Answering Distance Queries in directed graphs using fast matrix multiplication

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Answering Distance Queries in directed graphs using fast matrix multiplication. Seminar in Algorithms Prof. Haim Kaplan Lecture by Lior Eldar 1/07/2007. Structure of Lecture. Introduction & History Alg1 – APSP Alg2 – preprocess & query Alg3 – Hybrid Summary. Problem Definition.

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Answering Distance Queries in directed graphs using fast matrix multiplication

Seminar in Algorithms

Prof. Haim Kaplan

Lecture by Lior Eldar

1/07/2007

Structure of Lecture
• Introduction & History
• Alg1 – APSP
• Alg2 – preprocess & query
• Alg3 – Hybrid
• Summary
Problem Definition
• Given a weighted directed graph, we are requested to find:
• APSP - All pairs shortest paths – find for any pair
• SSSP - Single Source shortest paths – find all distances from s.
• A hybrid problem comes to mind:
• Preprocess the graph faster than APSP
• Answer ANY two-node distance query faster than SSSP.
• What’s it good for?
Previously known results – APSP
• Undirected graphs
• Approximated algorithm by Thorup and Zwick:
• Preprocess undirected weighted graph in

expected time.

• Generate data structure of size
• Answer any query in O(1)
• BUT: answer is approximate with a factor of 2k-1.
• For non-negative integer weights at most M – Shoshan and Zwick developed an algorithm of run time
• Directed graphs – Zwick - runs in
Previously known results - SSSP
• Positive weights:
• Directed graphs with positive weights – Dijkstra with
• Undirected graphs with positive integer edge weights – Thorup with
• Negative weights – much harder:
• Bellman-Ford
• Goldberg and Tarjan – assumes edge weight values are at least – N.
New Algorithm by Yuster / Zwick
• Solves the hybrid pre-processing-query problem for:
• Directed graphs
• Integer weights from –M to M
• Achieves the following performance:
• Pre-processing
• Faster than previously known APSP (Zwick) so long as the number of queries is
• Better than SSSP performance (Goldberg&Tarjan) for dense graphs with small alphabet – gap of
Beyond the numbers…
• An extension of this algorithm allows complete freedom in optimization of the pre-processing - query problem.
• to optimize an algorithm for an arbitrary number of queries q, we want: preprocessing time + q * query time to be minimal.
• This defines the ratio between query time and pre-processing time - completely controlled by the algorithm inputs.
• Meaning: if we know in advance the number of queries we can fine-tune the algorithm as we wish.
Before we begin - scope
• Assumptions:
• No negative cycles
• Inputs:
• Directed Weighted Graph G=(V,E,w)
• Weights are –M,…0,…,M
• Outputs:
• Data structure such that – given any two nodes – produces the shortest distance between them (and not the path itself) – with high probability.
Matrix Multiplication
• The matrix product C=AB, where A is an matrix, B is , and C is matrix, is defined as follows:
• Define: the minimal number of algebraic operations for computing the matrix product.
• Define as the smallest exponent such that
• Theorem by Coppersmith and Winograd:
Distance Products
• The distance product , where A is an matrix, B is , and C is matrix, is defined as follows:
• Recall: if W is an n x n matrix of the edge weights of a graph then is the distance matrix of the graph.
• Lemma by Alon: can be computed almost as fast as “regular” matrix multiplication:
State-of-the-art APSP
• Randomized algorithm by Zwick that runs in time
• Intuition:
• Computation of all short paths is intensive.
• BUT: long paths are made up of short paths: once we pay the initial price we can leverage this work to compute longer paths with less effort.
• Strategy: Giving up on certainty - with a small number of distance updates we can be almost sure that any long-enough path has at least one representative that is updated.
Basic Operations
• Truncation
• Replace any entry larger than t with
• Selection
• Extract from D the elements whose row indices are in A, and column indices are in B.
• Min-Assignment
• Assign to each element the smallest between the two corresponding elements of D and D‘.
Pseudo-code
• Simply sample nodes and multiply decimated matrices…
On matrices and nodes…
• Column-decimated matrix

Distance between any two nodes

D

Shortest directed path from any node to any node in B

On matrices and nodes…(2)
• Row-decimated matrix

Distance between any two nodes

Shortest directed path from any node in B to any node

What do we prove?
• Lemma: if there is a shortest path between nodes i and j in G that uses at most edges, then after the -th iteration of the algorithm, with high probability we have
• Meaning: at each iteration we update with high probability all the paths in the graph of a certain length. This serves as a basis for the next iteration.
Proof Outline
• By Induction:
• Base case: easy – the input W contains all paths of length
• Induction step:
• Suppose that the claim holds for and show that it also holds for
• Take any two nodes that their shortest distance is at least . The -th iteration matrix product will (almost certainly) plug in their shortest distance at location (i,j) of D.

i

k

j’

j

i’

Why?
• Set
• The path p from i to j is at least 2s/3.
• This divides p into three subsections:
• Left – at most s/3
• Right – at most s/3
• Middle – exactly s/3
The Details
• The left and right “thirds” - help attain the induction step.
• The path p(i,k) and p(k,j) are short enough – at most 2s/3  good for previous step:
• The middle “third” – ensures the fault probability is low enough.
• Prob(no k is selected) =
• Probability still goes to 0 (as n tends to infinity) after computation of
• entries
• iterations
So…
• Assuming all previous steps were good enough:
• With high probability each long-enough path has a representative in B
• The update of the D using the product

plugs in the correct result.

• Note that:
• Each element is first limited to s*M
• This is necessary for the fast-matrix-multiplication algorithm
Complexity
• Where does the trick hide?
• The matrix alphabet increases linearly with iteration number
• The product size decreases with iteration number
• For each iteration :
• Alphabet size: s*M
• Product complexity: , where
• Total:
• Disregarding the log function, and optimizing between fast and naïve matrix products we get:
Fast Product versus Naive

*assuming small M

Complexity Behavior
• For a given matrix alphabet M, we find the cross-over point between the matrix algorithms.
• For high r (>M-dependent threshold) we use FMM
• Complexity dependent on M
• For low r (<threshold) we use naïve multiplication
• Complexity not dependent on M
• Q: How does complexity change over the iteration number?
Pre-processing algorithm
• Motivation:
• We rarely query all node-pairs
• Strategy:
• Replace the costly matrix product

with 2 smaller products:

• Generate data structure such that each query costs only
Starting with the query…
• Pseudo-code:
• What is a sufficient trait of D, such that the returned value will be, with high probability
• Answer: with high probability, a node k on the path from i to j should have:
New matrix type
• Row&Column-decimated matrix

Query data structure for any two nodes

D

Query data-structure for any 2 nodes in B

What do we prove?
• Lemma 4.1: If or , and there is a shortest path from i to j in G that uses at most edges, then after the -th iteration of the preprocessing algorithm, with high probability we have .
• Meaning: D has the necessary trait: for any path p, if we iterate long enough, then with high probability, for at least one node k (in p(i,j)) the entries d(i,k), d(k,j) will contain shortest paths. Hence, “query” will return the correct result.
Proof Outline - preprocess
• By Induction:
• Base case: easy – B=V, and the input W contains all paths of length .
• Induction step:
• Suppose that the claim holds for and show that it also holds for
• Take any two nodes that their shortest distance is at most . The l-th iteration matrix products (2) will (almost certainly) plug in their shortest distance at location (i,j) of D provided that EITHER or

.

i

k

j’

j

i’

Why?
• Set
• The path p from i to j is at least 2s/3.
• This divides p into three subsections:
• Left – at most s/3
• Right – at most s/3
• Middle – exactly s/3
The Details
• Assume that .
• With high probability ( ) there will be k in p(i,j), such that (remember why?)
• Both are also in ,since
• We therefore attain the induction step:
• The path p(i,k) and p(k,j) are short enough – at most 2s/3  good for previous step.
• The end-points of these paths (k) are in
• Therefore their shortest distance is in D
• The second product then updates correctly. (assumption critical here)
Where’s the catch?
• In APSP, we assure that:
• At every iteration l we compute the shortest path of length at most .
• BUT: we had to update all pairs each time
• In the preprocess algorithm, we assure:
• At every iteration l, we compute the shortest path of length at most only for a selected subset.
• BUT: this subset covers all possible subsequent queries, with high probability.
Complexity
• Matrix product: instead of operations we only get
• As before, for each iteration , the alphabet size is s*M.
• Total complexity:
• No matrix-product switch here!
Performance
• For small M, as long as the number of queries is less than we get better results than APSP.
• For small M:
• The algorithm overtakes Goldberg’s algorithm, if the graph is dense
• For a dense-enough graph , we can run many SSSP queries and still be faster:
The larger picture
• We saw:
• Alg1: heavy pre-processing, light query
• Alg2: light pre-processing, heavy query
• Alg3: ?

Query-oriented (APSP)

Preprocess- oriented (pre-process)

The Third Way
• Suppose we know in advance the we require no more than queries.
• We use the following:
• Perform iterations of the APSP algorithm
• Perform iterations of the pre-process algorithm
• Take the matrix B from the last step of step 1. The product returns in any shortest-distance query.
Huh?
• After the first stage  D holds all the shortest path of all “short” paths, of lengths at most with high probability.
• When the second starts stage it can be sure that the induction holds for all
• The second stage takes care of the “long” paths, with respect to querying. Meaning:
• If the path is long it will have a representative in one of the second-phase iterations
• If it is too-short – it will fall under the jurisdiction of the first stage.
Complexity
• The first stage ( updates) costs at most
• The second stage costs only
• The query costs
• For example – if want to answer a distance query in , we can pre-process in time
• Q: Why couldn’t we sample B in the query step of Alg2 – the one that initially costs O(n)?
• A: Because if the path is too short – we will have no guarantee that it will have a representative in B. Alg3 solves this because short distances are computed rigorously.
• Conclusion: the less we sample out of V when we query, the more steps we need to run APSP to begin with.
Final Procedure
• Given q queries, determine the query complexity using .
• This assumes M is small enough so that we use fast product. Otherwise compare to
• Execute alg3 using steps of APSP and steps of pre-process
• Query all q queries.
Summary
• For the problem we defined: directed graph, with integer weights, whose absolute value is at most M, we have seen:
• Alg1: State-of-the-art APSP in
• Alg2: State-of-the-art SSSP in
• Alg3: A method to calibrate between the two, for a known number of queries.