYNAMICPROGRAMMINGAPSP replaces this iterative method for computing powers with repeated squaring, exactly like we saw at the beginning of the semester. The fast algorithm is simplified slightly by the fact that unless there are negative cycles, every `funny power' after the V th is the same. There are faster methods for multiplying matrices, similar to Karatsuba's divide-and-conquer algorithm for multiplying integers. (Google for `Strassen's algorithm'.) Unfortunately, these algorithms us subtraction, and there's no `funny' equivalent of subtraction. (What's the inverse operation for min?) So at least for general graphs, there seems to be no way to speed up the inner loop of our dynamic programming algorithms. Fortunately, this isn't true. There is a beautiful randomized algorithm, due to Noga Alon, Zvi Galil, Oded Margalit*, and Moni Noar,1 that computes all-pairs shortest paths in undirected graphs in O(M (V ) log2 V ) expected time, where M (V ) is the time to multiply two V V integer matrices. A simplified version of this algorithm for unweighted graphs was discovered by Raimund Seidel.2
20.8
Floyd and Warshall's Algorithm
Our fast dynamic programming algorithm is still a factor of O(log V ) slower than Johnson's algorithm. A different formulation due to Floyd and Warshall removes this logarithmic factor. Their insight was to use a different third parameter in the recurrence. Number the vertices arbitrarily from 1 to V . For every pair of vertices u and v and every integer r, we define a path (u, v, r) as follows: (u, v, r) := the shortest path from u to v where every intermediate vertex (that is, every vertex except u and v) is numbered at most r.
N. Alon, Z. Galil, O. Margalit*, and M. Naor. Witnesses for Boolean matrix multiplication and for shortest paths. Proc. 33rd FOCS 417-426, 1992. See also N. Alon, Z. Galil, O. Margalit*. On the exponent of the all pairs shortest path problem. Journal of Computer and System Sciences 54(2):255262, 1997. 2 R. Seidel. On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences, 51(3):400-403, 1995. This is one of the few algorithms papers where (in the conference version at least) the algorithm is completely described and analyzed in the abstract of the paper.
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Lecture 20: All-Pairs Shortest Paths
If r = 0, we aren't allowed to use any intermediate vertices, so (u, v, 0) is just the edge (if any) from u to v. If r > 0, then either (u, v, r) goes through the vertex numbered r, or it doesn't. If (u, v, r) does contain vertex r, it splits into a subpath from u to r and a subpath from r to v, where every intermediate vertex in these two subpaths is numbered at most r - 1. Moreover, the subpaths are as short as possible with this restriction, so they must be (u, r, r - 1) and (r, v, r - 1). On the other hand, if (u, v, r) does not go through vertex r, then every intermediate vertex in (u, v, r) is numbered at most r - 1; since (u, v, r) must be the shortest such path, we have (u, v, r) = (u, v, r - 1).
u
intermediate nodes r
v
=
intermediate nodes r-1
u
-- int no erm de ed s iat r-1 e
or -- r
v
e iat ed rm r-1 e int des no
Recursive structure of the restricted shortest path (u, v, r).
This recursive structure implies the following recurrence for the length of (u, v, r), which we will denote by dist(u, v, r): dist(u, v, r) = w(u v) min dist(u, v, r - 1), dist(u, r, r - 1) + dist(r, v, r - 1) if r = 0 otherwise
We need to compute the shortest path distance from u to v with no restrictions, which is just dist(u, v, V ). Once again, we should immediately see that a dynamic programming algorithm will implement this recurrence in (V 3 ) time.
FLOYDWARSHALL(V, E, w): for all vertices u for all vertices v dist[u, v, 0] w(u v) for r 1 to V for all vertices u for all vertices v if dist[u, v, r - 1] < dist[u, r, r - 1] + dist[r, v, r - 1] dist[u, v, r] dist[u, v, r - 1] else dist[u, v, r] dist[u, r, r - 1] + dist[r, v, r - 1]
Just like our earlier algorithms, we can simplify the algorithm by removing the third dimension of the memoization table. Also, because the vertex numbering was chosen arbitrary, so there's no reason to refer to it explicitly in the pseudocode.
FLOYDWARSHALL2(V, E, w): for all vertices u for all vertices v dist[u, v] w(u v) for all vertices r for all vertices u for all vertices v if dist[u, v] > dist[u, r] + dist[r, v] dist[u, v] dist[u, r] + dist[r, v]
Now compare this algorithm with FASTSHIMBELAPSP. Instead of O(log V ) passes through all triples of vertices, FLOYDWARSHALL2 only requires a single pass, but only because it uses a different nesting order for the three for-loops! 7
Algorithms
Lecture 20: All-Pairs Shortest Paths
Exercises
1. Let G = (V, E) be a directed graph with weighted edges; edge weights could be positive, negative, or zero. (a) How could we delete some node v from this graph, without changing the shortest-path distance between any other pair of nodes? Describe an algorithm that constructs a directed graph G = (V , E ) with weighted edges, where V = V \ {v}, and the shortest-path distance between any two nodes in H is equal to the shortest-path distance between the same two nodes in G, in O(V 2 ) time. (b) Now suppose we have already computed all shortest-path distances in G . Describe an algorithm to compute the shortest-path distances from v to every other node, and from every other node to v, in the original graph G, in O(V 2 ) time. (c) Combine parts (a) and (b) into another all-pairs shortest path algorithm that runs in O(V 3 ) time. (The resulting algorithm is not the same as Floyd-Warshall!) 2. All of the algorithms discussed in this lecture fail if the graph contains a negative cycle. Johnson's algorithm detects the negative cycle in the initialization phase (via Shimbel's algorithm) and aborts; the dynamic programming algorithms just return incorrect results. However, all of these algorithms can be modified to return correct shortest-path distances, even in the presence of negative cycles. Specifically, if there is a path from vertex u to a negative cycle and a path from that negative cycle to vertex v, the algorithm should report that dist[u, v] = -. If there is no directed path from u to v, the algorithm should return dist[u, v] = . Otherwise, dist[u, v] should equal the length of the shortest directed path from u to v. (a) Describe how to modify Johnson's algorithm to return the correct shortest-path distances, even if the graph has negative cycles. (b) Describe how to modify the Floyd-Warshall algorithm (FLOYDWARSHALL2) to return the correct shortest-path distances, even if the graph has negative cycles. 3. All of the shortest-path algorithms described in this note can also be modified to return an explicit description of some negative cycle, instead of simply reporting that a negative cycle exists. (a) Describe how to modify Johnson's algorithm to return either the matrix of shortest-path distances or a negative cycle. (b) Describe how to modify the Floyd-Warshall algorithm (FLOYDWARSHALL2) to return either the matrix of shortest-path distances or a negative cycle. If the graph contains more than one negative cycle, your algorithms may choose one arbitrarily.
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Lecture 20: All-Pairs Shortest Paths
4. Let G = (V, E) be a directed graph with weighted edges; edge weights could be positive, negative, or zero. Suppose the vertices of G are partitioned into k disjoint subsets V1 , V2 , . . . , Vk ; that is, every vertex of G belongs to exactly one subset Vi . For each i and j, let (i, j) denote the minimum shortest-path distance between vertices in Vi and vertices in Vj : (i, j) = min dist(u, v) u Vi and v Vj . Describe an algorithm to compute (i, j) for all i and j in time O(V 2 + kE log E). 5. Recall3 that a deterministic finite automaton (DFA) is formally defined as a tuple M = (, Q, q0 , F, ), where the finite set is the input alphabet, the finite set Q is the set of states, q0 Q is the start state, F Q is the set of final (accepting) states, and : Q Q is the transition function. Equivalently, a DFA is a directed (multi-)graph with labeled edges, such that each symbol in is the label of exactly one edge leaving any vertex. There is a special `start' vertex q0 , and a subset of the vertices are marked as `accepting states'. Any string in describes a unique walk starting at q0 ; a string in is accepted by M if this walk ends at a vertex in F . Stephen Kleene4 proved that the language accepted by any DFA is identical to the language described by some regular expression. This problem asks you to develop a variant of the FloydWarshall all-pairs shortest path algorithm that computes a regular expression that is equivalent to the language accepted by a given DFA. Suppose the input DFA M has n states, numbered from 1 to n, where (without loss of generality) the start state is state 1. Let L(i, j, r) denote the set of all words that describe walks in M from state i to state j, where every intermediate state lies in the subset {1, 2, . . . , r}; thus, the language accepted by the DFA is exactly L(1, q, n).
qF
Let R(i, j, r) be a regular expression that describes the language L(i, j, r). (a) What is the regular expression R(i, j, 0)? (b) Write a recurrence for the regular expression R(i, j, r) in terms of regular expressions of the form R(i , j , r - 1). (c) Describe a polynomial-time algorithm to compute R(i, j, n) for all states i and j. (Assume that you can concatenate two regular expressions in O(1) time.) 6. Let G = (V, E) be an undirected, unweighted, connected, n-vertex graph, represented by the adjacency matrix A[1 .. n, 1 .. n]. In this problem, we will derive Seidel's sub-cubic algorithm to compute the n n matrix D[1 .. n, 1 .. n] of shortest-path distances using fast matrix multiplication. Assume that we have a subroutine MATRIXMULTIPLY that multiplies two n n matrices in (n ) time, for some unknown constant 2.5
Automata theory is a prerequisite for the undergraduate algorithms class at UIUC. Pronounced `clay knee', not `clean' or `clean-ee' or `clay-nuh' or `dimaggio'. 5 The matrix multiplication algorithm you already know runs in (n3 ) time, but this is not the fastest algorithm known. The current record is 2.376, due to Don Coppersmith and Shmuel Winograd. Determining the smallest possible value of is a long-standing open problem; many people believe there is an undiscovered O(n2 )-time algorithm for matrix multiplication.
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Lecture 20: All-Pairs Shortest Paths
(a) Let G 2 denote the graph with the same vertices as G, where two vertices are connected by a edge if and only if they are connected by a path of length at most 2 in G. Describe an algorithm to compute the adjacency matrix of G 2 using a single call to MATRIXMULTIPLY and O(n2 ) additional time. (b) Suppose we discover that G 2 is a complete graph. Describe an algorithm to compute the matrix D of shortest path distances in O(n2 ) additional time. (c) Let D2 denote the (recursively computed) matrix of shortest-path distances in G 2 . Prove that the shortest-path distance from node i to node j is either 2 D2 [i, j] or 2 D2 [i, j] - 1. (d) Suppose G 2 is not a complete graph. Let X = D2 A, and let deg(i) denote the degree of vertex i in the original graph G. Prove that the shortest-path distance from node i to node j is 2 D2 [i, j] if and only if X [i, j] D2 [i, j] deg(i). (e) Describe an algorithm to compute the matrix of shortest-path distances in G in O(n log n) time.
c Copyright 2010 Jeff Erickson. Released under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License (http://creativecommons.org/licenses/by-nc-sa/3.0/). Free distribution is strongly encouraged; commercial distribution is expressly forbidden. See http://www.cs.uiuc.edu/~jeffe/teaching/algorithms for the most recent revision.
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